TitleTheoretical Study of Momentum Dependent Local-AnsatzVariational Approach to Correlated Electron System( Text_全文 )

Author(s) Patoary, MD. Atiqur Rahman

Citation

Issue Date 2013-09

URL http://hdl.handle.net/20.500.12000/28554

Rights

Doctoral Thesis of Philosophy

Theoretical Study of Momentum DependentLocal-Ansatz Variational Approach to Correlated

Electron System

September 2013

By

Md. Atiqur Rahman Patoary

Department of PhysicsGraduate School of Engineering and Science

University of the Ryukyus

Supervisor : Professor Yoshiro Kakehashi

Doctoral Thesis of Philosophy

Theoretical Study of Momentum DependentLocal-Ansatz Variational Approach to Correlated

Electron System

September 2013

By

Md. Atiqur Rahman Patoary

A dissertation submitted to the Graduate School ofEngineering and Science, University of the Ryukyus,

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

Department of PhysicsGraduate School of Engineering and Science

University of the Ryukyus

Supervisor : Professor Yoshiro Kakehashi

We, the undersigned, hereby, declare that we have read this thesis and we have attended thethesis defense and evaluation meeting. Therefore, we certify that, to the best of our knowledge thisthesis is satisfactory to the scope and quality as a thesis for the degree of Doctor of Philosophy inPhysics, Graduate School of Engineering and Science, University of the Ryukyus.

THESIS REVIEW & EVALUATION COMMITTEE MEMBERS

(Chairman) Prof. Yoshiro Kakehashi

(Committee) Prof. Takeshi Inaoka

(Committee) Prof. Chitoshi Yasuda

i

Abstract

We propose in this thesis a local-ansatz wavefunction approach with momentum dependentvariational parameters ( momentum-dependent local-ansatz = MLA ) in order to describe correlatedelectrons in the ground state. The idea is to choose the best local basis set obtained from the two-particle excited states in the momentum representation by projecting out those states onto the localsubspace and by controlling the amplitudes of the excited states in the momentum space. Within asingle-site approximation we calculate the ground-state energy and derive a self-consistent equa-tion for the variational parameters by minimizing the energy. We obtain an approximate solutionwhich interpolates between the weak Coulomb interaction limit and the atomic limit. We furtherdeveloped the theory to obtain the best value of the variational parameter self-consistently.

In order to verify the validity of the MLA, we perform the numerical calculations for the non-half-filled band as well as half-filled band in the Hubbard model on the hypercubic lattice in infinitedimensions. We confirm that the self-consistent scheme significantly improves the correlation en-ergy, the momentum distribution and quasiparticle weight.We also demonstrate that the theory im-proves the standard variational methods such as the local-ansatz approach (LA) and the Gutzwillerwavefunction approach (GW); the ground-state energy in the MLA is lower than those of the LAand the GW in the weak and intermediate Coulomb interaction regimes. The double occupationnumber is shown to be suppressed as compared with the LA. Calculated momentum distributionfunctions show a clear momentum dependence, which is qualitatively different from those of theLA and the GW. We also obtained the critical Coulomb interactionUc2 = 3.40 at which effectivemass of electrons diverges. The value is comparable to the best valueUc2 = 4.10 based on thenumerical renormalization group method.

We propose an improved MLA wavefunction which can describe the strong Coulomb interac-tion regime by modifying the starting wavefunction from theHartree-Fock (HF) type to an alloy-analogy (AA) type wavefunction. Numerical results based onthe half-filled band Hubbard modelon the hypercubic lattice in infinite dimensions show that the MLA-AA wavefunction yields theground-state energy lower than the GW in the strong Coulomb interaction regime. The MLA-AAyields the metal-insulator transition atUc = 3.26. Calculated double occupation number is smallerthan the result of the GW in the metallic regime, and is finite in the insulator regime as it shouldbe, while the GW gives the Brinkman-Rice atom. Furthermore, the momentum distribution ofMLA-AA shows a momentum-dependence in both the metallic andinsulator regions, on the otherhand the GW as well as the LA gives the constant value below andabove the Fermi level.

Finally, we generalize the variational theory of the MLA by introducing a hybrid (HB) wave-function as a starting wavefunction, whose potential can flexibly change from the HF type to theAA type by varying a weighting factor from zero to one. The MLA-HB scheme yields the ground-state energy lower than that of the GW in the whole Coulomb interaction regime, and shows thefirst-order transition atUc = 2.81 from the Fermi liquid to the non-Fermi liquid, indicating themetal-insulator transition. The MLA-HB reduces the doubleoccupancy more effectively than theGW and the LA in the weakU region. The resulting double occupancy jumps at the transitionpoint Uc = 2.81, and again monotonically decreases with increasingU . Finally the momentumdistribution of MLA-HB shows a distinct momentum dependence, which is qualitatively differentfrom that of the GW.

ii

DEDICATEDTO MY SONS

FARHAN & FARDIN

iii

ACKNOWLEDGEMENTS

First and foremost I would like to express my sincere and deepest gratitude to my supervisorProf. Yoshiro Kakehashi, Department of Physics, University of the Ryukyus, Japan for his ex-cellent supervisions, advice, continuous encouragement,and painstaking help during the courseof this work. This work would have been difficult to complete without his affectionate guardian-ship and inspiration for dedicated professional research.His perpetual energy and enthusiasm inresearch had motivated me.

I would like to thank my committee members, Prof. Takeshi Inaoka and Prof. Chitoshi Yasudafor their time, interest, helpful advice and thoughtful criticism.

I gratefully acknowledge the Ministry of Education, Culture, Sports, Science and Technologyof Japan for providing me the Japanese government (Monbukagakusho: Mext) scholarship to sup-port my Ph.D. work and life in Japan. I would like to thank the authority of the University of theRyukyus for providing me the funding which allowed me to attend conferences.

I would like to thank all the members of our group especially Shiroma and Nohara for extendingtheir helping hands to become accustomed to daily life in Japan. I got immense cooperation fromT. Shimabukuro at all times during my early works. I would like to thank T. Tamashiro and T.Nakamura for their various help and discussion. I benefited from discussions with them.

I pay my deep respect for my teachers Prof. Arun Kumar Basak, Prof. M. Alfaz Uddin andProf. A.K.A Fazlul Haque for their encouragement of higher education and research.

Among well-wishers, I am thankfully acknowledging the wishes of my friends and colleagues,Sumal Chandra, M. Nuruzzaman, Z. Ahmmed, M. Hassan, M. Rashid,and M. K. Uddin. I feltgreatly encouraged when all of them showed keen interest in the progress of my works.

I cannot find words to express my gratitude to my parents for their unending love and support.They have been the source of inspiration and towering pillarof prayer for my future. I will neverbe able to repay my debt to them by any means. I am thankfully acknowledging the prayers andencouragement of my father- and mother-in-law.

Finally, I wish to express my heart-felt appreciation to my wife whose cooperation and under-standing have been simply overwhelming and who let me devotemyself fully to this task over arather long period of time and waited for its completion ungrudgingly.

The Author

iv

Contents

1 Introduction 1

2 Hamiltonian and Wavefunction Method 62.1 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62.2 U = 0 andt = 0 in the Hubbard model . . . . . . . . . . . . . . . . . . . . . . . 82.3 Hartree-Fock approximation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 102.4 Variational wavefunction method . . . . . . . . . . . . . . . . . . .. . . . . . . . 11

2.4.1 Gutzwiller wavefunction . . . . . . . . . . . . . . . . . . . . . . . .. . . 122.4.2 Local-ansatz wavefunction . . . . . . . . . . . . . . . . . . . . . .. . . . 13

3 Momentum Dependent Local-Ansatz Approach with Hartree-Fock Wavefunction 163.1 Construction of momentum dependent local-ansatz wavefunction . . . . . . . . . . 163.2 The ground-state energy . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 17

3.2.1 Single-site approximation (SSA) . . . . . . . . . . . . . . . . .. . . . . . 183.2.2 Correlation energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

3.3 Determination of variational parametersηk′

2k2k

′

1k1

. . . . . . . . . . . . . . . . . . 203.3.1 Interpolating solution ofηk

′

2k2k

′

1k1

. . . . . . . . . . . . . . . . . . . . . . 213.3.2 Best choice ofη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Various physical quantities . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 253.4.1 Correlation energyǫc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.2 Electron number〈ni〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.3 Momentum distribution〈nkσ〉 . . . . . . . . . . . . . . . . . . . . . . . 293.4.4 Double occupation number〈ni↑ ni↓〉 . . . . . . . . . . . . . . . . . . . . 31

3.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 333.5.1 Effect of the best choice ofη . . . . . . . . . . . . . . . . . . . . . . . . 333.5.2 MLA in various physical quantities . . . . . . . . . . . . . . . .. . . . . 35

4 Momentum Dependent Local-Ansatz Approach with Alloy-Analogy Wavefunction 394.1 Alloy-analogy wavefunction . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 394.2 Local-ansatz+ alloy-analogy wavefunction approach with momentum dependent

variational parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 424.3 Numerical results: half-filled band Hubbard model . . . . .. . . . . . . . . . . . 51

5 Momentum Dependent Local-Ansatz Approach with Hybrid Wavefunction 545.1 Hybrid wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 545.2 Local-ansatz+ hybrid wavefunction approach with momentum dependent varia-

tional parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .595.3 Numerical results: half-filled band Hubbard model . . . . .. . . . . . . . . . . . 65

v

6 Summary and Discussions 69

A Appendix: Gutzwiller wavefunction 72

B Appendix: Wick’s theorem 76

C Appendix: Density of states in infinite dimensions 79

D Appendix: Fermi liquid theory 83

E Appendix: Coherent potential approximation 89

vi

Chapter 1

Introduction

Electron-electron interactions play an important role in condensed matter physics. In a simplesystem, they are taken into account as a mean-field potentialof an independent-particle system.In the Hartree-Fock (HF) approximation, for example, one replaces the Coulomb interactions be-tween electrons by the Coulomb and exchange potentials for anelectron. Solving the one-electronSchrodinger equation and putting electrons one by one from the bottom of the lowest level of one-electron energy eigen values up to the Fermi level accordingto the Pauli exclusion principle, onecan construct the ground state of the system. In solids, the energy eigen values form the so-calledenergy bands. Because of the formation of band gaps at the Brillouin zone boundary, one canexplain the basic properties of metal, semiconductor, and insulator (i.e., band insulator) on thebasis of the one-electron picture. The independent-particle approximation seems to be applicableespecially to the conduction electrons in simple metals such as the alkali metals (Li, Na, K), andalkaline earth metals (Ca, Sr, Ba). But for the systems containing 3d or 4f unfilled shell, it is knownthat there are various phenomena which cannot be explained by the independent-particle picture,because the electrons of these systems are considerably localized even in solids, and move in asmall area with strong electron-electron interactions.

The cohesive energies of 4d and 5d transition metals, for example, are much greater than inthe simple metals and follow a roughly parabolic variation as a function of the d electron fillingnumber. In the 3d transition metals, on the other hand, the parabolic behavior is known to breakdown, and show a deep minimum at Mn. Friedel pointed out that the parabolic behavior of 4d and5d series is explained by the band energy gain due to d electrons, while the minimum behaviorin the 3d series cannot be explained even if the effects of electron-electron interaction are takeninto account by means of the HF approximation. In order to explain the behavior of the cohesiveenergy of 3d transition metals, one has to take into account the energy which is missing in the HFapproximation. The latter is called the correlation energy, and is caused by electron correlations.

Electron correlations also play an important role in the stability of ferromagnetism. The HFapproximation overestimates the magnetic energy gain due to spin polarization, because it neglectsthe spin fluctuations in the residual interactions. This means that the ferromagnetism is artificiallystabilized in the HF approximation. If we apply the HF approximation to the real elements, mostof the elements in solids are known to show the ferromagnetism. This is not consistent with theexperimental data; experimentally there are only several elements showing the ferromagnetism.The spin fluctuations should destroy the most of the ferromagnetic order obtained by the HF ap-proximation when electron correlations are taken into account.

The third example showing the failure of the HF approximation is found in the Curie tempera-ture (TC) and associated susceptibility. The HF approximation overestimates the magnetic energy

1

and does not produce the magnetic entropy as found in the Heisenberg model because of the ne-glect of spin fluctuations. Detailed analysis based on the HFapproximation shows that the Curietemperatures are overestimated by a factor of ten (see Table1.1) and the susceptibility does notfollow the Curie-Weiss law aboveTC [1, 2]. In order to obtain the right results one has to take intoaccount the spin fluctuations at finite temperatures.

Table 1.1: The Curie temperaturesTC for Fe, Co, and Ni calculated by the HF approximation [1, 2].The experimental data (Expt.) are shown on the bottom line.

TC (K) Fe Co Ni

HF 12200 12100 4940Expt. 1040 1388 630

The metal-insulator transition in transition metal oxidesis one the oldest problem of electroncorrelations. Mott considered the electronic structure ofNiO. The material has the NaCl structure,so that the unit cell contains one Ni atom, and one O atom. The electron configuration of theNi28

(O8) atom is given by1s22s22p63s23p63d84s2 (1s22s22p4). The oxygen atoms are considered toform a closed shell in the compound taking electrons from Ni atoms, so that we haveNi2+ =1s22s22p63s23p63d8 andO2− = 1s22s22p6. In this case, the Fermi level should be on the d bandsaccording to the band theory. Thus, we can expect a metal because the 5-fold d bands overlap eachother in general. This feature has been verified in the band calculations as shown in Fig. 1.1. Theexperimental data however indicate that NiO is an insulator.

Figure 1.1: Density of states (DOS) of NiO in the paramagnetic state obtained by the band calcu-lation [3]. The vertical dashed line shows the Fermi level.

In order to explain the many-body phenomena mentioned above, various theories of electroncorrelations have been developed so far. These theories arebased on the variational method, theGreen function techniques [4], as well as many numerical techniques such as the exact diagonaliza-tion method [5] and the Monte-Carlo method [6]. The variational approach is one of the simplestmethods among them and has been applied to many systems as a practical tool. Although the vari-ational approach is limited to the ground state, there are various advantages as follows. First, the

2

method is simple and intuitive, so that one can construct a correlated wavefunction according to aphysical picture to the problem. Second, the wavefunction of the correlated state is directly given,so that one can obtain any physical quantity. Third, one could systematically improve the wave-function by adding the higher-order operators for correlations. Finally, it is possible to perform thefirst-principle calculation using more realistic Hamiltonian because of its simplicity.

The Gutzwiller wavefunction (GW) is one of the basic wavefunctions, because of its concep-tual simplicity and applicability to realistic systems [7,8, 9]. Assume that the Hamiltonian of thesystem is given by the single-band Hubbard model which takesinto account the on-site CoulombinteractionU on the same orbital. When the Coulomb repulsionU is large, the double occupancyon the same orbital should be suppressed to avoid the energy loss due to the Coulomb repulsionU [10, 11, 12]. The HF wavefunction does not describe such correlations because of the indepen-dent motion of electrons. Gutzwiller proposed a trial wavefunction which controls the probabilityamplitudes of doubly occupied states in the HF wavefunction, by making use of a projection oper-atorΠi(1 − (1 − g)ni↑ni↓). Hereniσ is the number operator for an electron on sitei with spinσ,variational parameterg reduces the amplitudes of doubly occupied states on local orbitals. Stoll-hoff and Fulde [13, 14, 15] proposed a method called the local-ansatz approach (LA), which issimpler than the GW in treatment. The LA wavefunction takes into account the states created bylocal two-particle operators such as the residual Coulomb interactions{Oi} = {δni↑δni↓}. Hereδniσ = niσ − 〈niσ〉0, 〈niσ〉0 being the average electron number on sitei with spin σ in the HFapproximation. The theory has been applied to many systems such as molecules, transition metals,polyacetylene, transition metal oxides and semiconductors [16, 17].

Though the GW and the LA are applicable for various correlated electron systems, they arenot sufficient for the description of correlations from the weak to the strong interaction regimes.Indeed, the Hilbert space expanded by the local operators isnot sufficient to characterize preciselythe weakly-correlated states; the LA does not reduce to the second-order perturbation theory inthe weak correlation limit. The same difficulty also arises for the GW even in infinite dimensions.Moreover, in the strong Coulomb interaction regime, the GW yields the Brinkman-Rice atom (i.e.,no charge fluctuation on an atom) instead of the insulator solid in infinite dimensions [18].

The GW does not take into account exactly the intersite correlations. The variational wavefunc-tion proposed by Jastrow [19] describes the correlations;|ψJ〉 = exp

[∑i

∑j fij ni nj

]|φ0〉. Here

|φ0〉 represents the ground-state of non-interacting fermions,fij =∫d3xd3x′|ϕi(r)|2|ϕj(r

′)|2f(r−r′) is the variational parameters depending on sitesi andj, andni = ni↑ + ni↓ is the charge den-sity operator on sitei. Note thatϕi(r) denotes the atomic wavefunction on sitei and the functionf(r − r′) is a variational function of the displacementr − r′. The wavefunction|ψJ〉 describesthe intersite long-range density-density correlations. However, the applications are limited to theweakly correlated systems and the low-dimensional systems.

Baeriswyl, on the other hand, proposed a wavefunction calledBaeriswyl wavefunction (BW)which accurately describes electron correlations in the strong Coulomb interaction regime [20, 21,22, 23]. It is constructed by applying a hopping operatorT onto the atomic wavefunction|Ψ∞〉;|ΨBW〉 = e−ηT |Ψ∞〉. HereT = −∑i,j,σ tija

†iσajσ is the kinetic energy operator,tij denotes the

transfer integral between sitesi andj, a†iσ (aiσ) being the creation (annihilation) operator for anelectron on sitei with spinσ. The operatore−ηT with a variational parameterη describes electronhopping from the atomic state and suppresses the configurations with high kinetic energy. TheBW describes well the insulator state in the strong correlation regime. However, it is not easy todescribe the metallic state from this viewpoint.

The purpose of this thesis is to propose a new local-ansatz wavefunction with momentum de-

3

pendent variational parameters (MLA) to overcome the difficulties mentioned above,i.e., to obtainthe correct results in both the weak and strongU limit, and to interpolate the correlated electronstate between the two limits. On the basis of the Hubbard model and the MLA we calculate vari-ous physical quantitiesi.e., the ground state energy, the double occupation number, themomentumdistribution as well as the quasiparticle weight, and demonstrate that the MLA overcomes both theGW and the LA.

In the MLA [24, 25, 26], we consider two-particle operators in the momentum space withmomentum dependent parameters and project them onto the local orbitals. With use of such localoperators{Oi}, we construct the MLA wavefunction as|ΨMLA〉 =

∏i(1 − Oi)|φ0〉. Here|φ0〉 is

the HF wavefunction andi denotes sites of atoms. The best local basis set is chosen by controllingthe variational parameters in the momentum space. Using thevariational principle, we determinethe momentum-dependent variational parameters. The ground-state energy is obtained analyticallyfrom our wavefunction, and agrees with the result of the second-order perturbation theory in theweak interaction limit, and reduces to the correct atomic limit. We demonstrate that the MLAapproach much improves the GW as well as the LA in the weak and the intermediate Coulombinteraction regimes.

In order to describe the correlations in the strong Coulomb interaction regime, we proposein the next step an improved MLA wavefunction [27], which starts from the alloy-analogy (AA)wavefunction instead of the HF one. We call the wavefunctionthe MLA-AA. The concept of theAA approximation can be traced back to Hubbard’s original work on electron correlations [12]. Heconsidered that electrons move slowly from site to site in the strong Coulomb interaction regime,so that an electron on a site with (without) opposite-spin electron on the same site feels a potentialǫ0+U (ǫ0), whereǫ0 andU denote the atomic level and the on-site Coulomb interaction parameter,respectively. The AA wavefunction is the ground-state wavefunction for an independent-particleHamiltonian with such two kind of random potentials. We found numerically that the MLA-AAtheory describes the strongly correlated regime reasonably, and can go beyond the GW in the strongCoulomb interaction regime.

We propose in the next step a new MLA wavefunction [28] which describes reasonably thewhole Coulomb interaction regime on the same footing. We construct a hybrid (HB) wavefunctionand make use of it as the starting wavefunction for the MLA. The HB wavefunction is definedby the ground-state of the independent-particle Hamiltonian with a HB potential consisting of theHF potential with a weight1− w and the AA potential with a weightw, so that the wavefunctioncan vary from the HF type to the AA one via the new variational parameterw. Hereafter we callthe new wavefunction the MLA-HB. On the basis of the numericalresults of calculations for thehalf-filled band Hubbard model, we will clarify the validityof our theory, and demonstrate that theMLA-HB much improves both the GW and the LA, and describes electron correlations from theweak to the strong Coulomb interaction regime.

The dissertation is organized as follows. In the following chapter we describe the single-bandHubbard model to investigate electron correlations in solids and the fundamentals of the HF ap-proximation to the Hubbard model. We also describe the key concept of the variational methodand introduce the basic wavefunctions such as the GW as well as the LA. In Chapter 3, we con-struct the MLA wavefunction on the basis of the HF wavefunction. We obtain the ground-stateenergy within a single-site approximation (SSA) and derivethe self-consistent equation for themomentum-dependent variational parameters. We develop a method to obtain the best value ofvariational parameters solving the equation. To examine the validity of the MLA wavefunction wepresent the results of numerical calculations for the half-filled band as well as non-half-filled band

4

Hubbard model on the hypercubic lattice in infinite dimensions. We verify that the MLA approachimproves both the LA and the GW in the weak and the intermediate Coulomb interaction regimes.

The concept of the AA Hamiltonian and the wavefunction are presented in Chapter 4. Wedescribe the correlated MLA-AA wavefunction which starts from the AA wavefunction. We obtainthe ground-state energy within the SSA and derive the self-consistent equation for the variationalparameters as well as its solution. We present the numericalresults for the half-filled band Hubbardmodel in infinite dimensions. We discuss the ground-state energy, the double occupation number,and the momentum distribution. It is observed that the MLA-AA theory describes the stronglycorrelated regime reasonably, and thus the MLA-AA + MLA-HF can go beyond the GW in boththe weak and the strong Coulomb interaction regimes.

In Chapter 5, we introduce the HB Hamiltonian as well as the HB wavefunction. We willexamine the properties of the HB wavefunction, calculatingthe ground-state energy, the doubleoccupation number and the momentum distribution in infinitedimensions. Next we present thecorrelated MLA-HB wavefunction which starts from the HB wavefunction. We obtain the ground-state energy within the SSA, and derive the self-consistentequation for the momentum dependentvariational parameters and its solution. We present our results of numerical calculations for thehalf-filled band Hubbard model in infinite dimensions. We discuss the ground-state energy, thedouble occupation number, the momentum distribution, and the quasiparticle weight as a functionof the Coulomb interaction energy parameter. We verify that the MLA-HB approach improves boththe GW and the LA in the whole Coulomb interaction regime. In particular, we demonstrate thatthe momentum distribution calculated by the MLA-HB shows a distinct momentum dependence,and is qualitatively different from those obtained by the GWand the LA which show the constantvalues below and above the Fermi level. The last Chapter 6 is devoted to summary and discussions.We discuss our numerical results in comparison with other methods. We also discuss the furtherdevelopments of the theory.

5

Chapter 2

Hamiltonian and Wavefunction Method

2.1 Hubbard model

The Hubbard model [10, 11] is a basic model Hamiltonian for electrons in solids. It is thesimplest model in which electrons hop from site to site with on-site Coulomb repulsion energy. Thekinetic energy tends to delocalize electrons, while the on-site Coulomb interaction tends to localizeelectrons. The model is considered to describe the metal-insulator transition, the magnetism, aswell as the superconductivity in the strongly correlated electron system. In this section we derivethe Hubbard model from the basic Hamiltonian.

The Hamiltonian of a solid is given in the second quantization as follows:

H =

∫ ∑

σ

Ψ†σ(r) h Ψσ(r)dr +

1

2

∑

σσ′

∫Ψ†

σ(r)Ψ†σ′(r

′)e2

| r − r′ |Ψσ′(r′)Ψσ(r)drdr′ . (2.1)

HereΨσ(r) denotes the field operator for the spinσ in the Fock space andh is the one electronHamiltonian. Let us assume thatN atoms are condensed and form a solid. When atomic distanceis large enough the overlaps of the atomic wavefunction are small and electrons are bound aroundeach atom. Then, one-electron energy eigen-function of solids is considered to be a superpositionof eigen-function of each atom. According to the tight-binding approximation, one electron eigenstateψk(r) is obtained by the linear combination of local atomic orbitals.

ψk(r) =∑

i

φi(r −Ri)〈i|k〉. (2.2)

Here we considered the electrons in the unfilled shells and assumed one orbital on each atom.φi(r −Ri) denotes the atomic orbital of the atom atRi.

Therefore, one can expand the field operatorΨσ(r) in the second quantization by means oflocal orbitals{φi} , namely,

Ψσ(r) =∑

k

akσψk(r) =∑

i

aiσφi(r), (2.3)

Ψ†σ(r) =

∑

k

a†kσψ∗k(r) =

∑

i

a†iσφ∗i (r). (2.4)

Here{akσ, a†k′σ′} = δkk′δσσ′ , and{akσ, ak′σ′} = {a†kσ, a†

k′σ′} = 0. We assume that the local

orbitals are orthogonal to each other. Using the orthonormal property of{φi} , we obtainaiσ =

6

∑k akσ〈i|k〉 , a†iσ =

∑k a

†kσ〈i|k〉∗. The operators have the same commutation relations as in

the case of{akσ}, {a†kσ}. Therefore,a†iσ(aiσ) denotes the creation (annihilation) operator for anelectron on sitei with spin σ. Then the quantityniσ = a†iσaiσ expresses the electron densityoperator on sitei for spinσ.

The Hamiltonian with use of localized orbitals is obtained by substituting the field operatorΨσ(r) in Eqs. (2.3) and (2.4) into Eq. (2.1).

H = H0 +HI , (2.5)

H0 =∑

ijσσ′

〈i|h|j〉a†iσajσ′ , (2.6)

HI =1

2

∑

i1,i2,i3,i4,σ,σ′

⟨i1i2

∣∣∣∣∣e2

|r − r′|

∣∣∣∣∣i3i4

⟩a†i1σa

†i2σ′ai3σ′ai4σ . (2.7)

The matrix element in the HamiltonianH0 is given by〈i|h|j〉 = (H)ij = ǫ0δij + tij(1− δij). Hereǫ0 is the atomic level,tij is the transfer integral between sitesi andj. Therefore, the non-interactingpartH0 is given by

H0 =∑

iσ

ǫ0niσ +∑

i

∑

j 6=i

∑

σ

tija†iσajσ . (2.8)

In the interaction partHI , the intra-atomic Coulomb integral(i1 = i2 = i3 = i4) shouldbe much larger than the others. Therefore, we only take into account the intra-atomic Coulombinteraction.

Ui =

⟨ii

∣∣∣∣∣e2

|r − r′|

∣∣∣∣∣ii⟩

=

∫drdr′ e

2φ∗i (r −Ri)φ

∗i (r

′ −Ri)φi(r′ −Ri)φi(r −Ri)

|r − r′| . (2.9)

This is called the intra-atomic Coulomb energy parameter. The interaction part of the HamiltonianHI is then written as

HI =∑

i

Uini↑ni↓. (2.10)

Therefore, the Hamiltonian for many-electron system with use of the local orbitals is given asfollows:

H = H0 +HI =∑

iσ

ǫ0niσ +∑

ijσ

′tija

†iσajσ + U

∑

i

ni↑ni↓ . (2.11)

The Hamiltonian (2.11) is known as the Hubbard model, and wasproposed by Gutzwiller andHubbard independently [7, 8, 9, 10, 11]. The first term describes the atomic energy. The secondterm is the kinetic energy describing electrons hopping between nearest-neighbor sitesi andj. Thethird term is the on-site Coulomb interaction energy, it goesthrough all the sites and adds an energyU if the site is doubly occupied. We adopt in the following the Hubbard model for the descriptionof correlated electrons in solid.

7

2.2 U = 0 and t = 0 in the Hubbard model

Let us consider first the atomic limit of the Hubbard model to clarify its basic nature. Foran atom, we have 4 atomic states: the empty state(n↑ = 0, n↓ = 0), the single electron states(n↑ = 1, n↓ = 0) and (n↑ = 0, n↓ = 1), and the doubly occupied state(n↑ = 1, n↓ = 1).Associated energies are given as0, ǫ0, ǫ0, and2ǫ0 + U , respectively. In the atomic limit for a solidwheretij = 0, electron numberni on each atom is no longer constant, though the total number ofelectronsN is given;N =

∑iσ niσ. The eigenstates are given by|Ψ〉 = |{niσ}〉, i.e., a set of the

electron numbers with spinσ on sitei. The eigenenergy for the state is given by

E({niσ}) =∑

i

(ǫ0ni + Uni↑ni↓). (2.12)

It should be noted that the energy of the system increases byU when the number of doublyoccupied statesD is increased by one. The ground-state energyE0 of the atomic limit is obtainedby minimizing the energy with respect to the number of doubleoccupancy in solids. Assume thatthe number of lattice points is given byL. whenN < L, the ground-state energy is obtained asE0 = ǫ0N by choosingD = 0. Magnetic moments on sites with an electron are active in this caseas shown in Fig. 2.1. Because there areL!/N !(L − N)! electron configurations on theL latticepoints, the ground state is[2NL!/N !(L−N)!]-fold degenerate.

Figure 2.1: Electron configurations for less than half-filling (upper figure) and for more than half-filling (lower figure) in the atomic limit

At the half filling, all the atoms are occupied by an electron so that spin degrees of freedomby 2N remain; the degenerated wave functions are given by|1s1z 1s2z 1s3z · · · 〉 when the wavefunction |{niσ}〉 is written as|n1s1zn2s2zn3s3z · · · 〉 by using the chargeni = ni↑ + ni↓ and thespinsiz = (ni↑ − ni↓)/2.

When the electron numberN is larger thanL, it is no longer possible to keepD = 0; theminimum value ofD is given byD = N − L. The ground state energy is then given byE0 =ǫ0N + U(N − L). The ground state is[22L−NL!/(2L − N)!(N − L)!]-fold degenerate becausethere areL!/(2L − N)!(N − L)! configurations for choosing2L − N(< L) sites with the singleelectron fromL lattice sites and there are22L−N spin degrees of freedom for each configuration.Note that the spins on2L−N sites are active in this case.

When there is no Coulomb interaction (U = 0), on the other hand, electrons are generallyitinerant. The Hamiltonian is given as

H =∑

ijσ

(H0)ij a†iσajσ . (2.13)

8

Here(H0)ij = ǫ0δij + tij(1 − δij). The noninteracting Hamiltonian is diagonalized by a unitarytransformationaiσ =

∑k akσ〈i|k〉 (a†iσ =

∑k a

†kσ〈i|k〉∗) so that

H =∑

kσ

ǫknkσ , (2.14)

whereǫk =∑

ij〈k|i〉(H0)ij〈j|k〉 is an eigenvalue for the tight-binding one-electron HamiltonianmatrixH0, and the set{〈j|k〉} (j = 1, · · · , L) is the eigen vector forǫk.

The eigen states for noninteracting HamiltonianH are given by|Ψ〉 = |{nkσ}〉, i.e., a set ofelectrons with momentumk and spinσ. The eigenenergy is given by

E({nkσ}) =∑

kσ

ǫknkσ , (2.15)

where the c-numbernkσ takes on the value of0 or 1. The ground state is obtained by puttingelectrons on the energy levels from the bottom to the Fermi level ǫF according to the Pauli principle,

|φ0〉 =[ ǫk<ǫF∏

kσ

a†kσ

]|0〉, (2.16)

so that the ground-state energy is given by

E0({nkσ}) =ǫk<ǫF∑

kσ

ǫk . (2.17)

Alternatively, defining the density of states per atom per spin as

ρ(ǫ) =1

L

∑

k

δ(ǫ− ǫk) , (2.18)

we can express the ground-state energy per atom as

E0 = 2

∫ ǫF

−∞

ǫρ(ǫ)dǫ . (2.19)

A non-interacting electron system is in general metallic unless the electrons in the atom form aclosed shell. The electrons in such systems are mobile. Thisis because one can add an electron atthe energy level just above the Fermi level by applying infinitesimal electric field. Note that spinsof itinerant electrons are also mobile.

We may expect that there is a transition from metal to insulator at half filling when the intra-atomic Coulomb interaction is increased. Assume that there is a band for a non-interacting systemwhose band width isW . The center of the gravity of the noninteracting band is assumed to belocated atǫ0. When the Coulomb interactionU is increased, each atom tends to be occupied byone electron, and electron hopping to neighboring sites tends to be suppressed in order to reducethe on-site Coulomb interaction energy. In the strongly correlated region, an electron should havepotentialǫ0+U on a site having an opposite-spin electron because of the increment of the Coulombinteraction energy due to double occupation, while an electron has a potentialǫ0 on an empty site.We then expect one more band with the band width of order ofW aroundǫ0 + U . The density ofstates as excitation spectrum is expected to split into two bands atUc ∼ W (see Fig. 2.2 ). The

9

formation of a gap at the Fermi level implies the existence ofan insulator. The insulating statetherefore may be realized by the electron correlations when

U > W . (2.20)

This is Hubbard’s alloy-analogy picture to the metal-insulator transition [12]. The metal-insulatortransition due to electron correlations as mentioned aboveis commonly known as the Mott transi-tion. The split bands are named the upper and lower Hubbard bands, respectively. The insulatorcaused by the electron correlations is referred as the Mott insulator.

W

ε0+U

ε0

Figure 2.2: The upper and lower Hubbard bands created by on-site Coulomb interactionU

2.3 Hartree-Fock approximation

Although the Hubbard model is a simple model Hamiltonian, itinvolves a difficulty in solv-ing the many-electron problem inherent in the on-site Coulomb interaction. Thus, approximatemethods have been employed to discuss the physics involved in the Hamiltonian. In this section,we make an independent-particle approximation called the Hartree-Fock (HF) approximation, andderive the effective Hamiltonian.

We can rewrite the interaction part of Eq. (2.11) as follows:

ni↑ni↓ = ni↑〈ni↓〉0 + ni↓〈ni↑〉0 − 〈ni↑〉0〈ni↓〉0 + δni↑δni↓ . (2.21)

Hereδniσ = niσ − 〈niσ〉0, 〈∼〉0 denotes an average with respect to a wavefunctionφ0 for non-interacting electrons:〈φ0|(∼)|φ0〉. δni↑δni↓ is the fluctuation of local charge on the same site.In the HF approximation we neglect the fluctuation term, and replace the original HamiltonianHwith an effective HamiltonianHHF for independent-particle system.

HHF =∑

iσ

(ǫ0 + U〈ni−σ〉0)niσ +∑

ijσ

tija†iσajσ −

∑

i

U〈ni↑〉0〈ni↓〉0 . (2.22)

The potential in the first term on the r.h.s. doubly counts theCoulomb interaction. Therefore, thelast term

∑i U〈ni↑〉0〈ni↓〉0 is subtracted to avoid the double counting of the interaction energy.

The wavefunctionφ0 is chosen to be the ground-state of the HF HamiltonianHHF.

10

The original Hamiltonian (2.11) is then expressed by the HF Hamiltonian and the residualinteraction.

H = HHF + U∑

i

Oi . (2.23)

HereOi = δni↑δni↓ .The HF approximation is applicable to the system with small Coulomb interactionU because

the first-order correction to the HF energy vanishesi.e., ∆E = 〈φ0|HI|φ0〉 = 0. The theory obeysthe variational principle. The energy in the HF theory always yields an upper bound for the exactground-state energyE0 i.e.,

〈φ0|H|φ0〉 = 〈φ0|HHF|φ0〉 ≥ E0 . (2.24)

It gives the physical results that follow any conservation laws. The HF theory generally tends tooverestimate the symmetry breaking in the Hubbard model since correlations between the electronsare totally ignored. Therefore, the HF theory cannot provide us with a quantitative understandingof the ferromagnetism and the metal-insulator transition as discussed in the introduction.

2.4 Variational wavefunction method

One should take into account electron correlations becausethe HF ground-state energy is over-estimated as we have mentioned in the introduction. We describe in this subsection the variationalmethod to determine approximately the ground-state energyand the wavefunction for correlatedelectrons.

The variational principle is based on the fact that for any (arbitrary) trial function |ψ〉 wechoose, the energyE is always larger than the exact energyE0:

E =〈ψ|H|ψ〉〈ψ|ψ〉 ≥ E0 . (2.25)

To verify this, the trail wavefunction can formally be expanded in terms of the exact eigen statesof H:

|ψ〉 =∑

n

cn|φn〉, (2.26)

withH|φn〉 = En|φn〉 . (2.27)

Since|ψ〉 is normalized;〈ψ|ψ〉 =∑

n |cn|2 = 1. Therefore the expectation value ofH for thefunction|ψ〉 is given by

E = 〈H〉 =

⟨∑

m

cmφm|H∑

n

cnφn

⟩

=∑

m

∑

n

c∗mcnEn 〈φm|φn〉 =∑

n

En|cn|2 . (2.28)

The ground-state energy is, by definition, the smallest eigenvalue, thus,En ≥ E0. Then we obtain

E ≥ E0 , (2.29)

11

which proves Eq. (2.25).The equality condition occurs only when|ψ〉 is proportional to the true ground state|ψ0〉. The

variational equation is written as follows.

δE[ψ] = 0 . (2.30)

If |ψ〉 depends on parameterα, E[ψ] also depends onα. The variational ansatz (2.30) enables usto varyα so as to minimizeE[ψ]. The minimum value ofE[ψ] provides us with an upper limit forthe true energy of the system.

The Gutzwiller wavefunction and the Local-Ansatz wavefunction are popular as a basic wave-function in correlated electron calculations because of their simplicity and analytic character. Inthe following subsections we will give a brief outline of these wavefunctions.

2.4.1 Gutzwiller wavefunction

Let us consider the wavefunction for the Hubbard Hamiltonian. The Coulomb interactionreduces the double occupancy on the same site to avoid the energy loss. Taking into account thispoint Gutzwiller [7, 8, 9] proposed a trial wavefunction so-called Gutzwiller wavefunction (GW).

|ΨGW〉 =[∏

i

(1− (1− g)ni↑ni↓)]|φ0〉 . (2.31)

Hereniσ is the number operator for electron on sitei with spinσ. The wavefunction describeson-site electron correlations by making use of a projectionoperatorni↑ni↓ onto the HF state|φ0〉.The variational parameterg (0 ≤ g ≤ 1) has to be determined variationally by minimizing theenergy. It controls the amplitudes of the doubly occupied state in the HF approximation. Whenthe variational parameterg = 1, the state corresponds to an uncorrelated state; on the other hand,g = 0 state corresponds to the atomic state in which all the doublyoccupied states have beenremoved from the HF wavefunction.

The total energy for the GW is given as follows (see Appendix A).

E(g) =∑

σ

qσ

(Nσ∑

k

ǫk

)+ UD . (2.32)

Here the band narrowing factorqσ is given as

qσ =

(√(Nσ −D)(L−N +D) +

√D(N−σ −D)

)2

Nσ(L−Nσ). (2.33)

Nσ denotes the total electron number for the spinσ,D is the number of doubly occupied sites, andL denotes the number of lattice points.

In the non-magnetic state at half-filling(N = L), the energy per site is simplified as follows.

ǫ(g) = −q|ǫb|+ U d . (2.34)

Here

q = 16

(1

2− d

)d , (2.35)

12

g =d

12− d

, (2.36)

andd = D/L denotes the double occupation number per site.ǫb is the band energy per site, andis given by the non-interacting density of states per atom and per spinρ(ǫ) = L−1

∑k δ(ǫ− ǫk) as

follows.

ǫb = 2

∫ 0

−∞

ǫρ(ǫ)dǫ . (2.37)

Minimizing the energy (2.34) with respect tog (i.e.,d), we obtain

q = 1− U2

U2c

, (2.38)

d =1

4

(1− U

Uc

), (2.39)

and

g =1− U

Uc

1 +U

Uc

. (2.40)

HereUc = 8|ǫb| is a critical Coulomb interaction at whichq andd vanish. The ground-state energyis given by

ǫ(g) = −1

8Uc

(1− U

Uc

)2

. (2.41)

WhenU > Uc, there is another solution:g = 0 or d = 0 which yields the minimum energyǫ(g) = 0. Note that electrons are completely localized atU = Uc; q = d = 0. This implies thatmetal-insulator transition occurs atU = Uc. This is known as the Brinkman-Rice transition [18].

It is obvious that the GW is exact both in the small Coulomb interactions limit(U → 0), andin atomic limit (U → ∞). In spite of the simplicity the GW (2.31) is known to describevariousaspects of the strongly correlated electrons. However, it does not yield the correct weak Coulombinteraction regime even in infinite dimensions.

2.4.2 Local-ansatz wavefunction

The GW yields a physical picture for correlated electrons and has been applied to many prob-lems in the strongly correlated electron systems. However it was not so easy to apply the methodto the realistic Hamiltonian. An alternative approach which is simpler in treatment and applicableto the realistic Hamiltonian was proposed by Stollhoff and Fulde [13, 14, 15]. It is known as thelocal-ansatz (LA) wavefunction.

In the LA the Hilbert space for the correlated electrons is expanded by the local operators suchas the residual Coulomb interactions{Oi} = {δni↓δni↑} so that the wavefunction can describe theweak Coulomb interaction regime. The ansatz for the Hubbard model is written as

|ΨLA〉 =[∏

i

(1− ηLAOi)]|φ0〉 . (2.42)

13

HereηLA is the variational parameter as the amplitudes of the basis set expanded by{Oi}.The variational energy is expressed as

E = 〈φ0|H|φ0〉+ Ec . (2.43)

The energyEc denotes the correlation energy which cannot be described bythe HF approximation.In the single-site approximation, the correlation energy per atom is given by

ǫc =−2ηLA〈OiH〉0 + η2LA〈OiHOi〉0

1 + η2LA〈O2i 〉0

. (2.44)

Minimizing the energyǫc with respect to theηLA, we obtain

ηLA =−〈OiHOi〉0 +

√〈OiHOi〉20 + 4〈OiH〉20〈O2

i 〉02〈OiH〉0〈O2

i 〉0. (2.45)

Each element in the variational parameter and the correlation energy is given as follows.

〈OiHOi〉0 = 〈OiH0Oi〉0 + U〈O3i 〉0 , (2.46)

〈OiH0Oi〉0 =∑

σ

〈ni−σ〉0(1− 〈ni−σ〉0)[〈ni−σ〉0

∫dǫ ǫ ρiσ(ǫ)−

∫dǫ ǫ ρiσ(ǫ)

], (2.47)

〈O3i 〉0 = 〈O2

i 〉0(1− 2〈ni↑〉0)(1− 2〈ni↓〉0) , (2.48)

〈OiH〉0 = U〈O2i 〉0 , (2.49)

〈O2i 〉0 = 〈ni↑〉0(1− 〈ni↑〉0)〈ni↓〉0(1− 〈ni↓〉0) . (2.50)

Here〈niσ〉0 is the average electron number on sitei with spinσ. ρiσ(ǫ) is the density of states forthe one electron energy eigenvalues for the non-interacting system.f(ǫ) is the Fermi distributionfunction at zero temperature.

To obtain the analytic expression of various physical quantities in half-filled band Hubbardmodel, we consider the hypercubic lattice with Gaussian density of statesρ(ǫ) = (1/

√π) exp(−ǫ2).

The correlation energy per atom for the LA is written as

Ec =

−U2

ηLA4

+4√π

(ηLA4

)2

1 +(ηLA

4

)2 . (2.51)

Here the variational parameterηLA is defined by

ηLA =

− 1√π+

√1

π+U2

16U

32

. (2.52)

14

The double occupation is given by

〈ni↑ni↓〉 =1

2

(1

2− ηLA/4

1 +(ηLA/4

)2

). (2.53)

The analytic expression of momentum distribution is definedas follows.

〈nkσ〉 =1

2(1 + Z)f(ǫkσ) +

1

2(1− Z)(1− f(ǫkσ)). (2.54)

Heref(ǫkσ) is the Fermi distribution function and theZ is the quasiparticle weight. It is written asfollows.

Z =1− 3η2LA/16

1 + η2LA/16. (2.55)

The Hilbert space expanded by the operators{Oi} is not sufficient to characterize exactly theweakly correlated region. It does not reduce to the second-order perturbation theory in the weakcorrelation limit.

15

Chapter 3

Momentum Dependent Local-AnsatzApproach with Hartree-Fock Wavefunction

As we have emphasized in the previous chapter, the LA does notreduce to the second-orderperturbation theory in the weak correlation limit. The samedifficulty also arises for the GW evenin infinite dimensions. To remove the difficulty in the weak Coulomb interaction regime and toimprove the correlated states in the intermediate Coulomb interaction regime, we introduce inthis chapter a local-ansatz wavefunction with momentum-dependent variational parameters, anddemonstrate that the new approach much improves the LA as well as the GW in the weak and theintermediate Coulomb interaction regimes.

3.1 Construction of momentum dependent local-ansatz wave-function

Let us expand the LA wavefunction with respect to the Coulomb interactionU to examine thebehavior in the weak Coulomb interaction limit.

|ΨLA〉 = |ψ0〉+ |ψ1〉LA + · · · , (3.1)

with|ψ1〉LA = −

∑

i

ηLAOi|φ0〉 . (3.2)

We can rewrite the termOi = δni↓δni↑ using the relationsaiσ =∑

k akσ〈i|k〉 and a†iσ =∑k a

†kσ〈k|i〉 as follows:

Oi = δni↓δni↑ =∑

k1k′1k2k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉 δ(a†k′2↓ak2↓)δ(a†k′1↑ak1↑). (3.3)

Therefore,

|ψ1〉LA = −∑

i

∑

k1k′1k2k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉 ηLA δ(a†k′2↓ak2↓)δ(a

†k′1↑ak1↑)|φ0〉. (3.4)

Here 〈i|k〉 = exp(−ik · Ri)/√N is an overlap integral between the localized orbital and the

Bloch state with momentumk, Ri denotes the atomic position ,N is the number of sites andδ(a†k′σakσ) = a†k′σakσ − 〈a†k′σakσ〉0.

16

Finally, we obtain the expanded form of the LA wavefunction in the weak Coulomb interactionlimit as follows:

|ΨLA〉 = |ψ0〉 −∑

i

∑

k1k′1k2k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉 ηLA δ(a†k′2↓ak2↓)δ(a

†k′1↑ak1↑)|φ0〉. (3.5)

The above expression does not agree with the result of the purtarbation theory.To clarify the wavefunction in the correct weak Coulomb interaction limit, we start from the

HF wavefunction|φ0〉 and expand the the ground-state wavefunction|ψ〉 using the Rayleigh-Schrodinger perturbation theory as follows.

|Ψ〉 = |φ0〉+ |ψ1〉+ · · · , (3.6)

|ψ1〉 = −∑

i

∑

k1k′1k2k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉 η(0)

k′2k2k′1k1δ(a†

k′2↓ak2↓)δ(a

†k′1↑ak1↑)|φ0〉 , (3.7)

η(0)

k′2k2k′1k1

= −U limz→0

fk′2k2k′1k1z − ǫk′1↑ + ǫk1↑ − ǫk′2↓ + ǫk2↓

. (3.8)

Herefk′2k2k′1k1 is a Fermi factor of two-particle excitations defined byfk′2k2k′1k1 = f(ǫk1↑)(1 −f(ǫk′1↑))f(ǫk2↓)(1 − f(ǫk′2↓)), f(ǫ) is the Fermi distribution function at zero temperature, andǫkσ = ǫkσ − µ. µ is the Fermi level. ǫkσ is the HF one-electron energy eigenvalue given byǫkσ = ǫ0 + U〈ni−σ〉0 + ǫk − σh, ǫk being the Fourier transform oftij.

Equation (3.7) compared with Eq. (3.4) manifests that one has to take into account the mo-mentum dependence of the variational parameters to obtain the exact result in the weak Coulombinteraction limit. We therefore propose the following wavefunction with momentum-dependentvariational parameters{ηk′2k2k′1k1}.

|Ψ〉 =∏

i

(1− Oi)|φ0〉 , (3.9)

Oi =∑

k1k2k′1k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉ηk′2k2k′1k1δ(a†k′2↓ak2↓)δ(a

†k′1↑ak1↑) . (3.10)

We call the new form of Eq. (3.9) the momentum-dependent local ansatz (MLA) [24, 25, 26]. TheoperatorOi is still localized on sitei because of the projection〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉. It shouldbe noted thatO†

i 6= Oi and OiOj 6= OjOi (i 6= j) in general. These properties do not causeany problem when we make a single-site approximation (SSA).The wavefunction|Ψ〉 reduces to|ΨLA〉 when{ηk′2k2k′1k1} become momentum-independent.

3.2 The ground-state energy

We have to determine the variational parameters on the basisof the variational principle in orderto determine the wavefunction. The correlation energyEc is defined by the difference between theexact ground-state energy and the HF one as follows.

Ec = 〈H〉 − 〈H〉0 =〈Ψ|H|Ψ〉〈Ψ|Ψ〉 . (3.11)

17

Here H = H − 〈H〉0. Although it is not easy to calculate the correlation energywith use ofthe new wavefunction (3.9), one can obtain the expression for the energy within the single-siteapproximation (SSA).

3.2.1 Single-site approximation (SSA)

We generalize here the problem to calculate the correlationenergy, and consider the average ofa physical quantityA with respect to the correlated wavefunction (3.9).

〈A〉 = 〈Ψ|A|Ψ〉〈Ψ|Ψ〉 =

AN

BN

. (3.12)

Here A = A − 〈A〉0, 〈∼〉0 being the average with respect to the HF wavefunction.AN andBN

are defined as follows:

AN =⟨[∏

i

(1− O†i )]A[∏

i

(1− Oi)]⟩

0, (3.13)

BN =⟨[∏

i

(1− O†i )][∏

i

(1− Oi)]⟩

0. (3.14)

ExpandingBN with respect to site1, we obtain

BN = B(1)N−1 −

⟨O†

1

[∏

i

(1)(1− O†

i )][∏

i

(1)(1− Oi)

]⟩0

−⟨[∏

i

(1)(1− O†

i )]O1

[∏

i

(1)(1− Oi)

]⟩0

+⟨O†

1

[∏

i

(1)(1− O†

i )]O1

[∏

i

(1)(1− Oi)

]⟩0, (3.15)

and

B(1)N−1 =

⟨[∏

i

(1)(1− O†

i )][∏

i

(1)(1− Oi)

]⟩0. (3.16)

Here the product∏

i

(1) means the product with respect to all sites excluding site 1.When we calculateBN using the Wick theorem (see appendix B), we neglect the contractions

between different sites. This is a SSA and then Eq. (3.15) is expressed as

BN =⟨(1− O†

1

)(1− O1

)⟩0B

(1)N−1. (3.17)

We can make the same approximation forAN . In this case, there are two-types of terms, the termsin which the operatorO1 is contracted toA and the other terms withO1 contracted to the operatorsOi (i 6= 1). We have then

AN =⟨(1− O†

1

)A(1− O1

)⟩0B

(1)N−1 +

⟨(1− O†

1

)(1− O1

)⟩0A

(1)N−1 , (3.18)

18

and

A(1)N−1 =

⟨[∏

i

(1)(1− O†

i )]A[∏

i

(1)(1− Oi)

]⟩0. (3.19)

Successive application of the recursive relations (3.17) and (3.18) leads to

AN =∑

i

⟨(1− O†

i

)A(1− Oi

)⟩0B

(i)N−1 , (3.20)

BN =⟨(1− O†

i

)(1− Oi

)⟩0B

(i)N−1 =

∏

i

⟨(1− O†

i

)(1− Oi

)⟩0. (3.21)

Taking the ratioAN/BN , we obtain.

〈A〉 =∑

i

⟨(1− O†

i

)A(1− Oi

)⟩0⟨(

1− O†i

)(1− Oi

)⟩0

. (3.22)

This is the general expression for an operatorA in the SSA.

3.2.2 Correlation energy

The expression of the correlation energy per atomǫc is obtained analytically by making use ofthe formula (3.22).

ǫc =〈(1− O†

i )H(1− Oi)〉0〈(1− O†

i )(1− Oi)〉0. (3.23)

We consider all the sites are equivalent to each other. Since〈H〉0 = 0, and〈O†i 〉0 = 〈Oi〉0 = 0, the

correlation energy is given as follows,

ǫc =−〈O†

i H〉0 − 〈HOi〉0 + 〈O†i HOi〉0

1 + 〈O†i Oi〉0

. (3.24)

Each term in the correlation energy (3.24) can be calculatedby making use of the Wick theorem(see Appendix B) as follows.

〈HOi〉0 = U∑

k1k2k′1k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉∑

j

〈k1|j〉〈j|k′1〉〈k2|j〉〈j|k′2〉ηk′2k2k′1k1 fk′2k2k′1k1 ,

(3.25)

〈O†i H〉0 = 〈HOi〉∗0 , (3.26)

〈O†i HOi〉0 =

∑

k1k2k′1k

′2

〈i|k′1〉〈k1|i〉〈i|k′2〉〈k2|i〉 η∗k′2k2k′1k1 fk′2k2k′1k1∑

k3k4k′3k

′4

〈k′3|i〉〈i|k3〉〈k′4|i〉〈i|k4〉

×(∆Ek′2k2k

′1k1δk1k3δk′1k′3δk2k4δk′2k′4 + Uk′2k2k

′1k1k

′4k4k

′3k3

)ηk′4k4k′3k3 , (3.27)

19

Uk′2k2k′1k1k

′4k4k

′3k3

= U∑

j

[〈j|k1〉〈k3|j〉f(ǫk3↑)δk′1k′3 − 〈k′1|j〉〈j|k′3〉[1− f(ǫk′3↑)]δk1k3

]

×[〈j|k2〉〈k4|j〉f(ǫk4↓)δk′2k′4 − 〈k′2|j〉〈j|k′4〉[1− f(ǫk′4↓)]δk2k4

], (3.28)

〈O†i Oi〉0 =

1

N4

∑

k1k2k′1k

′2

|ηk′2k2k′1k1 |2fk′2k2k′1k1 . (3.29)

Here∆Ek′2k2k′1k1

= ǫk′2↓ − ǫk2↓ + ǫk′1↑ − ǫk1↑ is the two particle excitation energy.The above expressions (3.25) and (3.28) contain nonlocal terms in the summation overj (i.e.,∑

j). We thus make additional SSA called theR=0 approximation [29, 30]. For example, we havein Eq. (3.25)

∑

j

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉〈k1|j〉〈j|k′1〉〈k2|j〉〈j|k′2〉 =1

N4

∑

j

ei(k1+k2−k′1−k′2)(Rj−Ri), (3.30)

and only take into account the local term (j = i), in theR=0 approximation. Within the approxi-mation,〈HOi〉0 = 〈O†

i H〉∗0 and〈O†i HOi〉0 reduce as follows:

〈HOi〉0 =U

N4

∑

k1k2k′1k

′2

fk′2k2k′1k1 ηk′2k2k′1k1 , (3.31)

〈O†i HOi〉0 =

1

N4

∑

k1k2k′1k

′2

fk′2k2k′1k1 η∗k′2k2k

′1k1

[∆Ek′2k2k

′1k1ηk′2k2k′1k1

+U

N2

{∑

k3k4

f(ǫk3↑)f(ǫk4↓) ηk′2k4k′1k3 −∑

k3k′4

f(ǫk3↑)[1− f(ǫk′4↓)] ηk′4k2k′1k3

−∑

k′3k4

[1− f(ǫk′3↑)]f(ǫk4↓) ηk′2k4k′3k1 +∑

k′3k′4

[1− f(ǫk′3↑)][1− f(ǫk′4↓)] ηk′4k2k′3k1

}].

(3.32)

3.3 Determination of variational parametersηk′2k2k

′1k1

Variational parameters{ηk′2k2k′1k1} are obtained by minimizing the correlation energyǫc, i.e.,Eq. (3.24) with Eqs. (3.29), (3.31), and (3.32). The self-consistent equations for{ηk′2k2k′1k1} in theSSA are given as follows.

(∆Ek′2k2k′1k1

− ǫc)ηk′2k2k′1k1

+U

N2

[∑

k3k4

f(ǫk3↑)f(ǫk4↓)ηk′2k4k′1k3 −∑

k3k′4

f(ǫk3↑)(1− f(ǫk′4↓))ηk′4k2k′1k3

−∑

k′3k4

(1− f(ǫk′3↑))f(ǫk4↓)ηk′2k4k′3k1 +∑

k′3k′4

(1− f(ǫk′3↑))(1− f(ǫk′4↓))ηk′4k2k′3k1

]= U . (3.33)

20

It should be noted that the variational parameters{ηk2′k2k1′k1} in Eq. (3.33) vanish whenU −→0, i.e., ηk′2k2k′1k1 ∼ O(U). Thus in the weakU limit, one can omit the second term on the l.h.s.(left-hand-side). We then obtain the solution in the weakU limit as

ηk′2k2k′1k1 =U

∆Ek′2k2k′1k1

. (3.34)

In the atomic limit the transfer integralstij disappear, and one electron energy eigenvalueǫkbecomesk-independent,ǫ0. Thus,∆Ek′2k2k

′1k1

vanishes. In this limit we can drop thek dependenceof ηk′2k2k′1k1 , i.e.,ηk′2k2k′1k1 −→ η. Then, we find ak-independent solution being identical with theLA.

η =−〈OiHOi〉0 +

√〈OiHOi〉20 + 4〈OiH〉20〈O2

i 〉02〈OiH〉0〈O2

i 〉0. (3.35)

It is not easy to find the solution of Eq. (3.33) for the intermediate strength of Coulomb in-teractionU . We therefore propose an approximate solution in the following subsections, whichinterpolates between the weak and the atomic limits.

3.3.1 Interpolating solution ofηk′

2k2k

′

1k1

In this subsection we consider an approximate solution which interpolates between the weakand the atomic limits. We do not make any approximation to thefirst term with∆Ek′2k2k

′1k1

, becauseit is mandatory to describe exactly the weak Coulomb interaction regime. As we have mentionedbefore, the second term on the l.h.s. of Eq. (3.33) does not affect the solution in the weaklycorrelated limit . Therefore we approximate the variational parameters{ηk′2k2k′1k1} in the secondterm with a momentum-independent parameterη, which is suitable in the atomic region. Theself-consistent equation (3.33) then reduces as follows.

(∆Ek′2k2k′1k1

− ǫc) ηk′2k2k′1k1 + U [η(1− 2〈ni↑〉0)(1− 2〈ni↓〉0)] = U. (3.36)

We have then an interpolate solution as follows.

ηk′2k2k′1k1 =U [1− η(1− 2〈ni↑〉0)(1− 2〈ni↓〉0)]

∆Ek′2k2k′1k1

− ǫc. (3.37)

Here∆Ek′2k2k′1k1

in the denominator is the two particle excitation energy given by∆Ek′2k2k′1k1

=ǫk′2↓ − ǫk2↓ + ǫk′1↑ − ǫk1↑, andǫc is the correlation energy per atom (3.24):

ǫc =−〈Oi

†H〉0 − 〈HOi〉0 + 〈Oi

†HOi〉0

1 + 〈Oi

†Oi〉

, (3.38)

The best value ofη should be determined variationally as will be discussed, but we might makeuse of that in the LA in a simpler version of numerical calculations and might adopt the correlationenergy in the LA forǫc as well. We call this the non-self-consistent MLA [24] in thefollowings.

21

3.3.2 Best choice ofη

The best value ofη, however, should be determined variationally in general. In this subsectionwe further develop the theory in whichη is determined best. According to the variational principle,the ground-state energyE0 satisfies the following inequality.

E0 ≤ E[Ψ] =〈Ψ|H|Ψ〉〈Ψ|Ψ〉 . (3.39)

HereΨ is a trial wavefunction.In the MLA, we choose the wavefunctionΨ = Ψ({ηk′2k2k′1k1}) and the corresponding energy

E({ηk′2k2k′1k1}) satisfies the inequalityE0 ≤ E({ηk′2k2k′1k1}). For the stationary valuesη∗k′2k2k′1k1 ,we have

E0 ≤ E({η∗k′2k2k′1k1}) ≤ E({ηk′2k2k′1k1}) . (3.40)

In the previous calculations of non-selfconsistent MLA [24], we obtained an approximateη∗k′2k2k′1k1

(3.37);

ηk′2k2k′1k1(η, ǫc) =Uη

∆Ek′2k2k′1k1

− ǫc. (3.41)

Hereη = 1− η(1− 2〈ni↑〉0)(1− 2〈ni↓〉0). (3.42)

When we adopt the form (3.41) as a trial set of amplitudes, we have an inequality as

〈E0〉 ≤ E({η∗k′2k2k′1k1}) ≤ E({ηk′2k2k′1k1(η, ǫc)}) . (3.43)

The above relation implies that the bestη is again determined from the stationary condition of thetrial energyE({ηk′2k2k′1k1(η, ǫc)}). Becauseǫc should satisfy the stationary conditionδǫc = 0 forthe valueη∗, η∗ is determined by the following condition

[∂ǫ({ηk′2k2k′1k1(η, ǫc)})

∂η

]

ǫc

= 0 . (3.44)

The self-consistent equation is obtained from Eq.(3.44) inthe same way as in Eq. (3.33)

1

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1∂ηk′2k2k′1k1

∂η(∆Ek′2k2k

′1k1

− ǫc)ηk′2k2k′1k1

+U

N6

∑

k1k′1k2k

′2

fk′2k2k′1k1∂ηk′2k2k′1k1

∂η

{∑

k3k4

f(ǫk3↑)f(ǫk4↓)ηk′2k4k′1k3

−∑

k′3k4

[1− f(ǫk′3↑)]f(ǫk4↓)ηk′2k4k′3k1 −∑

k3k′4

f(ǫk3↑)[1− f(ǫk′4↓)]ηk′4k2k′1k3

+∑

k′3k′4

[1− f(ǫk′3↑)][1− f(ǫk′4↓)]ηk′4k2k′3k1

}=

U

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1∂ηk′2k2k′1k1

∂η. (3.45)

Here∂ηk′2k2k′1k1/∂η is obtained from Eq. (3.41) as

∂ηk′2k2k′1k1∂η

=U

∆Ek′2k2k′1k1

− ǫc. (3.46)

22

Substituting the above expression into the self-consistent equation (3.45), we obtain

η =1

1 +UC

D

. (3.47)

Here

C =1

N6

∑

k1k′1k2k

′2

fk′2k2k′1k1(∆Ek′2k2k

′1k1

− ǫc)

×{∑

k3k4

f(ǫk3↑)f(ǫk4↓)

(∆Ek′2k4k′1k3

− ǫc)−∑

k′3k4

[1− f(ǫk′3↑)]f(ǫk4↓)

(∆Ek′2k4k′3k1

− ǫc)

−∑

k3k′4

f(ǫk3↑)[1− f(ǫk′4↓)]

(∆Ek′4k2k′1k3

− ǫc)+∑

k′3k′4

[1− f(ǫk′3↑)][1− f(ǫk′4↓)]

(∆Ek′4k2k′3k1

− ǫc)

}, (3.48)

and

D =1

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1(∆Ek′2k2k

′1k1

− ǫc). (3.49)

In the energy representation, each term is expressed as follows:

C =

∫[

6∏n=1

dǫn

][6∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]f(ǫ5↑)f(ǫ6↓)

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)(ǫ4 − ǫ6 + ǫ2 − ǫ5 − ǫc)

−∫[

6∏n=1

dǫn

][6∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)][1− f(ǫ5↑)]f(ǫ6↓)

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)(ǫ4 − ǫ6 + ǫ5 − ǫ1 − ǫc)

−∫[

6∏n=1

dǫn

][6∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]f(ǫ5↑)[1− f(ǫ6↓)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)(ǫ6 − ǫ3 + ǫ2 − ǫ5 − ǫc)

+

∫[

6∏n=1

dǫn

][6∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)][1− f(ǫ5↑)][1− f(ǫ6↓)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)(ǫ6 − ǫ3 + ǫ5 − ǫ1 − ǫc),

(3.50)

and

D =

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc. (3.51)

Here ǫnσ = ǫn + ǫσ and ǫσ = ǫ0 + U〈n−σ〉0 − µ is the atomic level measured from the chemicalpotential, andρ(ǫ) is the density of states for the one-electron energy eigenvalues for the non-interacting systemtij.

It should be noted that the self-consistent solution (3.47)is also obtained by solving approxi-mately the original self-consistent Eq.(3.33). In order todo this, first we divide the both sides ofEq.(3.33) by(∆Ek′2k2k

′1k1

− ǫc) , substitute the form (3.41), and we obtain Eq. (3.47) after taking

23

the average with respect tok1k′1k2k′2 with a weightfk′2k2k′1k1 . The variational principles onη tell us

that such a solution should be the best among possible approximate solutions. We also note thatan approximate form (3.42), which was obtained in the non-self-consistent case [24], is derivedfrom the solution (3.47). In fact, we rewrite Eq. (3.47) asη = 1 − UηC/D. By replacing theapproximate form (3.41) in the expression ofUηC with the momentum independent valueη, wereach Eq. (3.42).

The expressionC given by Eq. (3.50) consists of the multiple integrals up to the 6-folds.One can reduce these integrals up to the 2-folds using the Laplace transform [31]. The Laplacetransform can significantly reduce the number of integrals in the physical quantities which appearin our variational theory. It is written as follows:

1

z − ǫ4 + ǫ3 − ǫ2 + ǫ1 + ǫc= −i

∫ ∞

0

dt ei(z−ǫ4+ǫ3−ǫ2+ǫ1+ǫc) t . (3.52)

Herez = ω + iδ, andδ is an infinitesimal positive number.The termD in Eq. (3.51) is written as follows:

D =

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)

= − limz→0

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]

(z − ǫ4 + ǫ3 − ǫ2 + ǫ1 + ǫc). (3.53)

Now using the relation of the Laplace transform (3.52), we obtain

D = −i limz→0

∫ ∞

0

dte(z+ǫc)t

∫dǫ1e

iǫ1tρ(ǫ1)f(ǫ1↑)

×∫dǫ2e

−iǫ2tρ(ǫ2)[1− f(ǫ2↑)]

∫dǫ3e

iǫ3tρ(ǫ3)f(ǫ3↓)

×∫dǫ4e

−iǫ4tρ(ǫ4)[1− f(ǫ4↓)]

= −i∫ ∞

0

dteiǫcta↑(−t)b↑(t)a↓(−t)b↓(t). (3.54)

Here

aσ(t) =

∫dǫρ(ǫ)f(ǫ+ ǫσ) e

−iǫt , (3.55)

bσ(t) =

∫dǫρ(ǫ)[1− f(ǫ+ ǫσ)] e

−iǫt . (3.56)

The 4-fold integrals ofD in Eq. (3.51) reduce to the 1-fold integral in Eq. (3.54).In the same way, we can perform the Laplace transform of the elementC (Eq.(3.50)) as follows

C = −∫ ∞

0

dtdt′eiǫc(t+t′)[a↑(−t)b↑(t+ t′)a↓(−t)b↓(t+ t′)a↑(−t′)a↓(−t′)

−a↑(−t− t′)b↑(t)a↓(−t)b↓(t+ t′)b↑(t′)a↓(−t′)

−a↑(−t)b↑(t+ t′)a↓(−t− t′)b↓(t)a↑(−t′)b↓(t′)+a↑(−t− t′)b↑(t)a↓(−t− t′)b↓(t)b↑(t

′)b↓(t′)]. (3.57)

24

3.4 Various physical quantities

3.4.1 Correlation energyǫcThe ground-state correlation energy (3.24) per atom in the SSA is given by

ǫc =−〈O†

i H〉0 − 〈HOi〉0 + 〈O†i HOi〉0

1 + 〈O†i Oi〉

. (3.58)

The ground-state correlation energy is obtained by substituting the variational parameters (3.41)into Eq. (3.58). Each element in the energy is given as follows.

〈HOi〉0 = 〈O†i H〉∗0

= U2η

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc, (3.59)

〈O†i HOi〉0 = 〈O†

i H0Oi〉0 + U〈O†iOiOi〉0 , (3.60)

〈O†i H0Oi〉0

= U2η2∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2(ǫ4 − ǫ3 + ǫ2 − ǫ1)−1, (3.61)

〈O†iOiOi〉0

= U2η2∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2

×[∫

[6∏

n=5

dǫn

][6∏

n=5

ρ(ǫn)

]f(ǫ5↑)f(ǫ6↓)

(ǫ4 − ǫ6 + ǫ2 − ǫ5 − ǫc)

−∫[

6∏n=5

dǫn

][6∏

n=5

ρ(ǫn)

]f(ǫ5↑)[1− f(ǫ6↓)]

(ǫ6 − ǫ3 + ǫ2 − ǫ5 − ǫc)

−∫[

6∏n=5

dǫn

][6∏

n=5

ρ(ǫn)

][1− f(ǫ5↑)]f(ǫ6↓)

(ǫ4 − ǫ6 + ǫ5 − ǫ1 − ǫc)

+

∫[

6∏n=5

dǫn

][6∏

n=5

ρ(ǫn)

][1− f(ǫ5↑)][1− f(ǫ6↓)]

(ǫ6 − ǫ3 + ǫ5 − ǫ1 − ǫc)

], (3.62)

〈O†i Oi〉0 = U2η2

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2. (3.63)

25

It should be noted thatη in Eq. (3.47) is given as a function ofǫc, andǫc in Eq. (3.24) dependson η andǫc. Therefore, both equations have to be solved self-consistently. To determine the bestvalue ofη, we start fromǫc in the LA for example, and calculateη according to Eq. (3.47). Nextwe calculate various elements〈HOi〉0 (〈O†

i H〉∗0), 〈O†i HOi〉0, and〈O†

i Oi〉0 which are given by Eqs.(3.59), (3.60), and (3.63), respectively. Using these values we calculateǫc again according to Eq.(3.24). We repeat this cycle until the self-consistency ofǫc andη is achieved. We call this schemethe self-consistent MLA [25] .

Various terms of correlation energyǫc in Eq. (3.24) are expressed by means of the Laplacetransform as follows:

〈O†i Oi〉0 = −U2η2

∫ ∞

0

dtdt′eiǫc(t+t′)a↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b↓(t+ t′) , (3.64)

〈HOi〉0 = 〈O†i H〉∗0 = iU2η

∫∞

0dt eiǫct a↑(−t)a↓(−t)b↑(t)b↓(t) , (3.65)

〈O†i H0Oi〉0 = −U2η2

∫ ∞

0

dtdt′eiǫc(t+t′)[a↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b1↓(t+ t′)

−a↑(−t− t′)b↑(t+ t′)a1↓(−t− t′)b↓(t+ t′)

+a↑(−t− t′)b1↑(t+ t′)a↓(−t− t′)b↓(t+ t′)

−a1↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b↓(t+ t′)], (3.66)

〈O†iOiOi〉0 = −U2η2

∫ ∞

0

dtdt′eiǫc(t+t′)[a↑(−t)b↑(t+ t′)a↓(−t)b↓(t+ t′)a↑(−t′)a↓(−t′)

−a↑(−t)b↑(t+ t′)a↓(−t− t′)b↓(t)a↑(−t′)b↓(t′)−a↑(−t− t′)b↑(t)a↓(−t)b↓(t+ t′)b↑(t

′)a↓(−t′)+a↑(−t− t′)b↑(t)a↓(−t− t′)b↓(t)b↑(t

′)b↓(t′)]. (3.67)

Here

a1σ(t) =

∫dǫρ(ǫ)f(ǫ+ ǫσ) ǫ e

−iǫt , (3.68)

b1σ(t) =

∫dǫρ(ǫ)[(1− f(ǫ+ ǫσ)] ǫ e

−iǫt . (3.69)

3.4.2 Electron number〈ni〉

In order to obtain the expectation value of any physical quantity, we may use the Feynman-Hellmann theorem. It relates the derivative of the total energy with respect to a parameter, to theexpectation value of the derivative of the Hamiltonian withrespect to the same parameter.

Consider a system with HamiltonianH(λ) that depends on some parameterλ. LetΨ(λ) be aneigenfunction ofH(λ) with an eigenvalueE(λ).

H(λ)Ψ(λ) = E(λ)Ψ(λ). (3.70)

26

We start from the relation.E(λ) = 〈Ψ(λ)|H(λ)|Ψ(λ)〉. (3.71)

Differentiating both sides with respect toλ yields

∂E

∂λ= 〈∂Ψ(λ)

∂λ|H(λ)|Ψ(λ)〉+ 〈Ψ(λ)|∂H(λ)

∂λ|Ψ(λ)〉+ 〈Ψ(λ)|H(λ)|∂Ψ(λ)

∂λ〉. (3.72)

Generally,H(λ) is the Hermite operator. Therefore,

∂E

∂λ= E(λ)〈∂Ψ(λ)

∂λ|Ψ(λ)〉+ 〈Ψ(λ)|∂H(λ)

∂λ|Ψ(λ)〉+ E(λ)〈Ψ(λ)|∂Ψ(λ)

∂λ〉

= 〈Ψ(λ)|∂H(λ)

∂λ|Ψ(λ)〉+ E(λ)

[〈∂Ψ(λ)

∂λ|Ψ(λ)〉+ 〈Ψ(λ)|∂Ψ(λ)

∂λ〉]. (3.73)

Since|Ψ(λ)〉 is assumed to be normalized, we have from the normalization condition,〈Ψ(λ)|Ψ(λ)〉 = 1, the following identity.

〈∂Ψ(λ)

∂λ|Ψ(λ)〉+ 〈Ψ(λ)|∂Ψ(λ)

∂λ〉 = ∂(1)

∂λ= 0. (3.74)

Hence, the second term on the r.h.s. of Eq. (3.73) vanishes, and we reach

∂E(λ)

∂λ= 〈Ψ(λ)|∂H(λ)

∂λ|Ψ(λ)〉. (3.75)

The above relation is called the Feynman-Hellmann theorem [32, 33].The average electron number is obtained by making use of Feynman-Hellmann theorem (3.75)

on the Hubbard model (2.11).

∂〈H〉∂ǫ0

= 〈Ψ|∂H∂ǫ0

|Ψ〉 =∑

σ

〈niσ〉. (3.76)

Here〈ni〉 =∑

σ〈niσ〉 is the average electron number on sitei. In the above expression we adoptthe normalized variational wavefunction|Ψ〉. Then the l.h.s. of the above equation is given as

∂〈H〉MLA

∂ǫ0= 〈Ψ|∂H

∂ǫ0|Ψ〉MLA = 〈φ|∂H

∂ǫ0|φ〉+

〈Ψ|(∂H∂ǫ0

− 〈φ|∂H∂ǫ0

|φ〉)|Ψ〉MLA

〈Ψ|Ψ〉MLA

, (3.77)

or∂〈H〉MLA

∂ǫ0=

[∂〈φ|H|φ〉

∂ǫ0

]

v

+N

[∂ǫc∂ǫ0

]

v,η

. (3.78)

Here,v is the potential energy of one electron,[A]v means that the change ofA via potentialv inthe HF wavefunction|φ〉 should be neglected.[A]v,η implies that there is no change ofA via v andη. Therefore, the relation (3.76) reduces to the following formula.

〈ni〉 =[∂〈φ|H|φ〉

∂ǫ0

]

v

+N

[∂ǫc∂ǫ0

]

v,η

. (3.79)

To calculate the r.h.s. of the above equation, we separate the energy into the HF energy and thecorrelation energy parti.e., 〈H〉 = 〈H〉0 + Nǫc. The HF energy is given by〈H〉0 = 〈φ|H0|φ〉 −

27

∑i Ui〈ni↑〉0〈ni↓〉0. HereH0 =

∑i(ǫ0 +Ui〈ni−σ〉0)niσ +

∑ijσ tija

†iσajσ, is the HF Hamiltonian as

an independent particle.The first term on the r.h.s. of Eq. (3.79) is calculated as

[∂〈φ|H|φ〉LA

∂ǫ0

]

v

= 〈φ|∂H∂ǫ0

|φ〉 = 〈ni〉0. (3.80)

The correlation energy per atom (3.24) is given by

Nǫc =∑

i

−〈O†i H〉0 − 〈HOi〉+ 〈OiHOi〉0

1 + 〈O†i Oi〉0

. (3.81)

Thus, we have

N

[∂ǫc∂ǫ0

]

v,η

=

−[∂〈O†

i H〉0∂ǫ0

]

v

−[∂〈HOi〉0∂ǫ0

]

v

+

[∂〈O†

i HOi〉0∂ǫ0

]

v

1 + 〈O†i Oi〉0

. (3.82)

Here〈HOi〉0 = 〈H0Oi〉0 + Ui〈OiOi〉. Using the Wick theorem (see Appendix B), we calculate〈H0Oi〉0 as〈H0Oi〉0 = 〈O†

i H0〉0 = 0. Therefore, the first two terms of the numerator on the r.h.s.of Eq. (3.82) vanish and the third term is written as

〈O†i HOi〉0 = 〈O†

i H0Oi〉0 + U〈O†iOiOi〉0. (3.83)

The second term on the r.h.s. of the above equation does not explicitly include ǫ0. Hence, there isno need to differentiate this term with respect toǫ0. The derivative of the first term with respect toǫ0 gives

∂〈O†i H0Oi〉0∂ǫ0

= 〈O†i

∑

σ

niσOi〉0 , (3.84)

and thus

∂〈O†i HOi〉0∂ǫ0

= 〈O†i

∑

σ

niσOi〉0 . (3.85)

Hence Eq. (3.82) becomes

N

[∂ǫc∂ǫ0

]

v,η

=

∑σ〈O

†i niσOi〉0

1 + 〈O†i Oi〉0

. (3.86)

The final expression for the average electron number is obtained by using Eqs. (3.79), (3.80) and(3.86).

〈ni〉 = 〈ni〉0 +∑

σ〈O†i niσOi〉0

1 + 〈O†i Oi〉0

. (3.87)

Here〈ni〉0 is the HF term, and∑

σ〈O†i niσOi〉0 denotes the correlation contribution.

28

The term〈O†i niσOi〉0 can be written by means of energy representation as follows.

〈O†i niσOi〉0

= U2η2∫[

5∏n=1

dǫn

][5∏

n=1

ρ(ǫn)

]f(ǫ1−σ)[1− f(ǫ2−σ)]f(ǫ3σ)[1− f(ǫ4σ)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2

×[ [1− f(ǫ5σ)]

(ǫ5 − ǫ3 + ǫ2 − ǫ1 − ǫc)− f(ǫ5σ)

(ǫ4 − ǫ5 + ǫ2 − ǫ1 − ǫc)

], (3.88)

After the Laplace transform, we obtain

〈O†i niσOi〉0 = −U2η2

∫ ∞

0

dtdt′eiǫc(t+t′)[a−σ(−t− t′)b−σ(t+ t′)aσ(−t− t′)bσ(t)bσ(t

′)

−a−σ(−t− t′)b−σ(t+ t′)aσ(−t)bσ(t+ t′)aσ(t′)]. (3.89)

Another quantity〈O†i Oi〉0 in Eq. (3.87) is given by Eq. (3.63).

3.4.3 Momentum distribution 〈nkσ〉

Momentum distribution function〈nkσ〉 represents a distribution of electrons in the momentumspace. The ground state energy for normalized wavefunctionΨ is

〈H〉 = 〈Ψ|(H0 +∑

i

UiOi)|Ψ〉 . (3.90)

In the momentum representation, the HamiltonianH is expressed as.

H =∑

kσ

ǫkσnkσ +∑

i

UOi , (3.91)

H0 =∑

kσ

(ǫ0 − σh+ U〈ni−σ〉0)nkσ +∑

kσ

ǫknkσ. (3.92)

Hereǫkσ ≡ ǫk − σh+ U〈ni−σ〉0.By making use of the Feynman-Hellmann theorem (3.75), we obtain

〈nkσ〉 = 〈Ψ| ∂H∂ǫkσ

|Ψ〉 = ∂〈H〉∂ǫkσ

. (3.93)

We calculate for the variational wavefunction the derivative on the r.h.s. of Eq. (3.93).

∂〈H〉MLA

∂ǫkσ= 〈Ψ| ∂H

∂ǫkσ|Ψ〉MLA = 〈φ| ∂H

∂ǫkσ|φ〉+

〈Ψ|( ∂H∂ǫkσ

− 〈φ| ∂H∂ǫkσ

|φ〉)|Ψ〉MLA

〈Ψ|Ψ〉MLA

, (3.94)

or∂〈H〉MLA

∂ǫkσ=

[∂〈φ|H|φ〉∂ǫkσ

]

v

+N

[∂ǫc∂ǫkσ

]

v,η

. (3.95)

Herev is the potential energy of the one particle state. Therefore,

〈nkσ〉 =[∂〈φ|H|φ〉∂ǫkσ

]

v

+N

[∂ǫc∂ǫkσ

]

v,η

. (3.96)

29

The Hamiltonian is divided into the HF energy and the correlation energy part:〈H〉 = 〈H〉0+Nǫc.The HF part is given by〈H〉0 =

∑kσ ǫkσ〈nkσ〉0 −

∑i Ui〈ni↑〉0〈ni↓〉0. Hence

[∂〈φ|H|φ〉∂ǫkσ

]

v

= 〈φ| ∂H∂ǫkσ

|φ〉 = 〈nkσ〉0. (3.97)

The correlation part in Eq. (3.96) is given by

N

[∂ǫc∂ǫkσ

]

v,η

=

−[∂〈O†

i H〉0∂ǫkσ

]

v

−[∂〈HOi〉0∂ǫkσ

]

v

+

[∂〈O†

i HOi〉0∂ǫkσ

]

v

1 + 〈O†i Oi〉0

. (3.98)

The 1st- and 2nd-term of the numerator on the r.h.s. are givenas〈HOi〉0 = 〈O†i H〉∗0 = U〈OiOi〉.

SinceU〈OiOi〉 is independent ofǫkσ, both the terms vanish. The third term on the r.h.s. of Eq.(3.98) is given as〈O†

i HOi〉0 = 〈O†i H0Oi〉0 + Ui〈O†

iOiOi〉0. SinceUi〈O†iOiOi〉0 is independent of

ǫkσ, we can ignore the differentiation of it with respect toǫkσ. Therefore[∂〈O†

i HOi〉0∂ǫkσ

]

v

=

[〈O†

i

∂H

∂ǫkσOi〉0

]

v

= 〈O†i δnkσOi〉0. (3.99)

HereH0 =∑

kσ ǫkσδnkσ andδnkσ = nkσ−〈nkσ〉0. Equation (3.98) is therefore written as follows.

N

[∂ǫc∂ǫkσ

]

v,η

=∑

i

〈O†i nkσOi〉0

1 + 〈O†i Oi〉0

. (3.100)

We obtain the expression of〈nkσ〉 using Eqs. (3.96), (3.97), and (3.100) as follows.

〈nkσ〉 = 〈nkσ〉0 +∑

i

〈O†i nkσOi〉0

1 + 〈Oi

†Oi〉0

. (3.101)

Since〈O†i nkσOi〉0 should be independent of sitei for a system with one atom per unit cell, we

have∑

i〈O†i nkσOi〉0 = N〈O†

i nkσOi〉0, N being the total number of sites. Finally, we obtain theexpression of the momentum distribution〈nkσ〉 as follows.

〈nkσ〉 = 〈nkσ〉0 +N〈O†

i nkσOi〉01 + 〈Oi

†Oi〉0

. (3.102)

The momentum distribution function consists of the HF term〈nkσ〉0 and correlation contributionN〈O†

i nkσOi〉0.The termN〈O†

i nkσOi〉0 can be written in the energy representation as follows.

N〈O†i nkσOi〉0

= U2η2[[1− f(ǫkσ)]

∫[

3∏n=1

dǫn

][3∏

n=1

ρ(ǫn)

]f(ǫ1−σ)[1− f(ǫ2−σ)]f(ǫ3σ)

(ǫkσ − ǫ3 + ǫ2 − ǫ1 − ǫc)2

− f(ǫkσ)

∫[

3∏n=1

dǫn

][3∏

n=1

ρ(ǫn)

]f(ǫ1−σ)[1− f(ǫ2−σ)]f(ǫ3σ)

(ǫ3 − ǫkσ + ǫ2 − ǫ1 − ǫc)2

], (3.103)

30

In the Laplace transform representation, it is given as

N〈O†i nkσOi〉0 = U2η2

∫ ∞

0

dtdt′eiǫc(t+t′)a−σ(−t− t′)b−σ(t+ t′)

×[f(ǫkσ)bσ(t+ t′)eiǫk(t+t′) − [1− f(ǫkσ)]aσ(−t− t′)e−iǫk(t+t′)

]. (3.104)

Another quantity〈O†i Oi〉0 in Eq. (3.102) is given by Eq. (3.63).

3.4.4 Double occupation number〈ni↑ni↓〉

The double occupation number〈ni↑ni↓〉 denotes the probability of finding doubly-occupiedelectrons on the same site. In the HF approximation, this is constant irrespective ofU . It isobtained by applying the Feynman-Hellmann theorem to the Hubbard model (2.11):

〈ni↑ni↓〉 = 〈Ψ|∂H∂Ui

|Ψ〉 = ∂〈H〉∂Ui

. (3.105)

In the case of the MLA,

∂〈H〉MLA

∂Ui

= 〈Ψ|∂H∂Ui

|Ψ〉MLA = 〈φ|∂H∂Ui

|φ〉+〈Ψ|(∂H

∂Ui

− 〈φ|∂H∂Ui

|φ〉)|Ψ〉MLA

〈Ψ|Ψ〉MLA

. (3.106)

Therefore,

〈ni↑ni↓〉 =[∂〈φ|H|φ〉∂Ui

]

v

+N

[∂ǫc∂Ui

]

v,η

. (3.107)

Here v represents the potential energy of one electron. The HF energy is given by〈H〉0 =∑kσ ǫkσ〈nkσ〉 −

∑i Ui〈ni↑〉0〈ni↓〉0. The first term is given by

[∂〈φ|H|φ〉∂Ui

]

v

= 〈ni↑〉0〈ni↓〉0. (3.108)

The second term on the r.h.s. of Eq. (3.107) is given by

N

[∂ǫc∂Ui

]

v,η

=

−[∂〈O†

i H〉0∂Ui

]

v

−[∂〈HOi〉0∂Ui

]

v

+

[∂〈O†

i HOi〉0∂Ui

]

v

1 + 〈O†i Oi〉0

. (3.109)

First, we calculate the 1st-term and 2nd-term on the r.h.s. of the above expression.

〈HOi〉0 = Ui〈OiOi〉 =U

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1 ηk′2k2k′1k1 , (3.110)

〈O†i H〉0 = 〈HOi〉0

∗. (3.111)

Therefore, [∂〈HOi〉0∂Ui

]

v

=〈HOi〉0Ui

=1

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1 ηk′2k2k′1k1 , (3.112)

31

[∂〈O†

i H〉0∂Ui

]

v

=〈O†

i H〉0Ui

=1

N4

∑

k1k′1k2k

′2

fk′2k2k′1k1 η∗k′2k2k

′1k1. (3.113)

In the same way, Eqs. (3.112) and (3.113) can be written as follows.[∂〈O†

i H〉0∂Ui

]

v

=〈O†

i H〉0Ui

= 〈O†iOi〉0, (3.114)

[∂〈HOi〉0∂Ui

]

v

=〈HOi〉0Ui

= 〈OiOi〉0. (3.115)

Next, we calculate the third-term on the r.h.s. of Eq. (3.109).[∂〈O†

i HOi〉0∂Ui

]

v

=1

N6

∑

k1k2k′1k

′2

fk′2k2k′1k1 η∗k′2k2k

′1k1

+

[∑

k3k4

f(ǫk3↑)f(ǫk4↓) ηk′2k4k′1k3

−∑

k3k′4

f(ǫk3↑)[1− f(ǫk′4↓)] ηk′4k2k′1k3

−∑

k′3k4

[1− f(ǫk′3↑)]f(ǫk4↓) ηk′2k4k′3k1

+∑

k′3k′4

[1− f(ǫk′3↑)][1− f(ǫk′4↓)] ηk′4k2k′3k1

]. (3.116)

Here the Hamiltonian is divided into the HF and the correlation parts.

〈O†i HOi〉0 = 〈O†

i H0Oi〉0 + 〈O†i

∑

j

UjOjOi〉0, (3.117)

Therefore, [∂〈O†

i HOi〉0∂Ui

]

v

= 〈O†i

∂H0

∂Ui

Oi〉0 +[∂

∂Ui

〈O†i

∑

j

UjOjOi〉0]

v

. (3.118)

Using the HF Hamiltonian (3.92), we obtain∂H0/∂Ui =∑

σ〈ni−σ〉0niσ. Thus,[∂〈O†

i HOi〉0∂Ui

]

v

=∑

σ

〈ni−σ〉0〈O†i niσOi〉0 + 〈O†

iOiOi〉0, (3.119)

The correlation contribution in Eq. (3.109) becomes as follows.[∂ǫc∂Ui

]

v,η

=−〈O†

iOi〉0 − 〈OiOi〉0 + 〈O†iOiOi〉0 +

∑σ〈ni−σ〉0〈O†

i niσOi〉01 + 〈O†

i Oi〉0. (3.120)

Finally, the double occupation number〈ni↑ni↓〉 is obtained from Eqs. (3.107), (3.108), and(3.120) as follows.

〈ni↑ni↓〉 =〈ni↑〉0〈ni↓〉0

+−〈O†

iOi〉0 − 〈OiOi〉0 + 〈O†iOiOi〉0 +

∑σ〈ni−σ〉0〈O†

i niσOi〉01 + 〈O†

i Oi〉0. (3.121)

32

The expression,〈O†iOi〉0 + 〈OiOi〉0 is written in the energy representation as follows.

〈O†iOi〉0 + 〈OiOi〉0

= 2Uη

∫[

4∏n=1

dǫn

][4∏

n=1

ρ(ǫn)

]f(ǫ1↑)[1− f(ǫ2↑)]f(ǫ3↓)[1− f(ǫ4↓)]

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc), (3.122)

In the Laplace transformation, it is given by

〈O†iOi〉0 + 〈OiOi〉0 = 2iUη

∫ ∞

0

dt eiǫcta↑(−t)b↑(t)a↓(−t)b↓(t) . (3.123)

The other quantities〈O†iOiOi〉0, 〈O†

i niσOi〉0, and〈O†i Oi〉0 have been given by Eqs. (3.62),

(3.88), and (3.63), respectively.

3.5 Numerical results

We have performed the numerical calculations for the non-half-filled as well as half-filled bandsof the Hubbard model in order to examine the validity of the improved scheme of the MLA andthe effect of electron correlations. To calculate various physical quantities, we have adopted thehypercubic lattice in infinite dimensions, where the SSA works best. The density of states (DOS)for non-interacting system is given byρ(ǫ) = (1/

√π) exp(−ǫ2) in this case [24](See Appendix

C). The energy unit is chosen to be∫dǫρ(ǫ)ǫ2 = 1/2. The external magnetic fieldh is assumed to

be zero.

3.5.1 Effect of the best choice ofη

To calculate various quantities in the MLA, we solved the self-consistent equations (3.24) and(3.47) with use of the Laplace transforms of elements. In this sub-section, we compare the self-consistent results with the non self-consistent ones to clarify the role of the best choice ofη.

Figure 3.1 shows the calculated correlation energy as a function of Coulomb interaction. Thecorrelation energy for the non-selfconsistent MLA gives the lower correlation energy as com-pared with the LA. The correlation energy for the MLA with thebest choice ofη (i.e., the self-consistent MLA) is lower than that of the non-self-consistent MLA. The results indicate that theself-consistency ofη is significant for finding the best energy.

In Fig. 3.2 we show an example of the momentum distribution asa function of energyǫkσwhen electron number per atom (n) is deviated from 1. The MLA with non self-consistentη (3.42)shows a bump in the vicinity of the Fermi level, leading to an unphysical result. The MLA withself-consistentη yields a significant momentum dependence which shows monotonical decreaseof the distribution with increasingǫkσ.

We have also calculated the quasiparticle weightZ vs. Coulomb interaction energy curves athalf-filling. As shown in Fig. 3.3, we find that the best choiceof η increasesZ (i.e., decreases theinverse effective mass), so that the critical Coulomb interaction of the divergence of the effectivemass,Uc2 changes from3.21 to 3.40. The latter is closer to the numerical renormalization group(NRG) [34] valueUc2 = 4.10, which is believed to be the best at present.

From the above discussions on the results with and without self-consistentη, it is obviousthat self-consistent MLA improves the MLA theory without self-consistentη. In the followingdiscussions we adopt the MLA with the best choice ofη.

33

-0.8

-0.75

-0.7

-0.65

-0.6

-0.55

-0.5

-0.45

-0.4

-0.35

-0.3

4 4.5 5 5.5 6 6.5 7 7.5

Ec

U

n=0.8

MLA (sc-η) MLA(nsc-η)LA

Figure 3.1: The correlation energyEc vs. Coulomb interaction energy curve forn = 0.8. The thicksolid curve: the MLA with self-consistentη, the thin curve: the MLA with non self-consistentη,and the dashed curve: the LA.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.5 0 0.5 1

⟨nkσ

⟩

εkσ

n = 0.6U = 2.0

MLA(sc-η) MLA(nsc-η) LA

Figure 3.2: The momentum distribution as a function of energy ǫkσ for various theories withn =0.6 andU = 2.0. The solid curve: the MLA with the best choice ofη, the dashed curve: the MLAwithout the best choice ofη, and the dotted curve: the LA.

34

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5

Z

U

n=1.0

MLA(sc-η) MLA(nsc-η)

Figure 3.3: Quasiparticle-weight Z vs. Coulomb interactioncurves in the MLA with self-consistentη (solid curve), and without (dashed curve).

3.5.2 MLA in various physical quantities

We present in this section the numerical results on various physical quantities, and discuss thenew aspects of the MLA and related effects of electron correlations by comparing the MLA withthe LA.

We first represent in Fig. 3.4 the calculated correlation energy per atom as a function ofCoulomb interactionU . The energy in the MLA is lower than that of the LA over all Coulombinteraction energy parametersU and electron numbersn. These results imply that the MLA im-proves the LA. The magnitude of the correlation energy|ǫc| tends to increase with increasingU ,because with increasingU the correlation corrections increase asU2 for smallU and cancel theHF energy loss being linear inU for largeU . For a fixed value of the Coulomb interactionU , thegain of the correlation energy|ǫc| increases with increasingn, because there is a correlation energygain at each doubly-occupied site and the number of such sites increases with increasingn.

Figure 3.5 depicts the double occupation number〈n↑n↓〉 vs. Coulomb interaction curves forthe half-filled case as well as non-half-filled case. In the uncorrelated limit, the double occupancyis the same for both the LA and the MLA and it decreases with increasing Coulomb interactionU because electrons move on the lattice so as to suppress the loss of Coulomb energy due to thedouble occupation. For the half-filled case, the suppression of the double occupancy is prominentin the intermediate regime of Coulomb interaction. We find that the MLA wavefunction reducesthe double occupancy as compared with that of the LA, for the non-half-filled case, in the range0 < U . 5, while in the range5 . U the double occupancy in the MLA is larger than that of theLA. It implies that the LA with momentum-independentηLA overestimates the itinerant characterfor weak and intermediateU regions, while it overestimates the atomic character for largeU region.

The momentum-distribution function shown in Fig. 3.6 indicates more distinct difference be-tween the LA and the MLA. The distributions in the LA are constant below and above the Fermilevel irrespective ofU . The same behavior is also found in the GW [7, 8, 9]. The MLA curves

35

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0 2 4 6 8 10

Ec

U

n=1.0

n=0.8

n=0.6

n=0.4

n=0.2

MLA LA

Figure 3.4: The correlation energiesEc vs. Coulomb interaction energy parameterU in the MLA(solid curve) and the LA (dashed curve) for various electronnumbern.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

⟨n↑n

↓⟩

U

n=1.0

n=0.8

n=0.6

n=0.4

n=0.2

MLA LA

Figure 3.5: The double occupation number〈n↑n↓〉 vs. Coulomb interaction energyU curves in theMLA (solid curve) and the LA (dotted curve).

36

show a monotonical decrease of the distribution with increasingǫkσ, indicating a distinct momen-tum dependence of〈nkσ〉 via energyǫkσ, which is qualitatively different from both the LA and theGW.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1 -0.5 0 0.5 1

⟨nkσ

⟩

εkσ

U = 3

21

n=1.0

MLA LA

Figure 3.6: The momentum distribution as a function of energy ǫkσ for various electron numberwith constant Coulomb interaction energy parametersU = 3. The MLA: solid curves, the LA:dashed curves.

The quasiparticle weightZ (i.e. the inverse effective mass) is obtained from the jump at theFermi level in the momentum distribution according to the Fermi liquid theory [35, 36, 37, 38] (seeAppendix D). Calculated quasiparticle weight vs Coulomb interaction curves are shown in Fig. 3.7for the half-filled case. The quasiparticle weight in the LA changes asZ = (1 − 3η2LA/16)/(1 +η2LA/16) and vanishes atUc2(LA) = 24/

√3π (= 7.82). In the GW [18], the quasiparticle weight

changes asZ = 1 − (U/Uc2)2. The curve in the GW agrees with the LA curve for smallU . But

it deviates from the LA whenU becomes larger, and vanishes atUc2(GW) = 8/√π (= 4.51).

It should be noted that the GW curve strongly deviates from the curve in the NRG [34] which isconsidered to be the best. It is remarkable that the RPT-1 ( renormalized perturbation theory withfirst order approximation to the memory function ) is close tothe NRG throughUc2 = 3.705. Weobserve that the critical Coulomb interactionUc2 for the self-consistentη is 3.40 in the MLA whileUc2 in the non-self-consistentη yields3.21. The quasiparticle weight in the MLA much improvesthe LA as seen in Fig.3.7. We note that the wavefunction itself does not show the metal-insulatortransition atUc2 in the present approximation because the approximate expression of variationalparameters (3.37) has no singularity at finite value ofU . The values ofZ obtained by the LA andthe MLA should be regarded as an estimate from the metallic side.

In summary, we have developed a momentum dependent local-ansatz wavefunction approach(MLA) to the correlated electron systems in solids to solve best a self-consistent equation forvariational parameters at half-filling as well as non half-filling. With use of the improved varia-tional scheme we performed the numerical calculations for the Hubbard model on the hypercubiclattice in infinite dimensions. We verified that the self-consistent scheme significantly improvesthe correlation energy and the momentum distribution as compared with the non self-consistent

37

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

Z

U

n=1.0

RPT-1NRG LA GW MLA

Figure 3.7: Quasiparticle-weight vs. Coulomb interaction curves in various theories. The RPT-1:dashed curve [39], the NRG: thin solid curve [34], the LA: dotted curve, the MLA: solid curve,and the GW: dot-dashed curve.

scheme in the MLA. We also demonstrated that the theory improves the standard variational meth-ods such as the local-ansatz approach (LA) and the Gutzwiller wavefunction approach (GW); theground-state energy in the MLA is lower than those of the LA and the GW in the weak and inter-mediate Coulomb interaction regimes. The double occupationnumber is shown to be suppressedas compared with the LA. Calculated momentum distribution functions show a distinct momentumdependence, which is qualitatively different from those ofthe LA and the GW.

38

Chapter 4

Momentum Dependent Local-AnsatzApproach with Alloy-Analogy Wavefunction

As we have mentioned in the previous chapter the MLA wavefunction does suitably describeelectron correlations in the weak and the intermediate Coulomb interaction, but it cannot suppressloss of Coulomb repulsion in the strong interaction regime. To improve the difficulty we pro-pose in this chapter a new wavefunction [27] by changing the starting wavefunction from the HFwavefunction to the alloy-analogy (AA) wavefunction whichis suitable in the strong Coulombinteraction regime.

4.1 Alloy-analogy wavefunction

The concept of the alloy-analogy Hamiltonian can be traced back to Hubbard’s original workon electron correlations [12]. In the strong Coulomb interaction regime, electrons with spinσmove slowly from site to site due to electron correlations, and therefore should feel a potentialUinstead of the HF average potentialU〈ni−σ〉0, when the opposite spin electron is on the same site.Hubbard regarded this system as an alloy with different random potentialsǫ0 + U andǫ0 havingthe concentration〈ni−σ〉 (occupied) and[1 − 〈ni−σ〉] (unoccupied), respectively. Hereǫ0 denotesthe atomic level of electron. This is called the alloy-analogy (AA) approximation (see Fig. 4.1).The AA Hamiltonian is then defined by

HAA =∑

iσ

(ǫ0 − µ+ Uni−σ)niσ +∑

ijσ

tija†iσajσ − U

∑

i

(ni↑〈ni↓〉AA + ni↓〈ni↑〉AA)

+U∑

i

〈ni↑〉AA〈ni↓〉AA. (4.1)

Here〈∼〉AA denotes the AA average〈φAA|(∼)|φAA〉 with respect to the ground-state wavefunction|φAA〉 of the AA HamiltonianHAA. Since the motion of electrons with opposite spin are treatedto be static in the AA approximation, related operators{ni−σ} are regarded as a random staticCnumberni−σ (0 or 1). Each configuration of{niσ} is considered as a snapshot in time development.

The ground-state energyE satisfies the following inequality for any wavefunction|φAA〉.E ≤ 〈φAA|H|φAA〉 = 〈HAA〉AA , (4.2)

and thus

E ≤ 〈H〉AA . (4.3)

39

ε0

ε0+U

Figure 4.1: Alloy-analogy approximation

Here〈H〉AA ≡ 〈φAA|H|φAA〉 = 〈HAA〉AA and the upper bar denotes the configurational average.The AA ground-state energy per atom is obtained by taking theconfigurational average.

〈H〉AA = nµ+ 2

∫ 0

−∞

ǫ ρiσ(ǫ) dǫ+ U〈ni↑〉AA〈ni↓〉AA − U(ni↑〈ni↓〉AA + ni↓〈ni↑〉AA) . (4.4)

Heren is the electron number per atom.ρiσ(ǫ) is the local density of states (DOS) and is obtainedfrom the one-electron Green function as follows.

ρiσ(ǫ) = − 1

πImGiiσ(z ) . (4.5)

The Green functionGiiσ(z) is defined by

Giiσ(z) = [(z −Hσ)−1]ii . (4.6)

Here(Hσ)ij is the one-electron Hamiltonian matrix for the AA Hamiltonian (4.1), which is definedby

(Hσ)ij = ǫiσδij + tij(1− δij) , (4.7)

ǫiσ = ǫ0 − µ+ Uni−σ . (4.8)

The average electron number〈niσ〉AA for the AA Hamiltonian (4.1) is given as

〈niσ〉AA =

∫f(ǫ)ρiσ(ǫ) dǫ . (4.9)

Heref(ǫ) denotes the Fermi distribution function.To obtain the local DOS, we make use of the coherent potentialapproximation (CPA) [40, 41].

In the CPA, we replace the random potentials at the surrounding sites with coherent potentialsΣσ(z). The on-site impurity Green functionGiiσ(z) is then obtained as follows.

Giiσ(z) =1

Fσ(z)−1 − ǫiσ + Σσ(z). (4.10)

HereFσ(z) is the on-site Green function for the coherent system in which all the random potentialshave been replaced by the coherent ones, and is given by

40

Fσ(z) =

∫ρ(ǫ) dǫ

z − Σσ(z)− ǫ. (4.11)

Here ρ(ǫ) is the DOS per site for the noninteracting system. The coherent potentialΣσ(z) isdetermined from the self-consistent condition (see Appendix E).

G00σ(z) = Fσ(z) . (4.12)

The configurational average of the impurity Green function is given as

G00σ(z) =〈n−σ〉AA

Fσ(z)−1 − ǫ0 + µ− U + Σσ(z)+

1− 〈n−σ〉AA

Fσ(z)−1 − ǫ0 + µ+ Σσ(z)(4.13)

The average DOS in the second term on the right hand side (r.h.s.) of Eq. (4.4) is given by

ρiσ(ǫ) = − 1

πImG00σ(z ) . (4.14)

The double occupation numbers on the r.h.s. of Eq. (4.4) are obtained in the SSA as follows.

〈ni↑ni↓〉AA = 〈ni↑〉AA〈ni↓〉AA =∑

α

Pα〈n↑〉α〈n↓〉α . (4.15)

Hereα = 00, 10, 01, 11 denotes the on-site electron number configuration. Alternative notationν that ν = 0 (empty on a site),1 ↑ (occupied by an electron with spin↑ ), 1 ↓ (occupied by anelectron with spin↓ ) and2 (occupied by 2 electrons) is also useful. In this case, the probabilityPα for the configurationα is expressed asP0, P1↑, P1↓ andP2.

For the empty sitei.e., α = 00, the expressionǫiσ in Eq. (4.8) for the up spin (↑) and down spin(↓) state becomesǫ0. Therefore, the term〈n↑〉00〈n↓〉00 in Eq. (4.15) is redefined as〈n↑〉ǫ0〈n↓〉ǫ0 .Similarly, the rest of the terms forα = 10, 01, 11 are expressed as〈n↑〉ǫ0〈n↓〉ǫ0+U , 〈n↑〉ǫ0+U〈n↓〉ǫ0 ,and〈n↑〉ǫ0+U〈n↓〉ǫ0+U , respectively. The electron number is then given by

〈nσ〉ǫ0 =∫f(ǫ)ρǫ0 σ(ǫ) dǫ , (4.16)

ρǫ0 σ(ǫ) = − 1

πImGǫ0σ(z ) , (4.17)

Gǫ0σ(z) =1

Fσ(z)−1 − ǫ0 + µ+ Σσ(z), (4.18)

〈nσ〉ǫ0+U =

∫f(ǫ)ρǫ0+U σ(ǫ) dǫ , (4.19)

ρǫ0+U σ(ǫ) = − 1

πImGǫ0+Uσ(z ) , (4.20)

41

Gǫ0+Uσ(z) =1

Fσ(z)−1 − ǫ0 + µ− U + Σσ(z). (4.21)

Note thatP0 + P1↑ + P1↓ + P2 = 1, and the probability of finding an electron with spin↑ (↓)on a site is given byP↑(↓) = P1↑(1↓) + P2. Three statistical probabilitiesP0, P1↑, andP1↓ dependon the probabilityP2. In the following, we obtain an approximate expression ofP2.

Note that we made the following approximations in the derivation of the AA Hamiltonian.

ni↑ni↓ ≈ ni↑ni↓ + ni↓ni↑ − ni↑ni↓. (4.22)

Taking the quantum mechanical average as well as the configurational average of the double oc-cupation number of the above expression, we obtain the probability of the double occupationP2

(= 〈ni↑ni↓〉) as follows.

P2 = ni↑〈ni↓〉AA + ni↓〈ni↑〉AA − ni↑ni↓ (4.23)

The last term on the r.h.s. of Eq. (4.23) may be regarded as theprobabilityP2. Therefore, weobtain

P2 ≈1

2(ni↑〈ni↓〉AA + ni↓〈ni↑〉AA). (4.24)

The double occupation numbers on the r.h.s. of Eq. (4.4) are obtained in the SSA as follows.

ni↑〈ni↓〉AA + ni↓〈ni↑〉AA = P↑〈n↓〉ǫ0+U + P↓〈n↑〉ǫ0+U (4.25)

The electron number〈nσ〉ǫ0+U is defined by Eq. (4.19).The momentum distribution in the AA scheme is given by

〈nkσ〉AA =

∫f(ǫ)ρkσ(ǫ) dǫ , (4.26)

ρkσ(ǫ) = − 1

πImFkσ(z ) , (4.27)

Fkσ(z) =1

z − Σσ(z)− ǫk. (4.28)

Hereǫk is the eigenvalue oftij with momentumk.In summary, in order to determine the AA energy〈H〉AA in Eq. (4.4) we first determine the

coherent potentialΣσ self-consistently by using the Eqs. (4.11), (4.12), and (4.13); then calculatethe average DOS (Eq. (4.14) ) as well as the double occupationnumber by Eqs. (4.15) and (4.25).

4.2 Local-ansatz+ alloy-analogy wavefunction approach withmomentum dependent variational parameters

We have constructed the MLA wavefunction in Sec. 3.1, which is given as follows (see Eq.(3.9)).

|ΨMLA〉 =∏

i

(1− Oi)|φ0〉. (4.29)

42

In the MLA wavefunction we considered the HF wavefunction asa starting wavefunction. Thetheory can describe electron correlations in the weak and the intermediate Coulomb interactionregimes. Hereafter we refer the wavefunction (4.29) to the MLA-HF wavefunction. The MLA-HFwavefunction is not suitable for the description of strongly-correlated electrons as we discussedin Chap. 3. In order to describe the strong interaction regimewe adopt in this section the AAwavefunction|φAA〉 as a starting wavefunction and propose a new ansatz, which wecall the MLA-AA wavefunction, as follows.

|ΨMLA−AA〉 =∏

i

(1− Oi)|φAA〉. (4.30)

Note that we introduced modified local operators{Oi} with variational parametersηκ′2κ2κ

′1κ1

in theabove expression, which is defined as follows.

Oi =∑

κ′2κ2κ

′1κ1

〈κ′1|i〉〈i|κ1〉〈κ′2|i〉〈i|κ2〉 ηκ′2κ2κ

′1κ1δ(a†

κ′2↓aκ2↓)δ(a

†κ′1↑aκ1↑) . (4.31)

Herea†κσ andaκσ are the creation and annihilation operators which diagonalize the HamiltonianHAA, and δ(a†κ′σaκσ) = a†κ′σaκσ − 〈a†κ′σaκσ〉AA. It should be noted that the MLA-AA wave-function reduces to the MLA-HF by replacing the random potential Uni−σ with the HF one,i.e.,U〈φ0|ni−σ|φ0〉, so thatΨMLA−AA andΨMLA−HF might be mutually connected to each other via asuitable parameter which interpolates between the two wavefunctions.

The inequality condition for the ground-state energyE of any wavefunction|Ψ〉 is written asfollows.

E ≤ 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 = 〈H〉AA +Nǫc . (4.32)

Here〈H〉AA denotes the energy for the AA wavefunction.ǫc is the correlation energy per atomdefined by

Nǫc =〈Ψ|H|Ψ〉〈Ψ|Ψ〉 , (4.33)

with H = H − 〈H〉AA. Since it depends on the electron configuration{niσ} via the AA potential,we have to take the configurational average at the end. To determine the variational parameters,we minimize the ground-state energy.

We adopt here the single-site approximation (SSA) to obtainthe correlation energy. It is givenby using the relation (3.22) as follows.

ǫc =−〈O†

i H〉AA − 〈HOi〉AA + 〈O†i HOi〉AA

1 + 〈O†i Oi〉AA

. (4.34)

Each element in the correlation energy (4.34) can be calculated by making use of Wick’s theo-rem as follows.

〈HOi〉AA = U∑

κ′2κ2κ

′1κ1

〈κ′1|i〉〈i|κ1〉〈κ′2|i〉〈i|κ2〉∑

j

〈κ1|j〉〈j|κ′1〉〈κ2|j〉〈j|κ′2〉ηκ′2κ2κ

′1κ1fκ′

2κ2κ′1κ1

,

(4.35)

43

〈O†i H〉AA = 〈HOi〉∗AA , (4.36)

〈O†i HOi〉AA =

∑

κ′2κ2κ

′1κ1

〈i|κ′1〉〈κ1|i〉〈i|κ′2〉〈κ2|i〉 η∗κ′2κ2κ

′1κ1fκ′

2κ2κ′1κ1

∑

κ′4κ4κ

′3κ3

〈κ′3|i〉〈i|κ3〉〈κ′4|i〉〈i|κ4〉

×(∆Eκ′

2κ2κ′1κ1δκ1κ3δκ′

1κ′3δκ2κ4δκ′

2κ′4+ Uκ′

2κ2κ′1κ1κ

′4κ4κ

′3κ3

)ηκ′

4κ4κ′3κ3

,

(4.37)

Uκ′2κ2κ

′1κ1κ

′4κ4κ

′3κ3

= U∑

j

[〈j|κ1〉〈κ3|j〉f(ǫκ3↑)δκ′1κ

′3− 〈κ′1|j〉〈j|κ′3〉(1− f(ǫκ′

3↑))δκ1κ3 ]

×[〈j|κ2〉〈κ4|j〉f(ǫκ4↓)δκ′2κ

′4− 〈κ′2|j〉〈j|κ′4〉(1− f(ǫκ′

4↓))δκ2κ4 ] , (4.38)

〈O†i Oi〉AA =

∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2 |ηκ′2κ2κ

′1κ1

|2 fκ′2κ2κ

′1κ1

. (4.39)

Herefκ′2κ2κ

′1κ1

is a Fermi factor of two-particle excitations which is defined by fκ′2κ2κ

′1κ1

= f(ǫκ1↑)(1 − f(ǫκ′

1↑))f(ǫκ2↓)(1 − f(ǫκ′

2↓)), f(ǫ) is the Fermi distribution function at zero temperature,

ǫκσ = ǫκσ − µ, andǫκσ is the one-electron energy eigenvalue for the AA Hamiltonian. Moreover,∆Eκ′

2κ2κ′1κ1

= ǫκ′2↓− ǫκ2↓ + ǫκ′

1↑− ǫκ1↑ is a two-particle excitation energy.

In the correlation energy, the expressions (4.35) and (4.38) contain nonlocal terms via summa-tion overj (i.e.,

∑j). We thus make additional SSA that we only take into account the local term

(j = i), so that〈HOi〉AA(= 〈O†i H〉∗AA) and〈O†

i HOi〉AA reduce as follows. The procedures arethe same as in the MLA-HF presented in Chap. 3.

〈HOi〉AA = U∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2 ηκ′2κ2κ

′1κ1

fκ′2κ2κ

′1κ1

, (4.40)

〈O†i HOi〉AA =

∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2

×η∗κ′2κ2κ

′1κ1

fκ′2κ2κ

′1κ1

[∆Eκ′

2κ2κ′1κ1ηκ′

2κ2κ′1κ1

+U{∑

κ3κ4

|〈κ3|i〉|2|〈κ4|i〉|2f(ǫκ3↑)f(ǫκ4↓) ηκ′2κ4κ

′1κ3

−∑

κ′3κ4

|〈κ′3|i〉|2|〈κ4|i〉|2(1− f(ǫκ′3↑))f(ǫκ4↓) ηκ′

2κ4κ′3κ1

−∑

κ3κ′4

|〈κ3|i〉|2|〈κ′4|i〉|2f(ǫκ3↑)(1− f(ǫκ′4↓)) ηκ′

4κ2κ′1κ3

+∑

κ′3κ

′4

|〈κ′3|i〉|2|〈κ′4|i〉|2(1− f(ǫκ′3↑))(1− f(ǫκ′

4↓)) ηκ′

4κ2κ′3κ1

}]. (4.41)

The variational parameters{ηκ′2κ2κ

′1κ1

} are obtained by minimizing the correlation energyǫc,i.e., Eq. (4.34) with Eqs. (4.39), (4.40), and (4.41). The self-consistent equations for{ηκ′

2κ2κ′1κ1

}

44

in the SSA are given as follows.

(∆Eκ′2κ2κ

′1κ1

− ǫc)ηκ′2κ2κ

′1κ1

+ U

[∑

κ3κ4

|〈κ3|i〉|2|〈κ4|i〉|2f(ǫκ3↑)f(ǫκ4↓) ηκ′2κ4κ

′1κ3

−∑

κ′3κ4

|〈κ′3|i〉|2|〈κ4|i〉|2(1− f(ǫκ′3↑))f(ǫκ4↓) ηκ′

2κ4κ′3κ1

−∑

κ3κ′4

|〈κ3|i〉|2|〈κ′4|i〉|2f(ǫκ3↑)(1− f(ǫκ′4↓)) ηκ′

4κ2κ′1κ3

+∑

κ′3κ

′4

|〈κ′3|i〉|2|〈κ′4|i〉|2(1− f(ǫκ′3↑))(1− f(ǫκ′

4↓)) ηκ′

4κ2κ′3κ1

]= U . (4.42)

To solve the equation approximately, we make use of the same idea as in the MLA-HF (see Sec.3.3.1); we obtain the following solution which interpolatebetween the weak Coulomb interactionlimit and the atomic limit.

ηκ′2κ2κ

′1κ1

(η, ǫc) =Uη

∆Eκ′2κ2κ

′1κ1

− ǫc. (4.43)

Hereη = 1− η(1− 2〈ni↑〉0)(1− 2〈ni↓〉0).The ground-state correlation energy is obtained by substituting the variational parameters (4.43)

into Eq. (4.34).

〈HOi〉AA = 〈O†i H〉∗AA

= U2 η

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc, (4.44)

〈O†i HOi〉AA = 〈O†

i H0Oi〉AA + U〈O†iOiOi〉AA , (4.45)

〈O†i H0Oi〉AA = U2η2

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

(ǫ4 − ǫ3 + ǫ2 − ǫ1)−1(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2,

(4.46)

〈O†iOiOi〉AA = U2η2

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc

×[ ∫

dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ5)f(ǫ6)

ǫ4 − ǫ6 + ǫ2 − ǫ5 − ǫc−∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ5)(1− f(ǫ6))

ǫ6 − ǫ3 + ǫ2 − ǫ5 − ǫc

−∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)(1− f(ǫ5))f(ǫ6)

ǫ4 − ǫ6 + ǫ5 − ǫ1 − ǫc+

∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)(1− f(ǫ5))(1− f(ǫ6))

ǫ6 − ǫ3 + ǫ5 − ǫ1 − ǫc

],

(4.47)

45

〈O†i Oi〉AA = U2η2

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2.

(4.48)

Hereρσ(ǫ) is the local DOS for the one-electron energy eigenvalues of the AA Hamiltonian matrix(4.7).

We determineη in Eqs. (4.44)-(4.48) variationally using the following variational principle.

E ≤ 〈H〉({η∗κ′2κ2κ

′1κ1

}) ≤ 〈H〉({ηκ′2κ2κ

′1κ1

(η, ǫc)}) . (4.49)

Here{η∗κ′2κ2κ

′1κ1

} are the exact solution for the Eq. (4.42). Minimizing the correlation energy withrespect toη (i.e., δǫc = 0), we obtain

η =1

1 +UC

D

. (4.50)

Here

C =∑

κ1κ′1κ2κ

′2

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2fκ′

2κ2κ′1κ1

(∆Eκ′2κ2κ

′1κ1

− ǫc)

×{∑

κ3κ4

|〈κ3|i〉|2|〈κ4|i〉|2f(ǫκ3↑)f(ǫκ4↓)

(∆Eκ′2κ4κ

′1κ3

− ǫc)

−∑

κ′3κ4

|〈κ′3|i〉|2|〈κ4|i〉|2[1− f(ǫκ′

3↑)]f(ǫκ4↓)

(∆Eκ′2κ4κ

′3κ1

− ǫc)

−∑

κ3κ′4

|〈κ3|i〉|2|〈κ′4|i〉|2f(ǫκ3↑)[1− f(ǫκ′

4↓)]

(∆Eκ′4κ2κ

′1κ3

− ǫc)

+∑

κ′3κ

′4

|〈κ′3|i〉|2|〈κ′4|i〉|2[1− f(ǫκ′

3↑)][1− f(ǫκ′

4↓)]

(∆Eκ′4κ2κ

′3κ1

− ǫc)

}, (4.51)

and

D =∑

κ1κ′1κ2κ

′2

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2fκ′

2κ2κ′1κ1

(∆Eκ′2κ2κ

′1κ1

− ǫc). (4.52)

46

In the energy representation, the coefficientsC andD are expressed as follows:

C =

∫ [ 6∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

×f(ǫ5)f(ǫ6)(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)−1(ǫ4 − ǫ6 + ǫ2 − ǫ5 − ǫc)

−1

−∫ [ 6∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

×(1− f(ǫ5))f(ǫ6)(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)−1(ǫ4 − ǫ6 + ǫ5 − ǫ1 − ǫc)

−1

−∫ [ 6∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

×f(ǫ5)(1− f(ǫ6))(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)−1(ǫ6 − ǫ3 + ǫ2 − ǫ5 − ǫc)

−1

+

∫ [ 6∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

×(1− f(ǫ5))(1− f(ǫ6))(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)−1(ǫ6 − ǫ3 + ǫ5 − ǫ1 − ǫc)

−1, (4.53)

and

D =

∫[ 4∏n=1

dǫn]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc. (4.54)

The expression ofC, i.e., Eq. (4.53) consists of 6-fold multiple integrals. Therefore we reducethese integrals to the 2-folds using the Laplace transform (3.52). The Laplace transformation ofCandD are given as follows.

C = −∫ ∞

0

dtdt′eiǫc(t+t′)[a↑(−t)b↑(t+ t′)a↓(−t)b↓(t+ t′)a↑(−t′)a↓(−t′)

−a↑(−t− t′)b↑(t)a↓(−t)b↓(t+ t′)b↑(t′)a↓(−t′)

−a↑(−t)b↑(t+ t′)a↓(−t− t′)b↓(t)a↑(−t′)b↓(t′)+a↑(−t− t′)b↑(t)a↓(−t− t′)b↓(t)b↑(t

′)b↓(t′)]. (4.55)

D = −i∫ ∞

0

dteiǫcta↑(−t)b↑(t)a↓(−t)b↓(t). (4.56)

Here

aσ(t) =

∫dǫρσ(ǫ)f(ǫ) e

−iǫt , (4.57)

bσ(t) =

∫dǫρσ(ǫ)(1− f(ǫ)) e−iǫt . (4.58)

The total energy per atom is obtained by taking the configurational average as follows (see Eq.(4.32) ).

〈H〉 = 〈H〉AA + ǫc . (4.59)

47

The AA energy〈H〉AA has been given by Eq.(4.4). The correlation energy can be obtained asfollows in the SSA.

ǫc =∑

α

Pα ǫcα . (4.60)

HerePα (α = 0, 1 ↑, 1 ↓, 2) are the probability for the on-site configurationα. ǫcα denotes thecorrelation energy for a given configurationα.

ǫcα =[−〈O†

i H〉AA − 〈HOi〉AA + 〈O†i HOi〉AA

1 + 〈O†i Oi〉AA

]α. (4.61)

The quantities〈HOi〉AA, 〈O†i HOi〉AA, and〈O†

i Oi〉AA are given by Eqs.(4.44), (4.45), and(4.48)in which the local densities of states have been given by the single-site ones,i.e., Eq. (4.14). TheLaplace transform of these expression are as follows:

〈O†i Oi〉AAα = −U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)[a↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b↓(t+ t′)

]α, (4.62)

〈HOi〉AAα = 〈O†i H〉∗AAα = iU2ηα

∫∞

0dt eiǫcαt

[a↑(−t)a↓(−t)b↑(t)b↓(t)

]α, (4.63)

〈O†i H0Oi〉AAα = −U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)[a↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b1↓(t+ t′)

−a↑(−t− t′)b↑(t+ t′)a1↓(−t− t′)b↓(t+ t′)

+a↑(−t− t′)b1↑(t+ t′)a↓(−t− t′)b↓(t+ t′)

−a1↑(−t− t′)b↑(t+ t′)a↓(−t− t′)b↓(t+ t′)]α, (4.64)

〈O†iOiOi〉AAα = −U2η2α

∫ ∞

0

dtdt′eiǫc(t+t′)[a↑(−t)b↑(t+ t′)a↓(−t)b↓(t+ t′)a↑(−t′)a↓(−t′)

−a↑(−t)b↑(t+ t′)a↓(−t− t′)b↓(t)a↑(−t′)b↓(t′)−a↑(−t− t′)b↑(t)a↓(−t)b↓(t+ t′)b↑(t

′)a↓(−t′)+a↑(−t− t′)b↑(t)a↓(−t− t′)b↓(t)b↑(t

′)b↓(t′)]α. (4.65)

Here,

ηα =1

1 +UCα

Dα

. (4.66)

Cα = −∫ ∞

0

dtdt′eiǫcα(t+t′)[a↑(−t)b↑(t+ t′)a↓(−t)b↓(t+ t′)a↑(−t′)a↓(−t′)

−a↑(−t− t′)b↑(t)a↓(−t)b↓(t+ t′)b↑(t′)a↓(−t′)

−a↑(−t)b↑(t+ t′)a↓(−t− t′)b↓(t)a↑(−t′)b↓(t′)+a↑(−t− t′)b↑(t)a↓(−t− t′)b↓(t)b↑(t

′)b↓(t′)]α. (4.67)

48

Table 4.1: The atomic levelǫiσ for the up-spin (↑) and the down-spin (↓) state in various on-siteconfigurations.

Configuration ǫi↑ ǫi↓

00 ǫ0 ǫ010 ǫ0 ǫ0 + U01 ǫ0 + U ǫ011 ǫ0 + U ǫ0 + U

Dα = −i∫ ∞

0

dteiǫcαt[a↑(−t)b↑(t)a↓(−t)b↓(t)

]α. (4.68)

As we have mentioned before, for the empty sitei.e., α = 00, the atomic levelǫiσ in Eq. (4.8)for the up-spin (↑) and down-spin (↓) state becomesǫ0. We summarize the results for all otherconfigurations in Table 4.1. Therefore we can expressa↑ anda↓ by aǫ0 andaǫ0+U . Therefore,

aβσ(t) =

∫dǫρβσ(ǫ)f(ǫ) e

−iǫt , (4.69)

bβσ(t) =

∫dǫρβσ(ǫ)(1− f(ǫ)) e−iǫt , (4.70)

a1βσ(t) =

∫dǫρβσ(ǫ)f(ǫ) ǫ e

−iǫt , (4.71)

b1βσ(t) =

∫dǫρβσ(ǫ)(1− f(ǫ)) ǫ e−iǫt . (4.72)

Hereβ = ǫ0 or ǫ0+U . The densities of statesρǫ0σ(ǫ) andρǫ0+Uσ(ǫ) have been given by Eqs. (4.17)and (4.20), respectively.

The double occupation number〈ni↑ni↓〉 is obtained from∂〈H〉/∂Ui. Making use of the single-site energy (3.24) and the Feynman-Hellmann theorem [32], and taking the configurational aver-age, we obtain the following expression.

〈ni↑ni↓〉 = 〈ni↑〉AA〈ni↓〉AA + 〈ni↑ni↓〉c . (4.73)

Here the AA contribution of the double occupancy〈ni↑〉AA〈ni↓〉AA has been given by Eq.(4.15).The second term is the correlation contribution given as follows.

〈ni↑ni↓〉c =∑

α

Pα〈ni↑ni↓〉cα , (4.74)

〈ni↑ni↓〉cα =

[−〈O†

iOi〉AA − 〈OiOi〉AA + 〈O†iOiOi〉AA +

∑σ〈ni−σ〉AA〈O†

i niσOi〉AA

1 + 〈O†i Oi〉AA

]

α

, (4.75)

49

〈O†iOi〉AA + 〈OiOi〉AA = 2Uη

∫ [ 4∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)

×f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc, (4.76)

〈O†i niσOi〉AA = U2η2

∫ [ 5∏

n=1

dǫn

]ρ−σ(ǫ1)ρ−σ(ǫ2)ρσ(ǫ3)ρσ(ǫ4)ρσ(ǫ5)

×f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc

×[

1− f(ǫ5)

ǫ5 − ǫ3 + ǫ2 − ǫ1 − ǫc− f(ǫ5)

ǫ4 − ǫ5 + ǫ2 − ǫ1 − ǫc

]. (4.77)

The quantities〈O†iOiOi〉AA and〈O†

i Oi〉AA have been given by Eqs.(4.47) and(4.48), respectively.The Laplace transform of the expressions〈O†

iOi〉AAα+ 〈OiOi〉AAα and〈O†i niσOi〉AAα are written,

respectively, as follows.

〈O†iOi〉AAα + 〈OiOi〉AAα = 2iUηα

∫ ∞

0

dt eiǫcαt[a↑(−t)b↑(t)a↓(−t)b↓(t)

]α, (4.78)

〈O†i niσOi〉AAα = −U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)[a−σ(−t− t′)b−σ(t+ t′)aσ(−t− t′)bσ(t)bσ(t

′)

−a−σ(−t− t′)b−σ(t+ t′)aσ(−t)bσ(t+ t′)aσ(t′)]α. (4.79)

The quantities〈O†i Oi〉AAα and〈O†

iOiOi〉AAα have been given by Eqs. (4.62) and (4.65), respec-tively.

Similarly, the momentum distribution〈nkσ〉 is obtained from∂〈H〉/∂ǫk as follows.

〈nkσ〉 = 〈nkσ〉AA + 〈nkσ〉c . (4.80)

The AA contribution of the momentum distribution〈nkσ〉AA has been given by Eq.(4.26). Thecorrelation contribution〈nkσ〉c is expressed as follows.

〈nkσ〉c =∑

α

Pα〈nkσ〉cα , (4.81)

〈nkσ〉cα =

[N〈OinkσOi〉AA

1 + 〈O†i Oi〉AA

]

α

. (4.82)

Herenkσ = nkσ − 〈nkσ〉AA, the numerator on the r.h.s. is given by

N〈O†i nkσOi〉AA = U2η2

∫ [ 4∏

n=1

dǫn

]ρσ(ǫ1)ρ−σ(ǫ2)ρ−σ(ǫ3)ρkσ(ǫ4)f(ǫ2)(1− f(ǫ3))

×{

f(ǫ1)(1− f(ǫ4))

(ǫ3 − ǫ2 + ǫ4 − ǫ1 − ǫc)2− (1− f(ǫ1))f(ǫ4)

(ǫ3 − ǫ2 + ǫ1 − ǫ4 − ǫc)2

}. (4.83)

50

The DOS in the momentum representationρkσ(ǫ) has been given by Eq. (4.27) in the SSA. Thecorrelation contribution quantity〈O†

i Oi〉AA is given by Eq.(4.48). The Laplace transform of thequantityN〈O†

i nkσOi〉0α ≡ [N〈O†i nkσOi〉AA]α is given as follows.

N〈O†i nkσOi〉AAα = U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)a−σ(−t− t′)b−σ(t+ t′)

×[bσ(t+ t′)akσ(−t− t′)− aσ(−t− t′)bkσ(t+ t′)

]α. (4.84)

Here

akσ(t) =

∫dǫ ρkσ(ǫ)f(ǫ) e

−iǫt , (4.85)

bkσ(t) =

∫dǫ ρkσ(ǫ)

[1− f(ǫ)

]e−iǫt . (4.86)

Therefore, we calculate the correlation energyǫcα (Eq. (4.61)) self-consistently with use ofEqs. (4.44), (4.45), (4.48), and (4.50), and calculate the average correlation energyǫc (Eq. (4.60))as well as the average AA energy〈H〉AA (Eq. (4.4)). Then we obtain the ground-state energy〈H〉(Eq. (4.59)) in the MLA-AA.

4.3 Numerical results: half-filled band Hubbard model

To examine the validity of the MLA-AA, we have performed the numerical calculations forthe half-filled band Hubbard model with nearest neighbor transfer integral on the hypercubiclattice in infinite dimensions, where the SSA works best [39,42]. We assumed here the non-magnetic case. In this case, the density of states for non-interacting system is given byρ(ǫ) =(1/

√π) exp(−ǫ2) [42]. The energy unit is chosen to be

∫dǫρ(ǫ)ǫ2 = 1/2. The characteristic band

widthW is given byW = 2 in this unit.Figure 4.2 shows the calculated results of the ground-stateenergy vs Coulomb interaction en-

ergy curves. In the weak Coulomb interaction regime (U/W . 1), the total energy of the MLA-HFis lower than the GW. On the other hand, we observe that the MLA-AA gives lower energy in com-parison with the GW in the strong Coulomb interaction regime (U/W > 1). We obtain the criticalCoulomb interactionUc2 = 3.40 at which the effective mass diverges. But beforeU approachesUc2 we find that the AA state showing the insulating state is stabilized, and the metal-insulatortransition occurs at the critical Coulomb interactionUc = 3.26. The transition is of the first orderin the present approach, and is consistent with the result ofNRG method, although the calculatedUc2 is somewhat smaller than that obtained by NRG (i.e., Uc2 = 4.1 ) [34]. The MLA-HF leads tothe total energy lower than that of the GW up to the critical Coulomb interactionUc and the MLA-AA gives the same behavior in the rangeU > Uc. More important is that the MLA scheme giveslower energy than the GW for overall Coulomb interaction regime and therefore can overcomesthe limitation of the GW.

We present in Fig. 4.3 the double occupation number〈n↑n↓〉 as a function of Coulomb inter-action energyU at half-filling. It decreases from1/4 with increasing Coulomb interaction, so asto reduce the loss of Coulomb energyU . The MLA-HF state reduces the double occupancy morethan the GW in the weakly correlated region, and jumps to the MLA-AA state atUc. In the stronglycorrelated regime, the MLA-AA gives finite value of double occupancy, while the GW gives the

51

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 2 4 6 8 10

Ene

rgy

U

n=1.0

MLA-AAMLA-HF

GW

Figure 4.2: The energy vs Coulomb interaction energyU curves in the MLA-AA (solid curve),MLA-HF (dashed curve), and the GW (thin solid curve) for the electron numbern = 1.0.

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

⟨n↑n

↓⟩

U

n=1.0

MLA-HFMLA-AA

GW

Figure 4.3: The double occupation number〈n↑n↓〉 vs Coulomb interaction energyU curve for theelectron numbern = 1.0 in the MLA-HF (solid-dashed curve), the MLA-AA (dashed-solid curve),and the GW (thin solid curve). The arrow shows a jump from the metallic state to the insulator atUc = 3.26.

52

0

0.2

0.4

0.6

0.8

1

1.2

-1 -0.5 0 0.5 1

⟨nkσ

⟩

εkσ

12

3

45

6

n=1.0

MLAGW

Figure 4.4: The momentum distribution as a function of energy ǫkσ for various Coulomb interactionenergy parametersU = 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 at half-filling. The MLA: solid curve, the GW:dashed curve, and the HF: thin solid curve.

Brinkman-Rice atom state. This verifies an improvement over the GW in the strongly correlatedregime. The double occupancy for the MLA scheme atUc is 0.032, and is consistent with the resultof the Quantum Monte Carlo (≈ 0.024) [43], though the latter uses the semi-elliptical density ofstates.

The momentum distribution for the MLA shows a clear momentum-dependence as a functionof the HF one electron energyǫkσ as shown in Fig. 4.4. It decreases monotonically with increasingǫkσ and shows a jump at the Fermi energy in the metallic state. Thedistribution for the GWon the other hand, is constant below and above the Fermi level[7, 8, 9]. The jump decreaseswith increasingU , and disappears beyondUc, indicating the insulating state. The curve becomesflatter with further increase ofU . Note that it leads to a Fermi liquid state for the weak Coulombinteraction regime and non Fermi liquid state in the strong Coulomb interaction regime.

In summary, we have shown that the MLA can describe the strongCoulomb interaction regimesby changing the starting wavefunction from the HartreeFock(HF) type to an alloy-analogy (AA)type wavefunction. Numerical results based on the half-filled band Hubbard model on the hyper-cubic lattice in infinite dimensions show that the new wavefunction yields the ground-state energylower than the Gutzwiller wavefunction (GW) in the strong Coulomb interaction regime. Cal-culated double occupation number is smaller than the resultof the GW in the metallic regime,and is finite in the insulator regime. Furthermore, the momentum distribution shows a distinctmomentum-dependence in both the metallic and insulator regions, which are qualitatively differ-ent from those of the GW.

53

Chapter 5

Momentum Dependent Local-AnsatzApproach with Hybrid Wavefunction

In this chapter, we introduce a hybrid (HB) wavefunction whose potential can flexibly changefrom the Hartree-Fock type to the alloy-analogy type by varying a weighting factor from zero toone. On the basis of the HB function we propose a new variational theory of momentum-dependentlocal-ansatz (MLA). The theory [28] describes reasonably electron correlations from the weak tothe strong Coulomb interaction regime.

5.1 Hybrid wavefunction

We reconsider in this section the single-band Hubbard model[10, 11, 12].

H =∑

iσ

(ǫ0 − µ)niσ +∑

ijσ

tij a†iσajσ + U

∑

i

ni↑ni↓ . (5.1)

Hereǫ0 (µ) is the atomic level (chemical potential),tij is the transfer integral between sitesi andj. U is the intra-atomic Coulomb energy parameter.a†iσ (aiσ) denotes the creation (annihilation)operator for an electron on sitei with spinσ, andniσ = a†iσaiσ is the electron density operator onsitei for spinσ.

In the HF approximation (see Eq. (2.22)), we neglected the fluctuationsδni↑δni↓ and replacethe many-body Hamiltonian (5.1) with an effective HamiltonianHHF showing independent-particlemotion.

HHF =∑

iσ

(ǫ0 − µ+ U〈ni−σ〉0)niσ +∑

ijσ

tij a†iσajσ − U

∑

i

〈ni↑〉0〈ni↓〉0 . (5.2)

Here〈∼〉0 denotes the HF average〈φ0|(∼)|φ0〉, and〈niσ〉0 is the average electron number on sitei with spinσ. |φ0〉 denotes the ground-state HF wavefunction for the HamiltonianHHF.

In the AA approximation (see Eq. (4.1)) on the other hand, we considered the strong Coulombinteraction regime, where electrons with spinσ move slowly from site to site due to electron corre-lations. Instead of the HF average potentialU〈ni−σ〉0, electrons should feel there a potentialU or 0,depending on the occupation of an electron with opposite spin on the same site. Hubbard regardedthis system as an alloy with different random potentialsǫ0 + U andǫ0. The AA Hamiltonian was

54

defined as follows (see Eq. (4.1)).

HAA =∑

iσ

(ǫ0 − µ+ Uni−σ)niσ +∑

ijσ

tija†iσajσ − U

∑

i

(ni↑〈ni↓〉AA + ni↓〈ni↑〉AA)

+U∑

i

〈ni↑〉AA〈ni↓〉AA. (5.3)

Here〈∼〉AA denotes the AA average〈φAA|(∼)|φAA〉 with respect to the ground-state wavefunction|φAA〉 of the AA HamiltonianHAA. Since the electrons with opposite spin are treated to be staticin the AA approximation, related operators{ni−σ} are regarded as a random staticC numberni−σ

(0 or 1). Each configuration{niσ} is considered as a snapshot in time development.The HF Hamiltonian works best in the weakly correlated regime, while the AA Hamiltonian

works better in the strongly correlated regime. In order to obtain better wavefunction for anyinteraction strengthU , we introduce the following HB Hamiltonian which is a linearcombinationof the HF and the AA Hamiltonians.

HHB =∑

iσ

(ǫ0 − µ+ U〈ni−σ〉HB + Uni−σ)niσ +∑

ijσ

tij a†iσajσ

−(U − U)∑

i

〈ni↑〉HB〈ni↓〉HB − U∑

i

(ni↑〈ni↓〉HB + ni↓〈ni↑〉HB) . (5.4)

Here〈∼〉HB denotes the HB average〈φHB|(∼)|φHB〉 with respect to the ground-state|φHB〉 of theHB Hamiltonian,U = (1−w)U andU = wU . We introduced a variational parameterw. Note thatHHB (= (1 − w)HHF + wHAA) reduces to the HF Hamiltonian whenw = 0, whileHHB reducesto the AA whenw = 1.0.

The ground-state energyE satisfies the following inequality for a normalized wavefunction|φHB〉.

E ≤ 〈φHB|H|φHB〉 = 〈HHB〉HB . (5.5)

The HB ground-state energy per atom is obtained by taking theconfigurational average.

〈H〉HB = nµ+ 2

∫ 0

−∞

ǫ ρiσ(ǫ) dǫ− (U − U)〈ni↑〉HB〈ni↓〉HB − U(ni↑〈ni↓〉HB + ni↓〈ni↑〉HB) .(5.6)

Here we assumed the system with one atom per unit cell.〈H〉HB denotes the HB average〈φHB|H|φHB〉.The upper bar denotes the configurational average andn is the electron number per atom.ρiσ(ǫ) isthe local density of states (DOS) and is obtained from the one-electron Green function.

ρiσ(ǫ) = − 1

πImGiiσ(z ) . (5.7)

The Green functionGiiσ(z) is defined here by

Giiσ(z) = [(z −Hσ)−1]ij . (5.8)

Note thatz = ǫ + iδ, δ being the infinitesimal positive number.(Hσ)ij is the one-electron Hamil-tonian matrix for the HB Hamiltonian (5.4), which is defined by

(Hσ)ij = (ǫ0 − µ+ U〈ni−σ〉HB + Uni−σ)δij + tij(1− δij) . (5.9)

55

The average electron number〈niσ〉HB with respect to the HB Hamiltonian (5.4) is given as

〈niσ〉HB =

∫f(ǫ)ρiσ(ǫ) dǫ , (5.10)

f(ǫ) being the Fermi distribution function.Since the HB Hamiltonian contains a random potential, we obtain the local DOS making use of

the coherent potential approximation (CPA) [40, 41]. In the CPA, we replace the random potentialsat the surrounding sites with a coherent potentialsΣσ(z). The on-site impurity Green functionGiiσ(z) is then obtained as follows.

Giiσ(z) =1

Fσ(z)−1 − ǫ0 + µ− U〈ni−σ〉HB − Uni−σ + Σσ(z). (5.11)

HereFσ(z) is the on-site Green function for the coherent system in which all the random potentialshave been replaced by the coherent ones, and has been given byEq. (4.11):

Fσ(z) =

∫ρ(ǫ) dǫ

z − Σσ(z)− ǫ. (5.12)

Hereρ(ǫ) is the DOS per site for the noninteracting system.The coherent potentialΣσ(z) is determined from a self-consistent condition.

G00σ(z) = Fσ(z) . (5.13)

The configurational average of the impurity Green function is now given as

G00σ(z) =∑

α

PαGα00σ(z) . (5.14)

Hereα = 00, 10, 01, 11 denotes the on-site electron configuration (n0↑,n0↓). Alternative notationν = 0 (empty on a site),1 ↑ (occupied by an electron with spin↑ ), 1 ↓ (occupied by an electronwith spin↓ ) and2 (occupied by 2 electrons) is also useful. In this case, the probabilityPα for theconfigurationα is expressed asP0, P1↑, P1↓, andP2.

The impurity Green functions in Eq. (5.14) for each configuration are given as follows.

G0000σ(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n−σ〉00 + Σσ(z), (5.15)

G1000↑(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n↓〉10 + Σσ(z), (5.16)

G1000↓(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n↑〉10 − U + Σσ(z), (5.17)

G0100↑(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n↓〉01 − U + Σσ(z), (5.18)

56

G0100↓(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n↑〉01 + Σσ(z), (5.19)

G1100σ(z) =

1

Fσ(z)−1 − ǫ0 + µ− U〈n−σ〉11 − U + Σσ(z). (5.20)

Here the electron number in the denominator is given by

〈nσ〉α =

∫f(ǫ)ρασ(ǫ) dǫ , (5.21)

ρασ(ǫ) = − 1

πImGα

00σ(z) . (5.22)

Furthermore, the average DOS in the second term at the right hand side (r.h.s.) of Eq. (5.6) is givenby

ρiσ(ǫ) = − 1

πImG00σ(z) =

∑

α

Pα ρασ(ǫ) . (5.23)

It should be noted thatP0+P1↑+P1↓+P2 = 1, and the probability of finding an electron withspin↑ (↓) on a site is given byP↑(↓) = P1↑(1↓)+P2. Three statistical probabilitiesP0, P1↑, andP1↓

are therefore given by the probabilityP2.The expression ofP2 is derived as follows. We first make the following approximations for the

AA and the HF Hamiltonians, respectively.

n↑n↓ ≈ n↑n↓ + n↓n↑ − n↑n↓ (AA) , (5.24)

n↑n↓ ≈ n↑〈n↓〉+ n↓〈n↑〉 − 〈n↑〉〈n↓〉 (HF) . (5.25)

In the HB scheme, we approximate the averages〈∼〉 at the r.h.s. of the above expressions withthose of the HB Hamiltonian (5.4), and superpose Eqs. (5.24)and (5.25) with the weightw and (1−w), respectively. Taking the the quantum mechanical averageof the superposed double occupationnumber as well as the configurational average, we obtain the probability of the double occupationP2 (= 〈n↑n↓〉) in the HB approximation as follows.

P2 = w (n↑〈n↓〉+ n↓〈n↑〉 − n↑n↓) + (1− w) 〈n↓〉〈n↑〉= w n↑〈n↓〉+ (1− w) 〈n↑〉〈n↓〉+ w n↓〈n↑〉+ (1− w) 〈n↑〉〈n↓〉

−[w n↑n↓ + (1− w) 〈n↑〉〈n↓〉] . (5.26)

The last term at the r.h.s. of Eq. (5.26) may be regarded as theprobabilityP2. Therefore, we obtain

P2 ≈ 1

2

[(w n↑ + (1− w)〈n↑〉)〈n↓〉+ (w n↓ + (1− w)〈n↓〉)〈n↑〉

], (5.27)

i.e.,

P2 =1

2w (n↑〈n↓〉+ n↓〈n↑〉) + (1− w) 〈n↑〉〈n↓〉 . (5.28)

57

In the single-site approximation, the last term at the r.h.s. of Eq. (5.28) is expressed as follows.

〈n↑〉〈n↓〉 =∑

α

Pα〈n↑〉α〈n↓〉α

= P0〈n↑〉00〈n↓〉00 + P1↑〈n↑〉10〈n↓〉10 + P1↓〈n↑〉01〈n↓〉01 + P2〈n↑〉11〈n↓〉11 .(5.29)

In the non-magnetic case, we have

〈n↑〉〈n↓〉 = 〈n↑〉200 + (P↑ + P↓)(〈n↑〉10〈n↑〉01 − 〈n↑〉200)+P2(〈n↑〉200 − 2〈n↑〉10〈n↑〉01 + 〈n↑〉211) . (5.30)

Similarly,

n↑〈n↓〉+ n↓〈n↑〉 = (P↑ + P↓)〈n↑〉01 + 2P2 (〈n↑〉11 − 〈ni↑〉01) . (5.31)

Substituting Eqs.(5.30) and(5.31) into Eq.(5.28), we obtain the final expression ofP2.

P2 =(1− w)〈n↑〉200 + (P↑ + P↓){1/2w 〈n↑〉01 + (1− w)〈n↑〉10〈n↑〉01 − 〈n↑〉200}

1− w(〈n↑〉11 − 〈n↑〉01)− (1− w)(〈n↑〉200 − 2〈n↑〉10〈n↑〉01 + 〈n↑〉211). (5.32)

The double occupation numbers at the r.h.s. of Eq. (5.6) are obtained in the SSA as follows.

〈ni↑ni↓〉HB = 〈ni↑〉HB〈ni↓〉HB =∑

α

Pα〈n↑〉α〈n↓〉α , (5.33)

ni↑〈ni↓〉HB + ni↓〈ni↑〉HB = (P↑ + P↓)〈n↑〉01 + 2P2 (〈n↑〉11 − 〈n↑〉01) . (5.34)

The momentum distribution in the HB scheme is given by

〈nkσ〉HB =

∫f(ǫ)ρkσ(ǫ) dǫ , (5.35)

ρkσ(ǫ) = − 1

πImFkσ , (5.36)

Fkσ =1

z − Σσ(z)− ǫk. (5.37)

Hereǫk is the eigenvalue oftij with momentumk.In summary, to determine the ground-state energy of the HB wavefunction 〈H〉HB in Eq.

(5.6), we first determine the coherent potentialΣσ(z) self-consistently by making use of Eqs.(5.12),(5.13), and (5.14). In the self-consistent loop theimpurity Green function for each con-figuration and corresponding electron number are determined by Eqs. (5.15)-(5.20) and (5.22),respectively. The probabilityP2 in the self-consistent loop is also determined by Eq. (5.32).

58

5.2 Local-ansatz+ hybrid wavefunction approach with mo-mentum dependent variational parameters

The momentum dependent local-ansatz (MLA) wavefunction isbased on the local-ansatz (LA)proposed by Stollhoff and Fulde:|ΨLA〉 =

[∏i(1−ηLAOi)

]|φ0〉 [13, 14, 15]. HereOi = δni↑δni↓

are the residual interaction, the amplitudeηLA is determined variationally. The operators{Oi}expand the Hilbert space to describe the weak Coulomb interaction regime. The LA however doesnot yield the exact result in the weak interaction limit. TheMLA wavefunction (Eq. 3.9) whichdescribes exactly the weak Coulomb interaction limit was constructed in Chap. 3 as follows [24,25].

|ΨMLA〉 =∏

i

(1− Oi)|φ0〉 , (5.38)

Oi =∑

k1k′1k2k

′2

〈k′1|i〉〈i|k1〉〈k′2|i〉〈i|k2〉 ηk′2k2k′1k1δ(a†k′2↓ak2↓)δ(a

†k′1↑ak1↑) . (5.39)

Here〈i|k〉 = exp(−ik ·Ri)/√N is an overlap integral between the localized orbital and theBloch

state with momentumk, Ri denotes atomic position, andN is the number of sites.ηk′2k2k′1k1 is a

momentum dependent variational parameter.a†kσ (akσ) denotes a creation (annihilation) operatorfor an electron with momentumk and spinσ, andδ(a†k′σakσ) = a†k′σakσ − 〈a†k′σakσ〉0.

In this chapter, we generalize the wavefunction (5.38) to besuitable in both the strong and theweak Coulomb interaction regimes. The idea is to adopt the HB ground-state wavefunction|φHB〉for the HamiltonianHHB (5.4) as a starting wavefunction, and to apply a new correlator

∏i(1−Oi)

as follows.|ΨMLA−HB〉 =

∏

i

(1− Oi)|φHB〉. (5.40)

Note that the local operators{Oi} have been modified as follows.

Oi =∑

κ′2κ2κ

′1κ1

〈κ′1|i〉〈i|κ1〉〈κ′2|i〉〈i|κ2〉 ηκ′2κ2κ

′1κ1δ(a†

κ′2↓aκ2↓)δ(a

†κ′1↑aκ1↑) . (5.41)

Hereηκ′2κ2κ

′1κ1

is a variational parameter,a†κσ andaκσ are the creation and annihilation operators

which diagonalize the HamiltonianHHB (5.4), andδ(a†κ′σaκσ) = a†κ′σaκσ − 〈a†κ′σaκσ〉HB. It shouldbe noted that the MLA-HB wavefunction (5.40) reduces to the MLA-HF with the uniform po-tentialU〈ni−σ〉0 when the variational parameterw = 0, and reduces to the MLA-AA with therandom potentialUni−σ whenw = 1. The MLA-HB wavefunction interpolates between the twowavefunctions.

The ground-state energyE satisfies the following inequality for any wavefunction|Ψ〉.

E ≤ 〈Ψ|H|Ψ〉〈Ψ|Ψ〉 = 〈H〉HB +Nǫc . (5.42)

Here〈H〉HB denotes the energy for the HB wavefunction.ǫc is the correlation energy defined by

Nǫc =〈Ψ|H|Ψ〉〈Ψ|Ψ〉 , (5.43)

59

with H = H − 〈H〉HB. Since it depends on the electron configuration{niσ} via the AA potential,we have to take into account the configurational average at the end. To determine the variationalparameters, we minimize the ground-state energy.

It is not easy to calculate exactly the correlation energy with use of the HB wavefunction (5.40).Therefore, we adopt again the single-site approximation (SSA). The average of〈A〉 of an operatorA = A− 〈A〉HB with respect to the wavefunction (5.40) is then given as follows.

〈A 〉 =∑

i

〈(1− O†i )A(1− Oi)〉HB

〈(1− O†i )(1− Oi)〉HB

. (5.44)

The detailed derivation of the above formula has been given in Sec. 3.2.1. Making use of the aboveformula, the correlation energy per atom is obtained as follows.

ǫc =−〈O†

i H〉HB − 〈HOi〉HB + 〈O†i HOi〉HB

1 + 〈O†i Oi〉HB

. (5.45)

Each term in the correlation energy (5.45) can be calculatedby making use of Wick’s theoremas follows.

〈HOi〉HB = U∑

κ′2κ2κ

′1κ1

〈κ′1|i〉〈i|κ1〉〈κ′2|i〉〈i|κ2〉∑

j

〈κ1|j〉〈j|κ′1〉〈κ2|j〉〈j|κ′2〉ηκ′2κ2κ

′1κ1fκ′

2κ2κ′1κ1

,

(5.46)

〈O†i H〉HB = 〈HOi〉∗HB , (5.47)

〈O†i HOi〉HB

=∑

κ′2κ2κ

′1κ1

〈i|κ′1〉〈κ1|i〉〈i|κ′2〉〈κ2|i〉 η∗κ′2κ2κ

′1κ1fκ′

2κ2κ′1κ1

∑

κ′4κ4κ

′3κ3

〈κ′3|i〉〈i|κ3〉〈κ′4|i〉〈i|κ4〉

×(∆Eκ′

2κ2κ′1κ1δκ1κ3δκ′

1κ′3δκ2κ4δκ′

2κ′4+ Uκ′

2κ2κ′1κ1κ

′4κ4κ

′3κ3

)ηκ′

4κ4κ′3κ3

, (5.48)

Uκ′2κ2κ

′1κ1κ

′4κ4κ

′3κ3

= U∑

j

[〈j|κ1〉〈κ3|j〉f(ǫκ3↑)δκ′1κ

′3− 〈κ′1|j〉〈j|κ′3〉(1− f(ǫκ′

3↑))δκ1κ3 ]

×[〈j|κ2〉〈κ4|j〉f(ǫκ4↓)δκ′2κ

′4− 〈κ′2|j〉〈j|κ′4〉(1− f(ǫκ′

4↓))δκ2κ4 ] , (5.49)

〈O†i Oi〉HB =

∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2 |ηκ′2κ2κ

′1κ1

|2 fκ′2κ2κ

′1κ1

. (5.50)

Herefκ′2κ2κ

′1κ1

is a Fermi factor of two-particle excitations which is defined by fκ′2κ2κ

′1κ1

= f(ǫκ1↑)(1 − f(ǫκ′

1↑))f(ǫκ2↓)(1 − f(ǫκ′

2↓)), f(ǫ) is the Fermi distribution function at zero temperature,

ǫκσ = ǫκσ − µ, andǫκσ is the one-electron energy eigenvalue for the HB Hamiltonian. Moreover,∆Eκ′

2κ2κ′1κ1

= ǫκ′2↓− ǫκ2↓ + ǫκ′

1↑− ǫκ1↑ is a two-particle excitation energy.

The above expressions (5.46) and (5.49) contain nonlocal terms via summation overj (i.e.,∑j). We thus make additional SSA that we only take into account the local term (j = i), so that

〈HOi〉HB(= 〈O†i H〉∗HB) and〈O†

i HOi〉HB reduce as follows.

〈HOi〉HB = U∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2 ηκ′2κ2κ

′1κ1

fκ′2κ2κ

′1κ1

, (5.51)

60

〈O†i HOi〉HB =

∑

κ′2κ2κ

′1κ1

|〈κ′1|i〉|2|〈κ1|i〉|2|〈κ′2|i〉|2|〈κ2|i〉|2

×η∗κ′2κ2κ

′1κ1

fκ′2κ2κ

′1κ1

[∆Eκ′

2κ2κ′1κ1ηκ′

2κ2κ′1κ1

+U{∑

κ3κ4

|〈κ3|i〉|2|〈κ4|i〉|2f(ǫκ3↑)f(ǫκ4↓) ηκ′2κ4κ

′1κ3

−∑

κ′3κ4

|〈κ′3|i〉|2|〈κ4|i〉|2(1− f(ǫκ′3↑))f(ǫκ4↓) ηκ′

2κ4κ′3κ1

−∑

κ3κ′4

|〈κ3|i〉|2|〈κ′4|i〉|2f(ǫκ3↑)(1− f(ǫκ′4↓)) ηκ′

4κ2κ′1κ3

+∑

κ′3κ

′4

|〈κ′3|i〉|2|〈κ′4|i〉|2(1− f(ǫκ′3↑))(1− f(ǫκ′

4↓)) ηκ′

4κ2κ′3κ1

}].(5.52)

In order to obtain the variational parameters{ηκ′2κ2κ

′1κ1

}, we minimize the correlation energyǫc,i.e., Eq. (5.45) with Eqs. (5.50), (5.51), and (5.52). The self-consistent equations for{ηκ′

2κ2κ′1κ1

}in the SSA are obtained as follows.

(∆Eκ′2κ2κ

′1κ1

− ǫc)ηκ′2κ2κ

′1κ1

+ U

[∑

κ3κ4

|〈κ3|i〉|2|〈κ4|i〉|2f(ǫκ3↑)f(ǫκ4↓) ηκ′2κ4κ

′1κ3

−∑

κ′3κ4

|〈κ′3|i〉|2|〈κ4|i〉|2(1− f(ǫκ′3↑))f(ǫκ4↓) ηκ′

2κ4κ′3κ1

−∑

κ3κ′4

|〈κ3|i〉|2|〈κ′4|i〉|2f(ǫκ3↑)(1− f(ǫκ′4↓)) ηκ′

4κ2κ′1κ3

+∑

κ′3κ

′4

|〈κ′3|i〉|2|〈κ′4|i〉|2(1− f(ǫκ′3↑))(1− f(ǫκ′

4↓)) ηκ′

4κ2κ′3κ1

]= U .

(5.53)

It is not easy to find the solution of Eq. (5.53) for the intermediate strength of Coulomb in-teractionU . To solve the equation approximately, we consider an interpolate solution which isvalid in both the weak Coulomb interaction limit and the atomic limit. Note that the first termat the left hand side (l.h.s.) of Eq. (5.53) is dominant and the second term is negligible in theweak Coulomb interaction limit. In the atomic limit, the momentum dependence ofηκ′

2κ2κ′1κ1

isnegligible. Thus, we approximate{ηκ′

2κ2κ′1κ1

} in the second term at the l.h.s. of Eq. (5.53) with amomentum independent parameterη which is suitable for the atomic region. Solving the equation,we obtain

ηκ′2κ2κ

′1κ1

(η, ǫc) =Uη

∆Eκ′2κ2κ

′1κ1

− ǫc. (5.54)

Hereη = [1− η(1− 2〈ni↑〉HB)(1− 2〈ni↓〉HB)].The ground-state correlation energy is obtained by substituting the variational parameters (5.54)

into Eq. (5.45). Each element in the energy is given as follows.

〈HOi〉HB = 〈O†i H〉∗HB = AU2 η , (5.55)

61

〈O†i HOi〉HB = B U2η2 = 〈O†

i H0Oi〉HB + U〈O†iOiOi〉HB , (5.56)

〈O†i H0Oi〉HB = B1 U

2η2 , (5.57)

〈O†iOiOi〉HB = B2 U

2 η2 , (5.58)

〈O†i Oi〉HB = C U2η2 . (5.59)

Here

A =

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc, (5.60)

B = B1 + U B2 , (5.61)

B1 =

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

(ǫ4 − ǫ3 + ǫ2 − ǫ1)−1(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2, (5.62)

B2 =

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc

×[ ∫

dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ5)f(ǫ6)

ǫ4 − ǫ6 + ǫ2 − ǫ5 − ǫc−∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)f(ǫ5)(1− f(ǫ6))

ǫ6 − ǫ3 + ǫ2 − ǫ5 − ǫc

−∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)(1− f(ǫ5))f(ǫ6)

ǫ4 − ǫ6 + ǫ5 − ǫ1 − ǫc+

∫dǫ5dǫ6ρ↑(ǫ5)ρ↓(ǫ6)(1− f(ǫ5))(1− f(ǫ6))

ǫ6 − ǫ3 + ǫ5 − ǫ1 − ǫc

],

(5.63)

C =

∫[ 4∏n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

(ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc)2. (5.64)

Hereρσ(ǫ) is the local DOS for the one-electron energy eigenvalues of the HB Hamiltonian matrix(5.9).

The best value ofη should be determined variationally. In fact, when we adopt the approximateform (5.54) as a trial set of amplitudes, we have a following inequality

E ≤ 〈H〉(w, {η∗κ′2κ2κ

′1κ1

}) ≤ 〈H〉(w, {ηκ′2κ2κ

′1κ1

(η, ǫc)}) . (5.65)

Here{η∗κ′2κ2κ

′1κ1

} are the exact solution for the Eq. (5.53). The above relationimplies that the bestvalue ofη can be again determined from the stationary condition (i.e., δǫc = 0), so that we obtain

η =−B +

√B2 + 4A2CU2

2ACU2. (5.66)

62

The total energy per atom should be obtained by taking the configurational average.

〈H〉 = 〈H〉HB + ǫc . (5.67)

The HB contribution〈H〉HB has been given by Eq.(5.6). The correlation energy can be obtainedas follows.

ǫc =∑

α

Pα ǫcα . (5.68)

Hereǫcα denotes the correlation energy for a given on-site configuration α.

ǫcα =[−〈O†

i H〉HB − 〈HOi〉HB + 〈O†i HOi〉HB

1 + 〈O†i Oi〉HB

]α. (5.69)

The quantities〈HOi〉HB, 〈O†i HOi〉HB, and〈O†

i Oi〉HB are given by Eqs.(5.55), (5.56), and(5.59),respectively, in which the local DOS have been replaced by those of the single-site CPA,i.e., Eq.(5.22).

Laplace transforms of the physical quantities (3.59)-(3.63) are given as follows.

Aα = i

∫ ∞

0

dt eiǫcαt aα↑(−t)aα↓(−t)bα↑(t)bα↓(t) , (5.70)

B1α = −∫ ∞

0

dtdt′eiǫcα(t+t′)

×[aα↑(−t− t′)bα↑(t+ t′)aα↓(−t− t′)b1α↓(t+ t′)

−aα↑(−t− t′)bα↑(t+ t′)a1α↓(−t− t′)bα↓(t+ t′)

+aα↑(−t− t′)b1α↑(t+ t′)aα↓(−t− t′)bα↓(t+ t′)

−a1α↑(−t− t′)bα↑(t+ t′)aα↓(−t− t′)bα↓(t+ t′)], (5.71)

B2α = −∫ ∞

0

dtdt′eiǫcα(t+t′)

×[aα↑(−t)bα↑(t+ t′)aα↓(−t)bα↓(t+ t′)aα↑(−tα′)aα↓(−t′)

−aα↑(−t)bα↑(t+ t′)aα↓(−t− t′)bα↓(t)aα↑(−t′)bα↓(t′)−aα↑(−t− t′)bα↑(t)aα↓(−t)bα↓(t+ t′)bα↑(t

′)aα↓(−t′)+aα↑(−t− t′)bα↑(t)aα↓(−t− t′)bα↓(t)bα↑(t

′)bα↓(t′)], (5.72)

Cα = −∫ ∞

0

dtdt′eiǫcα(t+t′)aα↑(−t− t′)bα↑(t+ t′)aα↓(−t− t′)bα↓(t+ t′). (5.73)

Hereα denotes the local electron configuration(α = 0, 1 ↑, 1 ↓, 2), and

aασ(t) =

∫dǫ ρασ(ǫ)f(ǫ) e

−iǫt , (5.74)

bασ(t) =

∫dǫ ρασ(ǫ)[1− f(ǫ)] e−iǫt , (5.75)

63

a1ασ(t) =

∫dǫ ρασ(ǫ)f(ǫ) ǫ e

−iǫt , (5.76)

b1ασ(t) =

∫dǫ ρασ(ǫ)[1− f(ǫ)] ǫ e−iǫt . (5.77)

The double occupation number〈ni↑ni↓〉 is obtained from∂〈H〉/∂Ui. Making use of the single-site energy (5.45), the Feynman-Hellmann theorem [32] and taking the configurational average,we obtain the following expression.

〈ni↑ni↓〉 = 〈ni↑〉HB〈ni↓〉HB + 〈ni↑ni↓〉c , (5.78)

Here the HB contribution of the double occupancy〈ni↑〉HB〈ni↓〉HB has been given by Eq.(5.33).The second term is the correlation contribution given as follows.

〈ni↑ni↓〉c =∑

α

Pα〈ni↑ni↓〉cα , (5.79)

〈ni↑ni↓〉cα =

[−〈O†

iOi〉HB − 〈OiOi〉HB + 〈O†iOiOi〉HB +

∑σ〈ni−σ〉HB〈O†

i niσOi〉HB

1 + 〈O†i Oi〉HB

]

α

, (5.80)

〈O†iOi〉HB + 〈OiOi〉HB = 2Uη

∫ [ 4∏

n=1

dǫn

]ρ↑(ǫ1)ρ↑(ǫ2)ρ↓(ǫ3)ρ↓(ǫ4)

×f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc, (5.81)

〈O†i niσOi〉HB = U2η2

∫ [ 5∏

n=1

dǫn

]ρ−σ(ǫ1)ρ−σ(ǫ2)ρσ(ǫ3)ρσ(ǫ4)ρσ(ǫ5)

×f(ǫ1)(1− f(ǫ2))f(ǫ3)(1− f(ǫ4))

ǫ4 − ǫ3 + ǫ2 − ǫ1 − ǫc

×[

1− f(ǫ5)

ǫ5 − ǫ3 + ǫ2 − ǫ1 − ǫc− f(ǫ5)

ǫ4 − ǫ5 + ǫ2 − ǫ1 − ǫc

]. (5.82)

The quantities〈O†iOiOi〉HB and〈O†

i Oi〉HB are defined by Eqs.(5.58) and(5.59), respectively.The element (5.81) for the calculation of the double occupancy is expressed using the Laplace

transform as follows

〈O†iOi〉HBα + 〈OiOi〉HBα = 2iUηα

∫ ∞

0

dt eiǫcαtaα↑(−t)bα↑(t)aα↓(−t)bα↓(t) . (5.83)

The correlation contribution to the electron number (5.82)which appears in the calculation of thedouble occupation number is expressed as

〈O†i niσOi〉HBα = −U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)

×[aα(−σ)(−t− t′)bα(−σ)(t+ t′)aασ(−t− t′)bασ(t)bασ(t

′)

−aα(−σ)(−t− t′)bα(−σ)(t+ t′)aασ(−t)bασ(t+ t′)aασ(t′)]. (5.84)

64

Similarly, the momentum distribution〈nkσ〉 is obtained from∂〈H〉/∂ǫk as follows.

〈nkσ〉 = 〈nkσ〉HB + 〈nkσ〉c . (5.85)

The HB contribution of the momentum distribution〈nkσ〉HB has been given by Eq.(5.35). Thecorrelation contribution〈nkσ〉c is expressed as follows.

〈nkσ〉c =∑

α

Pα〈nkσ〉cα , (5.86)

〈nkσ〉cα =

[N〈OinkσOi〉HB

1 + 〈O†i Oi〉HB

]

α

, (5.87)

N〈O†i nkσOi〉HB = U2η2

∫ [ 4∏

n=1

dǫn

]ρσ(ǫ1)ρ−σ(ǫ2)ρ−σ(ǫ3)ρkσ(ǫ4)f(ǫ2)(1− f(ǫ3))

×{

f(ǫ1)(1− f(ǫ4))

(ǫ3 − ǫ2 + ǫ4 − ǫ1 − ǫc)2− (1− f(ǫ1))f(ǫ4)

(ǫ3 − ǫ2 + ǫ1 − ǫ4 − ǫc)2

}. (5.88)

Herenkσ = nkσ − 〈nkσ〉HB. The DOS in the momentum representationρkσ(ǫ) has been given byEq. (5.36) in the SSA. The correlation contribution quantity 〈O†

i Oi〉HB is given by Eq.(5.59).The correlation contribution to the momentum distributionfunction (5.88) is given by

N〈O†i nkσOi〉0α = U2η2α

∫ ∞

0

dtdt′eiǫcα(t+t′)aα(−σ)(−t− t′)bα(−σ)(t+ t′)

×[bασ(t+ t′)akσ(−t− t′)− aασ(−t− t′)bkσ(t+ t′)

]. (5.89)

Here

akσ(t) =

∫dǫ ρkσ(ǫ)f(ǫ) e

−iǫt , (5.90)

bkσ(t) =

∫dǫ ρkσ(ǫ)

[1− f(ǫ)

]e−iǫt . (5.91)

In summary, we calculate the correlation energyǫcα (Eq. (5.69)) self-consistently with use ofEqs. (5.55), (5.56), (5.59), and (5.66) for a given weightw, and calculate the average correlationenergyǫc (Eq. (5.68)) as well as the average HB energy〈H〉HB (Eq. (5.6)). Then we obtain thetotal energy〈H〉(w) (Eq. (5.67)) for a givenw. Varyingw from 0 to 1 numerically, we obtain theground-state energy〈H〉 in the MLA-HB.

5.3 Numerical results: half-filled band Hubbard model

We have performed the numerical calculations to investigate the validity of momentum depen-dent local-ansatz approach (MLA) with hybrid (HB) variational wavefunction (i.e., MLA-HB). Weadopted here the half-filled band Hubbard model on the hypercubic lattice in infinite dimensions,

65

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 2 4 6 8 10

Ene

rgy

U

n=1.0

HBMLA-HB

GWLA

Figure 5.1: The energy vs Coulomb interaction energyU curves in the HB (thin solid curve), theMLA-HB (solid curve), the GW (dot-dashed curve) and the LA (dotted curve) for the electronnumbern = 1.0.

where the SSA works best [39, 42], and considered the non-magnetic case. In this case, the densityof states (DOS) for non-interacting system is given byρ(ǫ) = (1/

√π) exp(−ǫ2) [42]. The energy

unit is chosen to be∫dǫρ(ǫ)ǫ2 = 1/2. The characteristic band widthW is given byW = 2 in this

unit.The results of the ground-state energy vs Coulomb interaction energy curves are shown in Fig.

5.1. The energy of the HB wavefunction (without correlation) linearly increases with increasingCoulomb interaction strengthU in the weakU regime. AtUc0 = 1.43, the system shows a transi-tion from the Fermi liquid (FL) state (w = 0) to a non-Fermi liquid (NFL) state (w 6= 0), and showsa kink at the critical Coulomb interactionUc = 2.31, indicating the metal-insulator transition. Thetransition is of the first order in the present approach. The HB wavefunction gives lower energyin comparison with the GW and the LA in the strong Coulomb interaction regime (U/W & 1.5).The MLA-HB wavefunction further lowers the energy. In the weak Coulomb interaction regime,the ground-state energy of the MLA-HB is the lowest among theHB, LA, GW, and the MLA-HB.The MLA-HB shows the first-order transition atUc = 2.81 from the FL to the NFL, indicating themetal-insulator transition. The MLA-HB scheme gives lowerenergy for overall Coulomb interac-tion and therefore overcomes the GW.

Figure 5.2 shows the double occupation number〈n↑n↓〉 as a function of Coulomb interactionenergyU at half-filling. In the case of the HB, the double occupancy is constant (1/4) up toUc0 = 1.43, and decreases rapidly up to the critical pointUc = 2.31, at which it jumps from0.110 to 0.060. In the strong Coulomb interaction regime the double occupancy decreases withincreasingU and vanishes in the atomic limit. In the case of the MLA-HB, thedouble occupationnumber decreases smoothly from1/4 with increasing Coulomb interaction so as to reduce the lossof Coulomb energyU . Note that the MLA-HB reduces more the double occupancy as comparedwith that of the HB, GW and LA in the weakU region. The double occupancy in the MLA-HBjumps from0.106 to 0.045 at the transition pointUc = 2.81, and again monotonically decreases

66

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

⟨n↑n

↓⟩

U

n=1.0

HBMLA-HB

GWLA

Figure 5.2: The double occupation number〈n↑n↓〉 vs Coulomb interaction energyU curves athalf-filling (n = 1.0) in the HB (thin solid curve), the MLA-HB (solid curve), the GW (dot-dashedcurve), and the LA (dotted curve).

0

0.2

0.4

0.6

0.8

1

-1 -0.5 0 0.5 1

⟨nkσ

⟩

εkσ

U = 12

2.81

345

n=1.0

MLA-HBGWHF

Figure 5.3: The momentum distribution as a function of energy ǫkσ for various Coulomb interactionenergy parametersU = 1.0, 2.0, 2.81, 3.0, 4.0 and5.0 at half-filling. The MLA-HB: solid curves,the GW: dashed curves, and the HF: thin solid curve.

67

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8

Z

U

n=1.0

MLA-HBGW LA NRG

Figure 5.4: Quasiparticle-weight vs. Coulomb interaction curves in various theories. The MLA-HB: solid curve, the GW: dot-dashed, the LA: dotted curve, andthe NRG: thin solid curve [34].

with increasingU . Note that the double occupancy in the MLA-HB remains finite in the strongUregime as it should be, while the GW gives the Brinkman-Rice atom.

The momentum distribution for the MLA-HB is shown in Fig. 5.3. It decreases monotonicallywith increasingǫkσ and shows a jump at the Fermi energy in the metallic regime. The jumpdecreases with increasingU , and disappears beyondUc. When we further increase the CoulombinteractionU the curve becomes flatter. Note that the momentum distributions for the GW areconstant below and above the Fermi level [7, 8, 9]. These results indicate that the MLA-HBimproves the GW.

The quasiparticle weightZ (i.e., inverse effective mass) is obtained from the jump at the Fermilevel in the momentum distribution according to the Fermi liquid theory [35, 36, 37, 38]. Calculatedquasiparticle weight vs Coulomb interaction curves are shown in Fig. 5.4. The GW and the LAcurves strongly deviate from the curve of the NRG [34] which isconsidered to be the best. TheMLA-HB is close to the NRG in the metallic regime, and vanishesbeyondUc = 2.81. It should benoted that the NRG also shows the first order transition at a critical Coulomb interactionUc beforeZ vanishes atUc2 = 4.1. [34] The values ofUc in the NRG, however, has not yet been published.

68

Chapter 6

Summary and Discussions

We have proposed a new local-ansatz wavefunction with momentum-dependent variationalparameters (i.e., MLA-HF). It is constructed by using the ‘flexible’ local operators which producethe two-particle excited states in the momentum space from the Hartree-Fock state and projectthose states onto the local excited states in the real space.The best wavefunction is chosen bycontrolling the momentum dependent variational parameters of the excited states in the momentumspace on the basis of the variational principle. We obtainedthe ground-state energy of the MLAwithin a single-site approximation. Minimizing the energy, we derived a self-consistent equationfor the variational parameters, and obtained an approximate solution which interpolates betweenthe weak Coulomb interaction limit and the atomic limit. The MLA self-consistently determinesboth the variational amplitudeη and the correlation energyǫc making use of variational principles.The correlation energy in the MLA agrees with the result of the second-order perturbation theoryin infinite dimensions in the weak Coulomb interaction limit and yields the correct atomic limit asit should be.

To examine the improvement and validity of the MLA-HF theory, we performed in Chap. 3 thenumerical calculations for the half-filled band as well as the non-half-filled band on the basis of theHubbard model on the hypercubic lattice in infinite dimensions. We have investigated the effects ofthe best choice of the variational parameterη on various physical quantities especially in the non-half-filled case. In the correlation energy calculations, the non-self-consistentη (i.e., Eq. (3.42))yields the same results as the self-consistent case in the weak Coulomb interaction regime. But inthe intermediate- and strong- Coulomb interaction regimes we found that the self-consistency ofthe variational parameter becomes significant; the self-consistentη yields the result being betterthan the non-self-consistent case. We observed that the self-consistent MLA yields reasonablemomentum distribution functions while the non-self-consistent one leads to a unphysical bumpin the vicinity of the Fermi level. From these facts we conclude that the self-consistency ofη isindispensable for understanding electron correlations.

Within the self-consistent MLA-HF [25], we have clarified the role of the momentum depen-dence of variational parameters in comparison with the original Local-Ansatz (LA). We demon-strated that the self-consistent MLA improves the LA irrespective of the Coulomb interaction en-ergy parameterU and electron numbern. The correlation energy in the MLA is lower than those ofthe LA and the GW in the weak and intermediate Coulomb interaction regimes. Thus we verifiedthat the MLA wavefunction is better than both the LA and the GWin these regimes. The doubleoccupation number is suppressed as compared with the LA bothin the same interaction regimes.We found that the calculated momentum distribution functions show a distinct momentum depen-dence. This is qualitatively different from the LA and the GWbecause both of them lead to the

69

momentum-independence of the distributions below and above the Fermi level. We also found thatthe quasiparticle weight in the MLA-HF is close to those of the NRG in the range0 < U . 2.5,while the LA and the GW much overestimate them. The results suggest that the MLA-HF is ap-plicable to the systems withU/W . 1.5, for example, to the systems like transition metals andalloys. HereW denotes the band width of the noninteracting system. In the MLA-HF calcula-tion [25] the critical Coulomb interactionUc2 at half-filling was obtained from the vanishment ofthe quasiparticle weights. It isUc2 = 3.21 for the non self-consistentη andUc2 = 3.40 for theself-consistentη, respectively. The latter is comparable toUc2 = 4.10, the best value in the NRG,while the LA and the GW give larger valuesUc2 = 7.82 and4.51, respectively.

We have proposed an improved momentum dependent local-ansatz wavefunction in Chap. 4i.e., the MLA-AA. It allows us to describe electron correlationsstarting from the alloy-analogy(AA) limits instead of the Hartree-Fock (HF) state. The MLA-HF describes the Fermi-liquid state,while the MLA-AA describes the insulator state. We have performed the numerical calculationsfor the half-filled band Hubbard model on the hypercubic lattice in infinite dimensions, and demon-strated that the ground state energy for the MLA-AA is lower than the GW in the strong Coulombinteraction regime [27]. It yields the metal-insulator transition atUc = 3.26. The double occupa-tion number is suppressed in the weak and intermediate Coulomb interaction regimes as comparedwith the GW, jumps atUc, and remains finite in the strongly correlated regime as it should be.Finally, we found the momentum distribution functions showing a distinct momentum dependencein both the metallic and insulator regimes. These results indicate that the MLA-AA approach [27]can overcome the limitations of the original MLA-HF [24, 25], and goes beyond the GW in thestrongU regime.

We have proposed in Chap. 5 a new hybrid (HB) wavefunction and combined it with themomentum dependent local-ansatz approach MLA (i.e., the MLA-HB) to describe the correlatedelectron system from the weak to the strong Coulomb interaction regime [28]. The HB wavefunc-tion is the ground-state for the HB Hamiltonian and was used as a starting wavefunction of theMLA-HB. The HB Hamiltonian was constructed by a superposition of the HF Hamiltonian andthe AA one. The weightw of superposition is regarded as a variational parameter. When we adoptw = 0 (1), the HB wavefunction reduces to the HF (AA) state. In the MLA-HB, the best wavefunc-tion is chosen by controlling the momentum dependent variational parameters for the two-particleexcited states as well as the HB parameterw. We obtained the ground-state energy of the MLA-HB within a single-site approximation, and derived an approximate solution for the self-consistentequations of the variational parameters, which interpolates between the weak Coulomb interactionlimit and the atomic limit.

To examine the improvement and validity of the theory, we have performed the numericalcalculations for the half-filled band Hubbard model on the hypercubic lattice in infinite dimensions.In the case of the HB wavefunction we clarified that the ground-state energy increases linearly inthe weakU regime and it shows a lower energy as compared with the GW and the LA in the strongU regime. The double occupation number is constant up to theU = 1.43 (i.e., 〈n↑n↓〉HB = 0.25)and then decreases rapidly to the critical valueUc = 2.31 at which the first-order metal-insulatortransition occurs. In the strongU regime the〈n↑n↓〉HB remains finite.

We have demonstrated that the ground-state energy of the MLA-HB [28] is lower than those ofthe HB, GW and the LA in the whole Coulomb interaction regime. Inthe weak and intermediateCoulomb interaction regimes, the double occupation number is suppressed as compared with theothers. It jumps atUc = 2.81 and remains finite in the strongly correlated regime as it should be.The momentum distribution functions show a distinct momentum dependence in both the weak and

70

the strongU regimes. Moreover, we found that the behavior of the quasiparticle weight is closeto the NRG one. The above mentioned results indicate that the MLA-HB approach overcomesthe limitations of the original MLA [24, 25], and describes reasonably correlated electrons fromthe weak to the strong Coulomb interaction regime, so that it goes beyond the GW in the wholeCoulomb interactionU regime.

In the MLA approach we have adopted the single-site approximation which neglects the non-local correlations. In order to take into account the correlations properly, one has to consider thecontractions between the intersite operators in the calculation of the energy with use of Wick’stheorem. Furthermore, one has to extend the correlator to the nonlocal case. For example, onecan modify the correlator

∏i(1 − Oi) as exp(−

∑(i,j) Oij)

∏i(1 − Oi). Here we introduced a

nonlocal operatorOij =∑

k1k′1k2k

′2〈k′1|i〉〈i|k1〉〈k′2|j〉〈j|k2〉ηk′2k2k′1k1δ(a

†k′2↓ak2↓)δ(a

†k′1↑ak1↑) for the

description of the intersite correlations. It is worth for us to extend our single-site theory to thecluster taking into account the nonlocal effects mentionedabove.

It should also be noted that the present form of the MLA ansatzis not necessarily enough todescribe the strongly correlated electrons. For example, it cannot suppress the double occupancyfor infinitely largeU when the electron number per atom becomes less than one. In order to solvethe problem one has to modify the correlators(1 − Oi) as (1 − Oi − ζδni). Hereζ is a newvariational parameter andδni = ni − 〈ni〉. The termδni is needed to cover the Hilbert spacerelated to the double occupancy. The additional variational parameterζ therefore should suppressthe double occupancy.

Although more quantitative theories based on the quantum monte carlo (QMC) method andthe numerical renormalization group (NRG) method have been developed, the MLA approachpresented in this work is sufficiently simple, analytic, andtherefore applicable to more complexsystems. It also allows us to calculate any static averages with use of the wavefunction. Thus itis possible to apply the MLA approach to the multiband Hubbard model and to combine it withthe first-principles tight-binding Hamiltonian. Such a theory should provide us with a useful toolfor understanding the properties of correlated electrons and their physics in the realistic systems.These improvements and extensions of the theory are left forfuture work.

71

Appendix A

Appendix: Gutzwiller wavefunction

In this Appendix we derive the expression of the total energyof the Gutzwiller wavefunction,which is used in Sect. 2.4.1. The Hartree-Fock (HF) ground-state is expressed by an independent-particle state as follows.

|φ0〉 =( N↑∏

k

a†k↑

)( N↓∏

k

a†k↓

)|0〉 . (A.1)

HereNσ denotes the total electron number for the spinσ. Making use of a unitary transform suchasa†kσ =

∑i a

†iσe

ikRi/√L, we obtain the wavefunction in the real space representation as follows.

|φ0〉 =∑

C(N↑,N↓)

det

(1√Leiki .Rl(j)

)det

(1√Leiki .Rm(j)

)( N↑∏

i

a†l(i)↑

)( N↓∏

i

a†m(i)↓

)|0〉 . (A.2)

HereL denotes the number of lattice.ki is the momentum of thei-th electron below the Fermilevel. A set of sites(l(1), l(2), · · · , l(N↑)) ((m(1),m(2), · · · ,m(N↓))) denote a configuration ofelectrons with up (down) spin on the lattice, and(Rl(1),Rl(2), · · · ,Rl(N↑)) denote the positionsof the sites. The determinants are defined for the matrices whose (i, j) element is given by(1/

√L)exp(iki .Rl(j )) and (1/

√L)exp(iki .Rm(j )), respectively. Furthermore,

∑C(N↑N↓)

meansthe sum over all the configurations of electrons on a lattice when electron numbers of up and downspins are given. It should be noted that the{l(i)}({m(i)}) are ordered asl(1) < l(2) < · · · <l(N↑) (m(1) < m(2) < · · · < m(N↓)) in the Fock-space.

In the HF wavefunction, doubly occupied sites appear irrespective of the Coulomb interactionstrengthU in various electron configurations on a lattice. Such a statewith doubly occupied sitescauses a loss of Coulomb interaction energy. In the correlated electron system, the probabilityamplitudes of the doubly occupied states must be reduced to decrease the total energy. In orderto describe the on-site electron correlations, Gutzwillerintroduced a correlated wavefunction asfollows.

|ΨGW〉 =[ L∏

i

(1− (1− g)ni↑ni↓)]|φ0〉 . (A.3)

Hereni↑ni↓ is a projection operator that chooses the doubly occupied state on sitei, andg is avariational parameter controlling the amplitude of doublyoccupied states in the HF wavefunction.Note that theg = 1 state corresponds to an uncorrelated state; and theg = 0 state corresponds tothe atomic state in which all the doubly occupied states havebeen removed.

72

Substituting (A.2) in to the GW (A.3), we obtain the real space representation as follows.

|Ψ〉 =∑

D

gD∑

C(D,N↑,N↓)

det

(1√Leiki .Rl(j)

)det

(1√Leiki .Rm(j)

)( N↑∏

i

a†l(i)↑

)( N↓∏

i

a†m(i)↓

)|0〉 .

(A.4)

HereD is the number of doubly occupied sites, andC(D,N↑, N↓) denotes the electron configura-tions on a lattice whenD,N↑, andN↓ are given.

The energy of the Hubbard Hamiltonian (2.11) withǫ0 = 0 is then given as follows.

E(g) =

∑ijσ tij〈Ψ|a†iσajσ|Ψ〉+ U〈Ψ|∑i ni↑ni↓|Ψ〉

〈Ψ|Ψ〉 . (A.5)

Each term at the r.h.s. is given as follows.

〈Ψ|Ψ〉 =∑

D

g2D∑

C(D,N↑,N↓)

(w↑(R − R

′

)∣∣∣ Rl(1) Rl(2) · · · Rl(N↑)

Rl(1) Rl(2) · · · Rl(N↑)

)

×(w↓(R − R

′

)∣∣∣ Rm(1) Rm(2) · · · Rm(N↓)

Rm(1) Rm(2) · · · Rm(N↓)

), (A.6)

〈Ψ|∑

i

ni↑ni↓|Ψ〉

=∑

D

D g2D∑

C(D,N↑,N↓)

(w↑(R − R

′

)∣∣∣ Rl(1) Rl(2) · · · Rl(N↑)

Rl(1) Rl(2) · · · Rl(N↑)

)

×(w↓(R − R

′

)∣∣∣ Rm(1) Rm(2) · · · Rm(N↓)

Rm(1) Rm(2) · · · Rm(N↓)

). (A.7)

Here the functionwσ(R − R′

) is defined by

wσ(R − R′

) =1

L

Nσ∑

kn

eikn .(R−R′

) . (A.8)

The Gutzwiller overlap function with functionwσ(R − R′

) in (A.6) and (A.7) is defined by

(f(x, y)

∣∣∣ x1 x2 · · · xny1 y2 · · · yn

)=

∣∣∣∣∣∣∣∣

f(x1, y1) f(x1, y2) · · · f(x1, yn)f(x2, y1) f(x2, y2) · · · f(x2, yn)

· · · · · · · · · · · ·f(xn, y1) f(xn, y2) · · · f(xn, yn)

∣∣∣∣∣∣∣∣. (A.9)

In order to calculate the electron hopping term in the numerator of (A.5), we classify the con-figurationC(D,N↑, N↓) into 4 parts according to the 4 electron configurations on sites i andj;C(D,N↑, N↓, i↑= j ↑= 0), C(D,N↑, N↓, i↑= 0, j ↑= 1), C(D,N↑, N↓, i↑= 1, j ↑= 0), andC(D,N↑, N↓, i↑= j↑= 1). When the electron hopping operatora†i↑aj↑ is applied toΨ, the con-figurationC(D,N↑, N↓, i↑= 0, j ↑= 1) remains among 4 types of configurations. The numberof doubly occupied states in each configuration ofa†i↑aj↑|Ψ〉 can changes fromD according to

73

the configuration of the down spin electron on sitesi and j. When we express the configura-

tion as(i, j) =(i↑ j↑i↓ j↓

), the number of the doubly occupied states ofa†i↑aj↑|Ψ〉 is given byD for

(i, j) =(0 10 0

), D − 1 for (i, j) =

(0 10 1

), D + 1 for (i, j) =

(0 11 0

), andD for (i, j) =

(0 11 1

). We

therefore obtain

〈Ψ|a†i↑aj↓|Ψ〉

=[∑

D

g2D∑

C

(D,N↑,N↓,(i,j)=

(0 10 0

))+∑

D

g2D−1∑

C

(D,N↑,N↓,(i,j)=

(0 10 1

))

+∑

D

g2D+1∑

C

(D,N↑,N↓,(i,j)=

(0 11 0

))+∑

D

g2D∑

C

(D,N↑,N↓,(i,j)=

(0 11 1

))

]

×(w↑(R − R

′

)∣∣∣ Rl(1) Rl(2) · · · Rl(N↑)

Rl(1) Rl(2) · · · Rl(N↑)

)

×(w↓(R − R

′

)∣∣∣ Rm(1) Rm(2) · · · Rm(N↓)

Rm(1) Rm(2) · · · Rm(N↓)

). (A.10)

The configuration in the above expression (A.10), for exampleC(D,N↑, N↓, (i, j) =

(0 10 0

))means

the electron configuration whenD,N↑, andN↓ are given, and there is no electron on sitei, but sitej is occupied by an up-spin electron.

The difficulty in the Gutzwiller variational method is how totake the sums with respect to theelectron configurations in each term of the energy containing the overlap function ofwσ(R − R

′

).Gutzwiller replaced these overlap functions with their average values. This is called the Gutzwillerapproximation. For example in the calculation of the norm〈Ψ|Ψ〉, we make the following approx-imation.

〈Ψ|Ψ〉 ≈∑

D

g2D∑

C(D,N↑,N↓)

[ 1

(∑

D

∑C(D,N↑,N↓)

)

∑

D

∑

C(D,N↑,N↓)

]

×(w↑(R − R

′

)∣∣∣ Rl(1) Rl(2) · · · Rl(N↑)

Rl(1) Rl(2) · · · Rl(N↑)

)

×(w↓(R − R

′

)∣∣∣ Rm(1) Rm(2) · · · Rm(N↓)

Rm(1) Rm(2) · · · Rm(N↓)

). (A.11)

We then obtain

〈Ψ|Ψ〉 ≈ W0(g)

W0(1)〈φ0|φ0〉 . (A.12)

Here〈φ0|φ0〉 = 1. W0(g) is a sum over all configurations of correlation weightg2D for doubleoccupation number. It is defined by

W0(g) =∑

D

∑

C(D,N↑,N↓)

g2D . (A.13)

In the same way, we obtain

〈Ψ|a†i↑aj↑|Ψ〉 ≈ W1↑(g)

W1↑(1)〈φ0|a†i↑aj↑|φ0〉 , (A.14)

74

〈Ψ|ni↑ni↓|Ψ〉 ≈ W2(g)

W2(1)〈φ0|ni↑ni↓|φ0〉 . (A.15)

Here

W1↑(g) =∑

D

g2D∑

C

(D,N↑,N↓,(i,j)=

(0 10 0

))+∑

D

g2D−1∑

C

(D,N↑,N↓,(i,j)=

(0 10 1

))

+∑

D

g2D+1∑

C

(D,N↑,N↓,(i,j)=

(0 11 0

))+∑

D

g2D∑

C

(D,N↑,N↓,(i,j)=

(0 11 1

)), (A.16)

W2(g) =∑

D

∑

C(D,N↑,N↓)

D g2D . (A.17)

The weighting functionsW0(g), W1(g), andW2(g) are expressed by the following type of thehypergeometric functions.

F (α−N↑, β −N↓, L−N + γ; g2)

=∞∑

D=0

(L−N + γ − 1)! (α−N↑ +D − 1)! (β −N↓ +D − 1)! g2D

D! (L−N + γ +D − 1)! (α−N↑ − 1)! (β −N↓ − 1)!. (A.18)

HereN is the total number of electrons.α, β, andγ are integers of order of1. In the evaluation ofthe sums overD in these functions, we can adopt the maximum term approximation becauseD isa macroscopic variable. The weights for eachD term proportional to

(α−N↑ +D − 1)! (β −N↓ +D − 1)! g2D

D! (L−N + γ +D − 1)!. (A.19)

The representative value ofD that minimizes the above factor is given by

g2 =D(L−N↑ +D)

(N↑ −D)(N↓ −D). (A.20)

By making use of the representative valueD, we can express the total energy as follows.

E(g) =∑

σ

qσ

(Nσ∑

k

ǫk

)+ UD . (A.21)

Here the band narrowing factorqσ is given as

qσ =

(√(Nσ −D)(L−N +D) +

√D(N−σ −D)

)2

Nσ(L−Nσ). (A.22)

Equations (A.21) and (A.22) are identical with (2.32) and (2.33) in Sect. 2.4.1.

75

Appendix B

Appendix: Wick’s theorem

In order to calculate the operator products of many-body problems in Chapter 3, 4, and 5,we use the Wick’s theorem. In this appendix we describe the detailed derivation of the Wicktheorem. When we treat many-body problems, we encounter the average of the operator productswith respect to the non-interacting HamiltonianH0 as follows:

〈A1A2 . . . A2n〉0. (B.1)

The subscript zero denotes the thermal average of the independent-particle Hamiltonian. Here{Ai} are creation or annihilation operators that satisfy the following relations

AiAj + AjAi = (i, j). (B.2)

i.e.,

a†ka†k′ + a†k′a

†k = akak′ + ak′ak = 0, (B.3)

a†kak′ + ak′a†k = δkk′ . (B.4)

Note that,H0 =∑

k ǫknk. Thus,

〈a†ka†k′〉0 = 〈akak′〉0 = 0, (B.5)

〈a†kak′〉0 = 〈aka†k′〉0 = 0 (k 6= k′), (B.6)

and

〈a†kak〉0 = 〈nk〉0 =1

1 + eβǫk, (B.7)

〈aka†k〉0 = 1− 〈a†kak〉0 =1

1 + e−βǫk. (B.8)

Thus

〈AiAj〉0 =(ij)

1 + e±βǫi. (B.9)

Here+ for Ai = ak† and− for Ai = ak in the denominator.〈AiAj〉0 is called the contraction of a

pair (ij).Using this anti-commutation relation, one can exchangeA1 with A2 in the average

〈A1A2 . . . , A2n〉0 = (12)〈A3 . . . , A2n〉0 − 〈A2A1A3 . . . , A2n〉0. (B.10)

76

Using the same procedures, one can moveA1 to the end of the products as follows:

〈A1A2 . . . A2n〉0 = (12)〈A3 . . . A2n〉0 − (13)〈A2A4 . . . A2n〉0+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

+ (1, 2n)〈A2A4 . . . A2n−1〉0 − 〈A2A3 . . . A2nA1〉0. (B.11)

According to the relationak(β) = eβH0ake

−βH0 . (B.12)

Now differentiating the above equation with respect toβ

∂ak(β)

∂β= eβH0(H0ak − akH0)e

−βH0 . (B.13)

By making use ofH0 =∑

k′ ǫk′a

†

k′ak′ , nk

′ = a†k′ak′ and the anti-commutation relationaka

†

k′ =

δkk′ − a†k′ak intoH0ak − akH0, we obtain

H0ak − akH0 =∑

k′

ǫk′ (ak′†ak′ak − akak′

†ak′ )

=∑

k′

ǫk′ (ak′†ak′ak + ak′

†akak′ − ak′δkk′ )

= −ǫkak. (B.14)

Thus Eq. (B.13) is expressed by Eqs. (B.12) and (B.14)

∂ak(β)

∂β= −ǫkak(β). (B.15)

If we considerak(0) = ak then Eq. (B.15) becomes

ak(β) = ake−βǫk . (B.16)

Similarly,ak

†(β) = ak†eβǫk . (B.17)

From this, we haveake

−βH0 = e−βH0ake−βǫk , (B.18)

ak†e−βH0 = e−βH0ak

†eβǫk . (B.19)

According to the definition of thermal average,

〈A2 . . . A2nA1〉0 =tr(A2 . . .A2nA1e

−βH0)

tr(e−βH0). (B.20)

Thus, one can use the following relation in the trace of average (B.20)

Aie−βH0 = e±βǫke−βH0Ai. (B.21)

Here+ for Ai = a†k and− for Ai = ak Thus

〈A2 . . . A2nA1〉0 =e±βǫ1tr(A2 . . .A2ne

−βH0A1)

tr(e−βH0). (B.22)

77

Hence〈A2 . . . A2nA1〉0 = e±βǫ1〈A1A2 . . . A2n〉0. (B.23)

Here we made use of the identitytr(AB) = tr(BA).Substituting Eq. (B.23) into Eq. (B.11), we obtain,

(1 + e±βǫ1)〈A1A2 . . . A2n〉0 = (12)〈A3 . . . A2n〉0 − (13)〈A2A4 . . . A2n〉0+ · · ·+ (1, 2n)〈A2A3 . . . A2n−1〉0. (B.24)

Thus now using the Eq. (B.24), we obtain,

〈A1A2 . . . A2n〉0 = 〈A1A2〉〈A3 . . . A2n〉0 − 〈A1A3〉〈A2A4 . . . A2n〉0+ · · ·+ 〈A1A2n〉〈A2A3 . . . A2n−1〉0. (B.25)

This way, one can reduce the number of products in the average. Repeating the same procedurefor the remaining multiple products, we reach the Wick theorem

〈A1A2 . . . A2n〉0 =∑

{contractions}

(−1)δ(P )〈Ai1Ai2〉0〈Ai3Ai4〉0 . . . 〈Ai2n−1Ai2n〉0. (B.26)

Here the sum on the r.h.s. is taken over all possible pairs of contractions.(−1)δ(P ) takes+ or −depending on whether (i1, i2, i3, . . . . . . , i2n) is even or odd in permutation and

∑

{contractions}

=∑

.

i1 < i2, i3 < i4, . . . . . . , i2n−1 < i2n

i1 < i3 < i5 < . . . · · · < i2n−1

78

Appendix C

Appendix: Density of states in infinitedimensions

To perform the numerical calculation in Chapter 3, 4, and 5, weadopt the hypercubic latticein infinite dimensions. In this Appendix we describe the Gaussian density of states in infinitedimensions.

The tight-binding Hamiltonian with one atom per primitive unit cell is expressed as

Hij = ǫ0δij + tij(1− δij). (C.1)

Hereǫ0 is the atomic level andtij is the transfer integral between sitesi andj. The eigen valueequation is given by ∑

j

Hij〈j|k〉 = ǫk〈i|k〉. (C.2)

The energy eigen valueǫk is obtained by the Fourier transformation as

ǫk =∑

j

Hj0eikRj , (C.3)

i.e.ǫk = ǫ0 +

∑

j 6=0

tj0eikRj . (C.4)

let us consider the Hamiltonian in thed-dimensional hyper-cubic lattice. We have then thed-dimensionalk vectork = (k1, k2· · ·, kd) in the first Brillouin zone defined by

−πa< kn <

π

a(n = 1, 2, . . . , d). (C.5)

Herea represents a lattice constant. In addition, there are2d NN sites. We therefore obtain fortheNN hopping model in Eq. (C.4)

ǫk = ǫ0 + t

2d∑

j 6=0

eikRj . (C.6)

i.e.

ǫk = ǫ0 + 2td∑

n=1

cos kna . (C.7)

79

This is the energy eigen values for thed-dimensional simple-cubic lattice.Since the band widthW = 4d|t| diverges whend → ∞, we adopt the renormalized transfert

in the following, which is defined by

tij = − t√2d. (C.8)

Then the Eq. (C.7) becomes

ǫk = ǫ0 −2t√2d

d∑

n=1

cos kn . (C.9)

Here we adopted the unita = 1.The density of states (DOS) is calculated as follows.

ρ(ǫ) =1

N

∑

k

δ(ǫ− ǫk). (C.10)

Since there aredkn/(2π/L) states in the region[kn, kn + dkn] one can use the replacement

∑

kn

→∫

dkn2π/L

. (C.11)

HereL is the size of the crystal along thexn axis.Since there areL lattice points alongxn axis, we have a relationN = Ld.

Therefore,

ρ(ǫ) =

∫ [ d∏

n=1

dkn2π

]δ(ǫ− ǫk). (C.12)

In order to obtain the DOS in infinite dimensions, we introduce the characteristic functionΦd(s) of the (DOS) as follows.

Φd(s) ≡∫dǫeisǫρ(ǫ) =

∫ [ d∏

n=1

dkn2π

]eisǫk . (C.13)

i.e.,

Φd(s) = eisǫ0[ ∫ π

−π

dkn2π

e−iα cos kn

]d(C.14)

where

α =2ts√2d

. (C.15)

Let us consider the following integral.

∫ π

−π

dxe−iα cosx =∞∑

n=0

(−iα)nn!

∫ π

−π

cosn xdx. (C.16)

Here ∫ π

−π

dx = 2π , (C.17)

∫ π

−π

cos x dx = [sin x]π−π = 0 , (C.18)

80

∫ π

−π

cos2 x dx =

∫ π

−π

1 + cos 2x

2dx =

1

2[x+

1

2sin 2x]π−π = π , (C.19)

∫ π

−π

cos2n+1 x dx = 0 , (C.20)

∫ π

−π

cos2n x dx =

∫ π

−π

[

(1

2

)2n−1

+ [cos 2nx+

(2n1

)cos 2(2n− 1)x

+

(2n2

)cos 2(n− 2)x · · ·+

(2nn− 1

)cos 2x] +

(2nn

)(1

2

)2n

]dx,

=

(2nn

)(1

2

)2n

2π,

=(2n)!

(n!)22π

22n. (C.21)

Therefore, ∫ π

−π

dxe−iα cosx =∞∑

n=0

(−iα)2n(2n)!

(2n)!

(n!)22π

22n, (C.22)

i.e., ∫ π

−π

dxe−iα cosx = 2π∞∑

n=0

(−)n

(n!)2(α

2)2n. (C.23)

Where the Bessel functionJ0(z) is

J0(z) =∞∑

n=0

(−)n(z/2)2n

(n!)2. (C.24)

We obtain ∫ π

−π

dkn2π

e−iα cos kn = J0(α). (C.25)

Therefore,Φd(s) = eisǫ0 [J0(α)]

d. (C.26)

Let us expand the Bessel functionJ0(α) with respect tos.

J0(α) =∞∑

n=0

(−)n

(n!)2

(α2

)2n= 1−

(α2

)2+

1

4

(α2

)4+ · · · (C.27)

[J0(α)

]d=[1−

(α2

)2+

1

4

(α2

)4+ · · ·

]d,

= 1−(d1

)(α2

)2+

(d2

)[− (

α

2)2]2

+

(d1

)1

4

(α2

)4+ · · · ,

= 1− d(α2

)2+[12d(d− 1) +

1

4d](α

2

)4+ · · · ,

= 1− d(α2

)2+

1

4d(2d− 1)

(α2

)4+ · · · , (C.28)

81

Now assume the cumulant expansion of[J0(α)]d as

[J0(α)]d = exp

[c1

(α2

)2+ c2

(α2

)4+ · · · ],

=∞∑

n=0

1

n!

[c1

(α2

)2+ c2

(α2

)4+ · · ·

]n,

= 1 +[c1

(α2

)2+ c2

(α2

)4+ · · ·

]+

1

2

[c1

(α2)2 + c2

(α2

)4+ · · ·2],

= 1 + c1

(α2

)2+

1

2

(2c2 + c1

2)(α

2

)4+ · · · . (C.29)

Compare the above equation with Eq. (C.28), we obtain

c1 = −d.12(2c2 + c1

2) = 14d(2d− 1).

· · ·

i.e.,

c1 = −d.

c2 =1

4d(2d− 1)− 1

2c1

2 = −1

4d.

Therefore,

[J0(α)]d = exp

[− d(α2

)2− 1

4

(α2

)4+ · · ·

],

= exp[− d(t2s2

2d

)− 1

4

(t4s44d2

)+ · · ·

].

(C.30)

i.e.,

[J0(α)

]d= exp

[− d(t2s2

2d

)− 1

4

(t4s44d2

)+ · · ·

],

→ e−t2

2s2

(d→ ∞). (C.31)

Thus, Eq. (C.26) becomes,

Φd(s) = eisǫ0−

t2

2s2

. (C.32)

This is the characteristic function for the Gaussian DOS.

ρ(ǫ) =1√2π|t|

e−(ǫ− ǫ0)

2

2t2 . (C.33)

Thus we obtain the DOS on the hyper-cubic lattice in infinite dimensions. We used this type oflattice to perform the numerical calculation in Chapter 3, 4,and 5.

82

Appendix D

Appendix: Fermi liquid theory

In this appendix we derived the formula of the quasiparticleweightZ on the basis of the Fermiliquid theory, which is used in Sect. 3.5.2 and 5.3. The concept of the Fermi liquid was developedby Landau and has been extended byAbrikosov, Khalatnikov, Nozieres, Pines, Silin andothers [35, 36, 37, 38]. It explains why the low temperature properties of metals resemble so muchthose of a free electron system, i.e., with neglect electron-electron repulsions. It describes theelectronic parameters like the effective mass or density ofstates. A crucial part of that concept isthe notion of quasiparticles and quasiholes. In this Appendix we will explain the discontinuity ofthe momentum distribution at the Fermi surface in connection to the quasiparticle weight.

Let us suppose that we start from a system of non-interactingelectrons and that the interactionis slowly turned on. The basic assumption of the Fermi liquidtheory is that the classification ofthe energy levels remains unchanged. This implies that the energy levels must not cross as theinteraction sets in. The distribution functionnkσ helps to classify the excitation energies of non-interacting electron system. If we know thenkσ, we can easily calculate the energy of the system.In order to leave the classification of the energy levels unchanged when the interaction is turnedon, the energy of the interaction system must again be a function of the distribution functionnkσ.Whereas beforenkσ described the distribution of non-interacting electrons,it now describes thedistribution of the excitations which, followingLandau, are called quasiparticle.

The energy variationδE due to the change of the configuration is assumed to be expressed as

δE =∑

kσ

Ekσδnkσ +1

2

∑

kσk′σ′

f(kσ,k′σ′)δnkσδnk′σ′ + · · · . (D.1)

HereEkσ serves as a definition of the quasiparticle energy andδnkσ is the infinitesimal amount ofnkσ. The functionf(kσ,k′σ′), introduced by Landau, characterizes the electron-electron interac-tions.

Since the number of microscopic states do not change one may assume the following form ofthe entropyS for the interacting system under given configuration{nkσ}.

S = −∑

kσ

[nkσ lnnkσ + (1− nkσ) ln(1− nkσ)] . (D.2)

In the equilibrium state, the distribution{nkσ} which minimizes the number of microscopic statesW under given energyE and electron numberN will be realized.

S = lnWmax when E and N = constant,

83

i.e.,δ [S − βE − αN ] = 0 . (D.3)

HereδS = −

∑

kσ

[lnnkσ − ln(1− nkσ)] δnkσ , (D.4)

δE =∑

kσ

Ekσδnkσ , (D.5)

δN =∑

kσ

δnkσ . (D.6)

Thus

−∑

kσ

[ln

nkσ

1− nkσ

+ βEkσ + α

]δnkσ = 0 (D.7)

Note that the noninteracting particles obey Fermi-Dirac statistics, the quasiparticles do so too, andconsequently, the occupation probabilitynkσ of a quasiparticle state is given by

nkσ =1

eβ(Ekσ−µ) + 1. (D.8)

Hereµ = −β−1α denotes the chemical potential. The meaning ofβ is given as follows

δS = −∑

kσ

[lnnkσ − ln(1− nkσ)] δnkσ

=∑

kσ

β(Ekσ − µ)δnkσ

= βδE . (D.9)

TherforeδS

δE= β =

1

T. (D.10)

This means thatβ is the inverse temperature.The quasiparticle energyEkσ can be regarded as a renormalization of the non-interactingen-

ergyǫkσ.

Ekσ =ǫkσαkσ

. (D.11)

Hereαkσ is a renormalization factor. When we adopt the free electron model, the energyǫkσ iswritten as

ǫkσ =~2k2

2m. (D.12)

Using the same formula for the quasiparticle energyEkσ, we may write it as

Ekσ =~2k2

2m∗k

. (D.13)

Here we introduced an effective massm∗k. From Eqs. (D.11), (D.12), and (D.13), we obtain

αkσ =m∗

k

m. (D.14)

84

Thus the renormalization factorαkσ defined by (D.11) is called the mass enhancement factor.We can recognize that the mass enhancement factor enhances the low-temperature specific heat

by a factor ofα. According to the statistical mechanics, the low-temperature specific heat of Fermiliquid is written as

CV =

(∂E

∂T

)

V,N

=∂

∂T

(∑

kσ

Ekσnkσ

)−∑

kσ

∂Ekσ

∂Tnkσ . (D.15)

Neglecting the temperature dependence ofEkσ at low temperatures, we have an expression

∑

kσ

Ekσnkσ =∑

kσ

∫dωδ(ω − Ekσ)Ekσf(Ekσ)

=∑

kσ

∫dωδ(ω − Ekσ)ωf(ω)

=

∫dωρ(ω)ωf(ω) . (D.16)

Hereρ(ω) is called the quasiparticle energy density of states.

ρ(ω) ≡∑

kσ

δ(ω − Ekσ) . (D.17)

Therefore

CV =∂

∂T

∫dωρ(ω)ωf(ω) . (D.18)

Making use of the low-temperature expansion formula∫g(ω)f(ω)dω =

∫ µ

g(ω)dω +π2

6T 2g′(µ) + · · · . (D.19)

The one electron energy is expressed as follows.∫ωρ(ω)f(ω)dω =

∫ µ

ωρ(ω)dω + (µ− ǫF)ǫFρ(ǫF) +π2

6T 2 [ρ(µ) + µρ′(µ)] + · · · . (D.20)

The temperature changeµ− ǫF is obtained as follows:

N =∑

kσ

nkσ =

∫ρ(ω)f(ω)dω

=

∫ ǫF

ρ(ω)dω + (µ− ǫF)ρ(ǫF) +π2

6T 2ρ′(µ) + · · · , (D.21)

i.e.,

(µ− ǫF)ρ(ǫF) = −π2

6T 2ρ′(µ) + · · · . (D.22)

Thus∫ωρ(ω)f(ω)dω =

∫ ǫF

ρ(ω)dω +π2

6T 2 [ρ(µ) + (µ− ǫF)ρ

′(µ)] · · · . (D.23)

85

Therefore the expression of the specific heat of the Fermi liquid (D.18) in the lowest order atconstant volume, is written as

CV =1

3π2ρ(ǫF)T , (D.24)

ρ(ǫF) =∑

kσ

δ(ǫF − Ekσ) . (D.25)

WhenEkσ ≈ ǫkσ

α, (D.26)

we have

ǫF =ǫ0Fα, (D.27)

and

ρ(ǫF) =∑

kσ

δ

(ǫ0F − ǫkσ

α

)

=∑

kσ

αδ(ǫ0F − ǫkσ)

= α ρ(0)(ǫ0F) . (D.28)

Hereρ(0)(ǫ0F) =

∑

kσ

δ(ǫ0F − ǫkσ) . (D.29)

ThusCV = αC

(0)V , (D.30)

C(0)V =

1

3π2 ρ(0)(ǫ0F)T . (D.31)

Therefore the mass enhancement factorα enhances the low temperatures specific heat of the Fermiliquid by α = m∗/m.

We now look at the microscopic theory of the Landau Fermi liquid. The Green function on thecomplex plane is obtained by the analytic continuation of the temperature Green function.

Gkσ(z) =

∫ρkσ(ǫ)

z − ǫdǫ =

1

z − ǫkσ − Σkσ(z). (D.32)

Herez = ω + iδ, δ is the infinitesimal positive number. The functionΣkσ(z) is called the self-energy. The self-energyΣkσ(z) of a Fermi liquid has the general form

Σkσ(z) = Σkσ(0) +

(∂Σkσ

∂z

)

ω=0

ω + · · · . (D.33)

86

Let us linearize the self-energy around the Fermi level

Gkσ(z) =1

z − ǫkσ − Σkσ(0)− (∂Σkσ(0)/∂z) z + · · ·

=1

α′

kσz − ǫkσ − Σkσ(0)

+ · · ·

=1

(Reα′

kσ)z − ǫkσ − ReΣkσ(0) + i(zImα′

kσ − ImΣkσ(0))

=1/Reα

′

kσ

z − ǫkσ + ReΣkσ(0)

Reα′

kσ

+ izImα

′

kσ − ImΣkσ(0)

Reα′

kσ

. (D.34)

Hereα′

kσ= 1− ∂Σkσ(0)/∂ω. Thus

Gkσ(z) =αkσ

−1

z − Ekσ + iΓkσ

. (D.35)

Here we defined the mass enhancement factorαkσ by Reα′

kσ:

αkσ = 1− ∂ReΣkσ(0)

∂ω=m∗

k

m. (D.36)

In case of the phenomenological Fermi liquid theory, it is defined by Eq. D.14.The quasiparticle energyEkσ is defined by

Ekσ =ǫkσαkσ

=ǫkσ

1− ∂ReΣkσ(0)

∂ω

. (D.37)

Here we defined the energyǫkσ by ǫkσ = ǫkσ + ReΣkσ(0). The quasiparticle energy of the Fermiliquid from the microscopic point of view (Eq. D.37) is connected to the phenomenological oneby Eq. (D.11). The life timeΓkσ is defined as

Γkσ =zImα

′

kσ − ImΣkσ(0)

αkσ

. (D.38)

For the Fermi liquid state

Γkσ(0) = − ImΣkσ(0)

αkσ

= 0 . (D.39)

The quasiparticle weightZkσ is defined by

Zkσ = α−1kσ

=

(1− ∂ReΣkσ(0)

∂ω

)−1

=m

m∗k

. (D.40)

The retarded Green function near the Fermi level (w ≈ 0) is therefore written interms of thequasiparticle weightZkσ, the quasiparticle energyEkσ, and the life timeΓkσ as follows.

Gkσ(z) =Zkσ

z − Ekσ + iΓkσ

+ incoh. (D.41)

87

The second term on the right hand side denotes the other incoherent contribution.The imaginary part of the self-energy may be small near the Fermi level in the Fermi liquid

state. Note that the single particle DOSρkσ(ω) is connected to the quasi-particle DOSρQPkσ (ω) =

δ (ω − Ekσ) as

ρkσ(ω) = − 1

πImGkσ(z )

≈ − 1

πIm

Zkσ

z− Ekσ

= Zkσδ (ω − Ekσ) (D.42)

The momentum distribution〈nkσ〉 is expressed as

〈nkσ〉 =

∫dωf(ω)ρkσ(ω)

=

∫dωf(ω)[Zkσδ(ω − Ekσ) + incoh.]

= Zkσ θ(−Ekσ) + incoh. (D.43)

Thus, the quasiparticle weightZkσ describes the discontinuity in the momentum distribution func-tion of electrons on the Fermi surfacek = kF.

δ〈nkσ〉|k=kF = Zkσ =m

m∗k

(D.44)

We show this schematically in Fig. D.1. We have used this formula of the jump of the momentumdistribution at the Fermi level to measure the quasiparticle weight in subsection 3.5.2.

⟨nkσ⟩

kF

k

Zkσ

Figure D.1: DiscontinuityZkσ in the momentum distribution〈nkσ〉

88

Appendix E

Appendix: Coherent potentialapproximation

In order to understand the properties of the alloys one must know their electronic structure froma microscopic point of view. Electronic structure calculations in disorder alloys, however, is noteasy. The difficulty is that there is no translational symmetry in the system so that we cannot applythe Bloch theory. We present in this appendix the coherent potential approximation (CPA) [41] tocalculate the electronic structure of alloys in the single site approximation, which is uesd in chapter4.

Let us consider a substitutional binary alloy with A and B atom which are randomly distributed.The one electron Hamiltonian is given by

(Hσ)ij = ǫασδij + tij(1− δij) . (E.1)

Heretij is the transfer integral between sitesi andj. cα denotes the concentration of atomα.

cα =

{cA = 〈n−σ〉 (α = A)

cB = 1− 〈n−σ〉 (α = B) .(E.2)

n−σ denotes the average electron number per site. The atomic level ǫασ is defined by

ǫασ =

{ǫA = ǫ0 + U − µ (α = A)

ǫB = ǫ0 − µ (α = B) .(E.3)

ǫ0 is the atomic energy,U is the on the on-site Coulomb interaction energy parameter, andµ is thechemical potential. The Green functionGij(z) for the Hamiltonian is defined by

Gij(z) = [(z −Hσ)−1]ij . (E.4)

In the binary alloys we neglect the disorder of transfer integrals called the off-diagonal disorder.Because the atomic levelsǫασ are random variables, the Bloch theory is not applicable for thecalculation of the Green function. We make use of the coherent potential approximation (CPA) toobtain the Green function with use of the locator expansion [40]. Introducing the locator matrixLby means of

(L)ij = Liδij = (z − ǫασ)−1δij , (E.5)

89

we express the Green function matrix as

G = (L−1 − t)−1 = (1− Lt)−1L (E.6)

wheret denotes the transfer integral matrixtij. Thus,

G = L+ LtG (E.7)

Expanding the Green function with respect tot, we obtain

Gii(Z) = Li +∑

j 6=i

LitijLjtjiLi +∑

j 6=i

∑

k 6=j,i

LitijLjtjkLktkiLi + · · · . (E.8)

Here we have omitted the spin suffix for simplicity. The r.h.s. of (E.8) consists of the contributionfrom all the paths which start from sitei and end at the same sitei. They are expressed as followsby using the sum of all pathsSi which start from sitei and end at sitei without returning to siteion the way.

Gii(Z) = Li + LiSiLi + LiSiLiSiLi + · · · = (L−1i − Si)

−1 . (E.9)

AV

α

Figure E.1: Schematic representation of the coherent potential approximation. The central site isoccupied by an atomα (= A or B). The hatched sites are occupied by a coherent potential Σσ(z).〈 〉AV at the l.h.s. means a configurational average on the central site.

In the above expression, all the information outside the central atomi is in the self-energySi.To obtain the Green functionGii(z) in a single-site approximation, we approximate the randompotentials on the surrounding sites with an energy-dependent coherent potntialΣσ(z) (see the l.h. s. of Fig. E.1). We have then an impurity Green function foratomα on sitei from (E.9) asfollows.

Gασ(z) = (L−1ασ − Sσ)

−1 (E.10)

HereL−1ασ = z − ǫασ is the inverse locator on the central site with a type of atomα. Sσ is the self

energy in which all the atomic levels have been replaced by the coherent potentialΣσ(z). Notethat we have omitted the site indices for simplicity and haverecovered the spin suffix.

The self-energySσ is obtained from the coherent Green functionFσ(z) in which all the sitesare occupied by the coherent potential (see the r. h. s. of Fig. E.1)

Fσ(z) = (L −1σ − Sσ)

−1 . (E.11)

HereL −1σ (z) = z − Σσ(z). SubstitutingSσ obtained from(E.11) into (E.10), we obtain the

impurity Green function as follows.

Gασ(z) =(L−1ασ − L

−1σ (z) + F−1

σ (z))−1

. (E.12)

90

Since the coherent Green functionFσ(z) is defined byFσ(z) = [(z−Σσ(z)− t)−1]ii, it is obtainedfrom the following formula.

Fσ(z) =

∫ρ(ǫ)dǫ

z − Σσ(z)− ǫ. (E.13)

Hereρ(ǫ) is the density of states (DOS) for the energy eigen values of transfer matrixtij. We usethe above expression in Sect. 4.1 (Eq. 4.11).

The coherent potentialΣσ(z) is obtained from the condition that the configurational average ofthe impurity Green function(E.12) should be identical with the coherent Green function (see Fig.E.1).

G00σ(z) = Fσ(z) . (E.14)

Here

G00σ(z) =∑

α

cαGασ(z) . (E.15)

The equation (E.14) is known as the CPA equation which is essentially the same as Eq. (4.12) inSect. 4.1.

91

Bibliography

[1] Y. Kakehashi and M. Atiqur R. Patoary: J. Phys. Soc. Jpn.80 (2011) 034706.

[2] Y. Kakehashi: Modern Theory of Magnetism in Metals and Alloys (Springer Verlag Pub.,Berlin, 2013).

[3] T. Oguchi, K. Terakura, and A. R. Williams: Phys. Rev. B28 (1983) 6443.

[4] G. D. Mahan:Many Particle Physics (Plenum Pub., New York, 2000).

[5] A. Weiße and H.Fehske: in:Computational Many-Particle Physics, Lect. Notes Phys Ed. H.

Fehske, R. Schneider, and A. Weiße (Springer Verlag Pub., Berlin) 739(2008) chapter 18.

[6] K. Binder and D. W. Heermann:Monte Carlo Simulation in Statistical Physics An Introduc-tion (Springer Verlag Pub., Berlin, 2010).

[7] M. C. Gutzwiller: Phys. Rev. Lett.10 (1963) 159.

[8] M. C. Gutzwiller: Phys. Rev. A134(1964) 923.

[9] M. C. Gutzwiller: Phys. Rev. A137(1965) 1726.

[10] J. Hubbard: Proc. Roy. Soc. (London)A276 (1963) 238.

[11] J. Hubbard: Proc. Roy. Soc. (London)A277 (1964) 237.

[12] J. Hubbard: Proc. Roy. Soc. (London)A281 (1965) 401.

[13] G. Stollhoff and P. Fulde: Z. Phys. B26 (1977) 257.

[14] G. Stollhoff and P. Fulde: Z. Phys. B29 (1978) 231.

[15] G. Stollhoff and P. Fulde: J. Chem. Phys.73 (1980) 4548.

[16] P. Fulde:Electron Correlations in Molecules and Solids (Springer Verlag Pub., Berlin, 1995);

Correlated Electrons in Quantum Matter (World Scientific Pub., Singapore, 2012).

[17] P. Fulde: Adv. Phys.51 (2002) 909.

[18] W.F. Brinkman and T.M. Rice: Phys. Rev. B2 (1970) 4302.

[19] R. Jastrow: Phys. Rev.98 (1955) 1479.

[20] D. Baeriswyl: in:Nonlinearity in Condensed Matter, Ed. A. R. Bishop, D. K. Campbell, D.Kumar, and S. E. Trullinger, Springer-Verlag Series in Solids State Sciences69,(1987) 183.

92

[21] M. Dzierzawa, D. Baeriswyl, and L. M. Martelo: Helv. Phys. Acta70 (1997) 124.

[22] D. Baeriswyl: Found. Phys.30 (2000) 2033.

[23] B. Hetenyi: Phys. Rev. B82 (2010) 115104.

[24] Y. Kakehashi, T. Shimabukuro, and C. Yasuda: J. Phys. Soc. Jpn.77 (2008) 114702.

[25] M. Atiqur R. Patoary and Y. Kakehashi: J. Phys. Soc. Jpn.80 (2011) 114708.

[26] M A R. Patoary and Y. Kakehashi: J. Phys. Conf. Ser.391(2012) 012164.

[27] M. Atiqur R. Patoary, S. Chandra, and Y. Kakehashi: J. Phys. Soc. Jpn.82 (2013) 013701.

[28] M. Atiqur R. Patoary and Y. Kakehashi: J. Phys. Soc. Jpn.82 (2013) 084710.

[29] F. Kajzar and J. Friedel: J. de Phys.39 (1978) 397.

[30] G. Treglia, F. Ducastelle, and D. Spanjaard: J. de Phys.41 (1980) 281; G. Treglia, F.Ducastelle, and D. Spanjaard: J. de Phys.43 (1982) 341.

[31] H. Schweitzer and G. Czycholl: Z. Phys. B83 (1991) 93.

[32] H. Hellmann,Einfhrung in die Quantenchemie. Leipzig:Franz Deuticke(1937). p. 285 [inGerman]

[33] R. Feynman: Phys. Rev.56 (1939) 340.

[34] R. Bulla: Phys. Rev. Lett.83 (1999) 136.

[35] L. D. Landau: Sov. Phys. JETP.3 (1957) 920.

[36] A. A. Abrikosov and I. M. Khalatnikov: Reports Progr. Phys.22 (1959) 329.

[37] P. Nozieres and D. Pines:Theory Of Quantum Liquids, Vol. I (W. A. Benjamin, New York,1966).

[38] G. Baym and C. Pethick:Landau Fermi-Liquid Theory: Concepts and Applications (Wiley-interscience, New York, 1991).

[39] Y. Kakehashi and P. Fulde: Phys. Rev. B69 (2004) 045101.

[40] H. Shiba: Prog. Theor. Phys.46 (1971) 77.

[41] H. Ehrenreich and L. M. Schwartz, inSolid State Physics, edited by H. Ehrenreich, F. Seitzand D. Turnbull (Academic, New York, 1980), Vol.30.

[42] W. Metzner and D. Vollhardt: Phys. Rev. Lett.62 (1989) 324.

[43] N. Blumer:PhD thesis, University of Augsburg, 2002.

(www.physik.uni-augsburg.de/theo3/Theses/bluemer_color.pdf )

93