arX

iv:0

707.

3017

v3 [

cond

-mat

.str

-el]

5 N

ov 2

007

Diamagnetism of doped two-leg ladders and probing the nature of their commensurate phases

G. Roux,1,∗ E. Orignac,2 S. R. White,3 and D. Poilblanc11Laboratoire de Physique Theorique, IRSAMC, Universite Paul Sabatier, CNRS UMR5152,

118 Route de Narbonne, 31400 Toulouse, France.2Laboratoire de Physique de l’Ecole Normale Superieure de Lyon, ENS-Lyon, CNRS UMR5672,

46 Allee d’Italie, 69007 Lyon, France.3Department of Physics and Astronomy, University of California, Irvine, California 92697, USA.

(Dated: November 10, 2018)

We study the magnetic orbital effect of a doped two-leg ladder in the presence of a magnetic field componentperpendicular to the ladder plane. Combining both low-energy approach (bosonization) and numerical simu-lations (density-matrix renormalization group) on the strong coupling limit (t-J model), a rich phase diagramis established as a function of hole doping and magnetic flux.Above a critical flux, the spin gap is destroyedand a Luttinger liquid phase is stabilized. Above a second critical flux, a reentrance of the spin gap at highmagnetic flux is found. Interestingly, the phase transitions are associated with a change of sign of the orbitalsusceptibility. Focusing on the small magnetic field regime, the spin-gapped superconducting phase is robustbut immediately acquires algebraic transverse (i.e. alongrungs) current correlations which are commensuratewith the4kF density correlations. In addition, we have computed the zero-field orbital susceptibility for a largerange of doping and interactions ratioJ/t : we found strong anomalies at lowJ/t only in the vicinity of thecommensurate fillings corresponding toδ = 1/4 and1/2. Furthermore, the behavior of the orbital susceptibil-ity reveals that the nature of these insulating phases is different: while forδ = 1/4 a 4kF charge density waveis confirmed, theδ = 1/2 phase is shown to be a bond order wave.

PACS numbers: 74.20.Mn, 71.10.Pm, 75.40.Mg, 75.20.-g

I. INTRODUCTION

Ladder systems have proven to be remarkably interestingsystems, both as simple models exhibiting behavior simi-lar to 2D systems, and as systems exhibiting competitionbetween several types of ground states. Theoretical mod-els of doped ladders display a large superconducting (SC)phase1,2,3,4,5,6with d-wave pairing associated with the pres-ence of a spin gap, with a ground state which can be describedvariationally as a short-ranged resonating valence bond state.7

Charge density wave (CDW) correlations compete with thepairing correlations.5,6,8,9 The phase diagram of the isotropict-J model was sketched in Ref. 6 and displays, in addition tothis competition, insulating phases for the particular commen-surate dopingsδ = 1/4 and1/2. Another competition existsbetween the superconducting phase and an orbital antiferro-magnetic flux (OAF) phase5,10,11,12,13,14which has been ad-dressed in different ladder models by studying transverse cur-rent correlations which display a quasi-long range order intheOAF phase.

Ladders are also among the simplest systems through whicha magnetic flux can pass (see Fig. 1). When a magnetic fieldH is applied to an electronic system, it couples to both thespin of the electron, via Zeeman effect, and to the charge de-gree of freedom, via orbital effect. The total magnetic sus-ceptibility of a real material splits into various contributions15

χ = χspin+χorbcond+χ

orbcore, whereχspin is Pauli susceptibility and

χorbcond andχorb

core are the orbital susceptibilities of the conduc-tion and the core electrons.χorb

core must be evaluated from localatomic orbital and we will neglect it in what follows.χorb

cond isusually difficult to evaluate because one has to precisely de-scribe the evolution of the whole band structure with magneticfield.16,17 In the following, we will investigateχorb

cond within a

PSfrag replacements HH⊥

tJ

φ✛

✲φ✛

✲φ✛

✲φ✛

✲φ✛

✲φ✛

✲φ✛

✲φ✛

✲φ✛

✲

FIG. 1: (Color online) The isotropic t-J ladder under magnetic field.If the magnetic field has a component perpendicular to the ladderplane, a fluxφ passes through each plaquette. Below is the gauge ofthe Peierls substitution with opposite phases±φ/2 along the legs.

single-orbital model. When the magnetic field is applied par-allel to the plane of the ladder, the orbital effect is suppressedand only the Zeeman effect remains. The latter case has beendiscussed in details in this system and it was shown that adoping-controlled magnetization plateau and a large Fulde-Ferrell-Larkin-Ovchinnikov phase are obtained.18,19,20 Sincethe ladder possesses a spin gap, when the magnetic field is notin the ladder plane, the orbital contribution may dominate thespin contribution in the total susceptibility at low temperature.

Early numerical investigations of the t-J model with mag-netic orbital effect on ladders and 2D lattices revealed a strongeffect of the magnetic field on the magnetic and pairing prop-erties,21 but the results were limited to small systems. Abosonization study of a related model of spinless fermionicladders suggested the possibility of fractional excitations andof an OAF phase induced by the magnetic field.22,23 Carr andTsvelik9 studied the orbital effect of the magnetic field on theinterladder coupling using an effective model to describe a

2

single ladder. Lastly, it has been predicted24 that bosonic lad-ders could have commensurate vortex phases at commensu-rate fluxes which would represent a one-dimensional analogueof the two-dimensional vortex phase.

The purpose of the present paper is to consider the effectof a nonzero flux on the magnetic susceptibility on a singletwo leg ladder, and also to investigate the effect of strongerfluxes on the zero temperature phase diagram of the ladder. Tothis end, we combine the bosonization technique and density-matrix renormalization group25,26,27(DMRG, see Appendix Bfor details) to compute the phase diagram and physical prop-erties of a spin-1/2 fermionic ladder with orbital effect. Theresults are presented in two parts. The first part is devoted tothe analysis of the phase diagrams as a function of the fluxφ per plaquette in the weak- and strong-coupling limits. Thesecond part is focused on the physics of the spin-gapped phaseof doped ladder at small flux which is more relevant to real-istic magnetic fields. In particular, we discuss the stabilityof the insulating phases atδ = 1/4 and1/2 present in thephase diagram of the t-J model at zero flux. In the conclusion,we briefly give considerations on experiments which are con-nected to these results. For sake of clarity, we have relegatedsome of the technical details to appendices.

Including the flux

The magnetic flux couples to the kinetic part of the Hamil-tonian through Peierls substitution.17,28,29In what follows, dif-ferent hopping amplitudes along the chains (t‖) and betweenthe chains (t⊥) are considered, and the Hamiltonian is:

Ht = −t‖∑

i,σ

[

eiφ/2c†i+1,1,σci,1,σ + h.c.]

−t‖∑

i,σ

[

e−iφ/2c†i+1,2,σci,2,σ + h.c.]

−t⊥∑

i,σ

[

c†i,2,σci,1,σ + h.c.]

(1)

whereci,l,σ is the electron creation operator at sitei on leglwith spinσ. φ denotes the dimensionless flux per plaquette

φ =e

~

∮

�

A(x)dx =e

~H⊥a

2, (2)

with A(x) the vector potential which depends on the gaugechoice,a the lattice spacing. The magnetic field breaks time-reversal and chain exchange symmetries which, as expected,will have notable consequences on the current properties ofthe system. Symmetry and periodicity considerations allowus to limit the study to flux0 ≤ φ ≤ π. Exchanging chainsamounts to reversing the direction of the magnetic field. Moredetails on the gauge and flux quantization on a finite systemscan be found in Appendix B. The unit ofφ is 2πφ0 withφ0 = h/e = 4.1357× 10−15 T m2. For experimental consid-erations, a fluxφ = 0.01π already corresponds to a very highmagnetic field of H∼ 800 T for a typical valuea = 4 A ofthe Cu–O–Cu bond in a cuprate. The gauge chosen in Eq. (1)is represented on Fig. 1.

II. PHASE DIAGRAMS

A. Weak-coupling limit

We first introduce interactions between electrons using theHubbard ladder in magnetic flux. The Hamiltonian comprisesthe kinetic termHt incorporating the flux and the on-site re-pulsionU > 0:

H = Ht + U∑

i,p

ni,p,↑ni,p,↓ . (3)

According to the usual strategy,5 we will consider first thelimit U = 0 and study the non-interacting band structure.Then, we will turn onU ≪ t‖, t⊥ so that the band structureis not deformed and obtain the different phases in this weak-coupling limit. Let us begin with the discussion of the bandstructure.

The magnetic flux has a strong effect on the shape of thebands. Indeed, it mixes the bonding (0) and antibonding (π)bands which exist at zero flux. To emphasize the differencewith the zero field case, we call the two bands obtained atfinite flux the down (d) and up(u) bands. Results on the bandstructure are discussed in Refs. 22,23 and extended to a finiteand fixed filling in Appendix A. The band structure dependson the fluxφ and on the ratiot⊥/t‖ (see Fig. 2). We definetwo characteristic fluxesφc andφo, both of them dependent ont⊥/t‖, and such that aboveφc, a double well appears in bothbands and aboveφ0, a band gap opens between the bandsuandd. Four different possible shapes of the bands are obtainedaccording to the location of the flux with respect toφc andφo.The two critical flux lines crosses at(t⊥/t‖ =

√2, φ = π/2).

By filling these bands, one can show (see Appendix A) thatonly situations with either two or four Fermi points can occur.The location of these Fermi points and their respective Fermivelocities vary continuously with the flux. In the rest of the

0 1 2 30

π/2

π

PSfrag replacements

t⊥/t‖

√

2

A

B C

D

φ

φ0

φc

FIG. 2: The four typical shapes of the bands depending on the fluxand on the ratiot⊥/t‖. Critical fieldsφ0 (resp.φc) signal the appear-ance of a double-well (band gap). Note that the D phase alwayshasonly 2 Fermi points whatever the filling and that the C phasewhenφ = π always has 4 Fermi points.

3

0 π/2 π0

0.5

1

C0S1C1S2C0S0 (Band)C0S0 (Mott)

C2S2

C1S0

C1S0

C2S1

C1S1

0

c

PSfrag replacements

t⊥/t‖ = 1

n

φ

φ

φ

φ0

φc

FIG. 3: Phase diagram in the weak-coupling limit for an isotropicladder restricted to fillings0 ≤ n ≤ 1. Phases with 4 (resp. 2) Fermipoints fall generically into the C1S0 (resp. C1S1) class. Athalf-filling, interactions will drive the system into a Mott insulating phasefor φ < φ0 while a band insulating phase occurs whenφ > φc.Other phases can be found if the ratio of the Fermi velocity islarge(C2S1 and C2S2) and if thed-band Fermi wave-vector isπ/2 (C0S1and C1S2).

paper, we will work only at fixed electronic density (denotedby n) which will constrain the sum of the Fermi momenta asa result of the Luttinger theorem.30

Having obtained the non-interacting band structure, we addinteractions small enough not to perturb the band structure,following the strategy of Refs. 4,31,32,33,34. Adopting theusual notation CpSq for a phase withp gapless charge modesand q gapless spin modes, a system with two Fermi pointsand repulsive interactions is expected to be generically inaC1S1 phase, i.e. a Luttinger liquid state. With four Fermipoints and repulsive interactions, the system is generically ina C1S0 phase, i.e. a Luther-Emery liquid which is the univer-sality class of usual doped two-leg ladders. The critical fieldsat which the system changes from4 → 2 or 2 → 4 Fermipoints can by computed analytically (see Appendix A). Notethat from Refs. 4,33,34, in the case of 4 Fermi points, otherphases such as C2S1 or C2S2 appear when the difference be-tween the two Fermi velocities becomes sufficiently large toprevent runaway of some coupling constants in the RG flow.The large velocity difference implies that these phases areinthe vicinity of the transition region between the C1S0 and theC1S1 phase, where the Fermi velocity of the band that is emp-tying is going to zero. Moreover, this also implies that boththe C2S1 and C2S2 phases have a very small extent near thetransition region.4,33,34 The above considerations apply for asystem at a generic incommensurate filling. At commensu-rate filling, umklapp interactions can be relevant and lowerthe number of gapless modes.4,33,34More specifically, at half-filling, an insulating phase C0S0 of the Mott type (resp. Bandtype) is expected forφ < φ0 (resp.φ > φ0). It is also pos-

sible to have an umklapp interaction inside the bonding band(if its Fermi wavevector equalsπ/2) leading to either a C0S1or C1S2 phase.4 The phase diagram of an isotropic ladder re-sulting from these considerations is given on Fig. 3 where themain feature is a reentrance of the C1S0 phase at high flux.The high flux C1S0 phase has a band structure very similarto the one of the Hubbard chain with a next-nearest hoppingterm35,36 t′ for t′ & t/2 and the same competing orders asthe low flux C1S0 phase as we will see. This phase diagram isgeneric fort⊥/t‖ < 2 except thatφ0 < φc whent⊥/t‖ >

√2.

For t⊥/t‖ > 2, the C1S0 phase at low flux around half-fillingdisappears.

B. Strong-coupling limit: numerical results on the t-J model

We now let the interactions go to the strong-couplingregimeU ≫ t where the Hubbard model (3) reduces to thet-J model withJ ≃ 4t2/U . In this limit, we only use isotropiccouplingst = t‖ = t⊥ andJ = J‖ = J⊥ so that the t-JHamiltonian simply reads

Ht-J = PHtP + J∑

〈i,j〉

[Si · Sj −1

4ninj ] , (4)

whereSi is the spin operator andni = c†i,σci,σ is the elec-tronic density operator (leg index is omitted inSi andni). Pis the Gutzwiller projector which prevents double occupancyon a site. Observables are computed with DMRG for the rangeof doping0 < δ = 1 − n < 0.5. The phase diagram will bediscussed for the special caseJ/t = 0.5 for which the systemhas dominant superconducting fluctuations.2,6

1. Orbital susceptibility

The results for the non-interacting system of Appendix Ashow that the orbital susceptibility plotted in Fig. 15 changessign at the transitions from4 ↔ 2 Fermi points with sharpdiscontinuities (for0 < δ < 0.5). It is important to notethat the noninteracting orbital susceptibility contains contri-butions from all the occupied states and not just those at theFermi level which control the low-energy properties. There-fore, such connection of the change of sign of the orbital sus-ceptibility and a change in the number of Fermi point is notobvious. Nevertheless, we propose to extend this way of prob-ing the phase diagram to the interacting situation. Indeed,wecan compute the screening currentj‖ and its associated sus-ceptibilityχorb as a function of the fluxφ using the definitions

j‖(φ) = − 1

L

∂E0

∂φand χorb(φ) = − 1

L

∂2E0

∂φ2, (5)

in whichE0(φ) the ground-state energy andL is the lengthof the ladder. With this definition,χorb(φ) > 0 correspondsto orbital diamagnetism. The first relation is a consequenceof the Feynman-Hellman theorem and the second one resultsfrom the definition of the susceptibility as∂j‖/∂φ. This

4

0 π/2 πφ

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

C1S

0

C1S

1C

1S0

C1S

1

χorb (arb. units)

∆S/J

∆p/J

J/t = 0.5

FIG. 4: (Color online) Orbital susceptibility, charge and spin gapsfor J/t = 0.5 and on theδ = 0.063 line of Fig. 5. The zeros of thesusceptibility precisely probe the different two phase transitions.

screening current can be related to the mean value of thecurrent operatorsj1,2 along the two chains by noting thatj‖(φ) = 〈j2 − j1〉/2. Numerically, these quantities are com-puted directly from centered energy differences (to minimizediscretization effects)

j‖(φ) = −[E0(φ+ dφ) − E0(φ− dφ)]/(2Ldφ) (6)

χorb(φ) = [j‖(φ + dφ)− j‖(φ − dφ)]/(2dφ) , (7)

using the conditionsj‖(0) = j‖(π) = 0 (see Appendix B) andthe right and left derivatives forχorb(0) andχorb(π). Thesequantity are easy to compute numerically and are found tohave small finite size effects forJ/t = 0.5. The effect ofinteractions is to smooth the discontinuities at the transitionsbut we still expect that the sign-changes of the susceptibilitydo correspond to transitions between C1S0 and C1S1 phaseseven in the strong-coupling limit (see Fig. 4). The phase dia-gram obtained from this ansatz is consistent with the behaviorof other observables such as spin-spin correlation functions aswill be seen in the next paragraphs. Thus, we can sketch onFig. 5 a phase diagram similar to the one of Fig. 3 for the t-Jmodel withJ/t = 0.5. Compared with the weak-couplingphase diagram, the C1S1 phase is slightly wider at low dop-ing but thinner forδ ≃ 0.5. Even in the presence of stronginteractions, the overall shape of the phase diagram is not af-fected (although the precise location of the phase boundariesin theδ, φ plane does depend onU/t or J/t). Hence, for typ-ical densities0.5 < n < 1 in the isotropic t-J model, theleading effect which governs the phase diagram is the changeof the band structure under the applied flux. Lastly, one mustnote that the C1S0 phase persists at small flux at quarter-fillingδ = n = 0.5 in contrast to Fig. 3. This can be qualita-tively explained by noting that renormalization group studieson coupled chains have shown that interactions reduce the in-terchain hopping integralt⊥ with respect to its non-interactingvalue.37,38

0 π/2 π0.5

0.75

1

C1S0

C1S1

C1S0

0

0.25

0.5

PSfrag replacements

J/t = 0.5

δn

φ

φ0

φc

FIG. 5: Phase diagram of the t-J model forJ/t = 0.5, for whichthe system has dominant superconducting fluctuations at zero flux,determined from the zeros of the susceptibility on a system with L =32. Results are very similar to Fig. 3 with a transition to a C1S1phaseat intermediate flux and a reentrance to a C1S0 phase at large flux.

2. Elementary excitations : pairing energy and spin gap

The elementary excitations of a doped two-leg ladder are ei-ther the creation of a magnon, which cost is the spin gap∆S ,or the creation of two quasi-particles by breaking a Cooperpair, which cost defines the pairing energy∆p. FollowingRefs. 20,39, we compute them numerically from the defini-tions

∆S = E0(nh, Sz = 1)− E0(nh, S

z = 0) (8)

∆p = 2E0(nh − 1, Sz = 1/2)

−E0(nh, Sz = 0)− E0(nh − 2, Sz = 0) (9)

with E0(nh, Sz) the ground state energy of a system withnh

holes and spinSz along thez axis. Since one can alwayshave aSz = 1 state by breaking a Cooper pair, the condition∆S < ∆p must be satisfied in the thermodynamic limit. Forφ = 0 andJ/t = 0.5, it is known39 that the pair-breakingexcitation is larger than the magnon excitation (lowest tripletexcitation). These elementary gaps as a function of the fluxare displayed on Fig. 4. We observe a decrease of the pairingenergy withφ toward 0 at the critical flux corresponding tothe onset of the C1S1 phase. In the C1S1 phase, we have ametallic Luttinger liquid phase with zero pairing energy. Thecancellation of the pairing energy is thus the result of a bandemptying mechanism and should not be confused with a mag-netic superconducting critical fieldHc2 which would corre-spond to a high density of vortices in a 2D (albeit anisotropic)superconducting material. Indeed, we do not observe aHc1

superconducting critical field or commensurate vortex phasesas in the model of bosonic of Ref. 24. A situation correspond-ing to a trueHc2 critical field might rather be a small fluxthrough an array of coupled ladder. In this respect, the ap-proach of Ref. 9 copmutes correctly theHc2 field up to a fewapproximations.

The spin gap increases at low magnetic field until it crosses∆p (see Fig. 4). From local hole densities (data not shown),

5

1 10

x

10−9

10−6

10−3

S(x

)

0.5π0.63π0.75π0.81π

0 10 20 30 40 50 60

x

0.25π0.38π.........0.93ππ

0 π/2 πφ0

0.2

0.4∆

S/J

φ =

J/t = 0.5

ExpAlg

δ = 0.25

φ =L = 64

(a) (b)

FIG. 6: (Color online) Spin correlationsS(x) for fluxes in the C1S1phase(a) and in the C1S0 phases at low and high fluxes(b) in thephase diagram of Fig. 5 on theδ = 0.25 line. Insert: the behaviorof the spin gap computed on the same system using Eq. (8). Theseindependent observables confirm the reentrance of the C1S0 phase.

the domain of hole pairs slightly shrinks which can be inter-preted as a reinforcement of the spin-liquid background atlow magnetic field, in agreement with exact diagonalizationresults previously discussed.21 For larger flux but still in theC1S0 phase, the spin gap∆S becomes identical to the pair-ing energy∆p and both decrease towards zero as the fluxis increased. The energy difference∆p − ∆S can be inter-preted as the energy of a bound state of a magnon and a holepair. This magnetic resonant mode was discussed previouslyat zero flux by varying interactions39,40,41and its origin wasrelated with the opening of doping controlled magnetizationplateaus.19,20 Thus, the effect of adding Zeeman coupling atlow flux (φ . π/3 for J/t = 0.5) would give very similarresults to those of Refs. 19,20 since the bound state survivesto rather high magnetic flux. Finally, a small spin gap is re-covered at high flux (nearφ = π) in agreement with the weak-coupling limit predictions.

To gain further insights on these excitations we have com-puted the spin and pair correlation functions in the groundstate. The spin correlationsS(x) = 〈S(x) · S(0)〉 are short-range in a spin-gapped phase with a correlation lengthξ ∼1/∆S which gives complementary estimation of the evolu-tion of the spin gap, particularly important when the pairingenergy is smaller than the spin gap. From Fig. 6, we find asimilar increase of the spin gap (smallerξ) at small flux andalgebraic correlations in the C1S1 phase. The spin gap at highflux is again recover with short-range spin correlations.

The singlet operators on a rung defined by∆(x) =cx,1,↑cx,2,↓ − cx,1,↓cx,2,↑ gives the pairing correlationsP (x) = 〈∆(x)∆†(0)〉. While P (x) remains algebraic in theC1S0 and C1S1 phases, its overall magnitude follows the pair-ing energy and is strongly reduced in the C1S1 phase. Thesecorrelations can be discussed from the bosonization approachusing the conventions and definitions of Appendix C. We findthat the low-energy dominant term at wave-vectorq = 0 in

1 10 70

x

10−8

10−6

10−4

10−2

P(x

)

φ = 0.25πφ = 0.75πφ = 0.94π δ = 0.063

J/t = 0.5L = 64

1/x 1.9

1/x0.9

1/x2

FIG. 7: (Color online) Superconducting correlationsP (x) for threevalues of the flux corresponding to each phases encountered on theδ = 0.063 line of the phase diagram of Fig. 5. The decrease ex-ponent is much higher in the C1S1 and high-flux C1S0 phases andthe overall magnitude is strongly reduced suggesting a metallic state.Note that correlations display oscillations associated with a4kF con-tribution.

the low-field C1S0 phase reads

∆(x) ∝∑

σ

σ[

(ad)2ψd,R,σψd,L,−σ + (bd)

2ψd,L,σψd,R,−σ

−(bu)2ψu,R,σψu,L,−σ − (au)

2ψu,L,σψu,R,−σ

]

.

For instance, the intraband terms read

ψp,R,σψp,L,−σ ∼ ei[θc++pθc−−σ(φs++pφs−)] , (10)

with p = ± for d/u. From previous results,5 we know thatin a C1S0 phase all the fields exceptφc+ are gapped, with〈θc−〉 = 0 and 〈φs+〉 = π/2, 〈φs−〉 = π/2. These termsare thus algebraic with a decay exponent1/(2Kc+) which isthe continuation of the zero-flux physics. We observe fromFig. 7 thatKc+ increases with the magnetic field but a preciseevaluation is numerically difficult.

In the C1S1 phase, superconducting correlations are ex-pected and found to be algebraic with an exponentK−1

c +Ks

but with a much smaller amplitude, in agreement with a metal-lic phase.

The physical properties of the high-flux C1S0 phase arevery similar to that of the low-field C1S0 phase. Followingthe analysis by Fabrizio35 (see Appendix C for notation), thepairing order parameter in this phase reads :

∆(x) ∝∑

σ

σ[

(a1)2ψ1,R,σψ1,L,−σ + (b1)

2ψ1,L,σψ1,R,−σ

+(b2)2ψ2,R,σψ2,L,−σ + (a2)

2ψ2,L,σψ2,R,−σ

]

.

which will give fluctuations with an exponent1/(2Kc+).Computing the density and transverse current order parame-ters shows that they have a2Kc+ decay exponent associatedwith the wave-vector2(kF,1 − kF,2) = 2πn. The compet-ing orders are thus the same as in the low field phase studied

6

in Sec. III. The fact that the SC signal is small and with alarge exponent in Fig. 7 suggests that the CDW fluctuationsdominates in this strong-coupling regime making the high fluxC1S0 phase a spin-gapped metallic phase with strong trans-verse current fluctuations.

III. LOW-FIELD PROPERTIES OF THE LUTHER-EMERYPHASE

This section discusses the properties of the C1S0 Luther-Emery phase at very low fluxes relevant to the experimentalaccessible magnetic fields.

A. Current densities and correlations

With open boundary conditions used in DMRG, we haveaccess to the local density of holes and currents. The localhole density readsh(x) = 1 − 〈n(x)〉 while the mean valuesof local parallel and transverse current operators are computedusing the definitions

j1‖(x) = it‖[eiφ/2c†x+1,1,σcx,1,σ − e−iφ/2c†x,1,σcx+1,1,σ]

j2‖(x) = it‖[e−iφ/2c†x+1,2,σcx,2,σ − eiφ/2c†x,2,σcx+1,2,σ]

j⊥(x) = it⊥[c†x,1,σcx,2,σ − c†x,2,σcx,1,σ] . (11)

They are related to the current operatorjp along the chainpvia jp = 1

L

∑

x jp‖ (x) and thus to the screening currentj‖(φ).

We have checked that Kirchhoff’s conservation law for chargecurrents is satisfied at each vertex of the lattice. At zero flux,no currents are present in the ladder. When the magnetic fieldis applied, time-reversal symmetry is broken and local cur-rents have a non-zero expectation value depicted in Fig. 8 fora2×32 ladder with 4 holes. First, the two hole pairs manifestthemselves by two domains (areas with open circles). The lo-cal screening currents develop inside these domains and notatthe edges of the ladder. Clearly, in the strong-coupling limitwhere double occupancy is prohibited, the only domains inwhich electrons can take advantage of the flux are the roomleft by holes. This results in a periodic pattern for hole cur-rents whose length scale is exactlyδ−1 (= 16 in Fig. 8).Such a length scale is different from the usual magnetic length√

~/eH governing orbits of Landau levels. A similar lengthscale has been found in the study of OAF phases14 while thecurrent pattern is different from the one observed under mag-netic field. Note that these diamagnetic currents are not re-lated to the Meissner effect expected in a superconductor orina bosonic ladder.24 Actually, hole pairs are delocalized on thetwo chains so that this pattern does not correspond to currentsof pairs but rather to currentsinsidepairs.

To study the nature of the current fluctuations, we havecomputed the transverse current correlations

J(x) = 〈j⊥(x)j⊥(0)〉 − 〈j⊥(x)〉〈j⊥(0)〉 , (12)

where we have subtracted the finite local expectations (asone would do with density correlations). The main result of F

IG.

8:(C

olor

onlin

e)L

ocal

hole

dens

ityan

dlo

calc

urre

nts

inth

egr

ound

stat

eof

the

C1S

0ph

ase

(at

low

flux)

.T

hear

eaof

the

circ

les

ispr

opor

tiona

ltoh(x)−

δw

ithh(x)

the

loca

lde

nsity

ofho

les.

An

exce

ssof

hole

isre

pres

ente

dby

empt

yci

rcle

sw

hile

ala

ckof

hole

sis

repr

esen

ted

byfu

llci

rcle

s.T

he

line

thic

knes

sis

prop

ortio

nalt

oth

ecu

rren

tst

reng

than

dth

ear

row

give

sth

edi

rect

ion.

Not

eth

atth

etr

ansv

erse

curr

ent

has

been

resc

aled

(by

afa

ctor(×5))

,the

mai

nsc

reen

ing

curr

ents

are

onth

ech

ains

.In

tere

stin

gly,

the

orbi

tsha

vea

leng

thsc

ale

δ−1

dete

rmin

edby

the

aver

age

hole

dens

ity.

❄ ✲✛❄ ✲✛

❄ ✲✛❄ ✲✛

❄ ✲✛❄ ✲✛

❄ ✲✛❄ ✲✛

❄ ✲✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛❄ ✲✛

❄ ✲✛❄ ✲✛

❄ ✲✛❄ ✲✛

❄ ✲✛❄ ✲✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻ ✲

✛

✻②

②①

✈t

❜❡

❢❣

❣❣

❢❞

❛r

ss

r❛

❞❢

❣❣

❣❢

❡❜

t✈

①②

②

②②

①✈

t❜

❡❢

❣❣

❣❢

❞❛

rs

sr

❛❞

❢❣

❣❣

❢❡

❜t

✈①

②②

Plo

toflo

calcu

rren

tsj ⊥

(x)

(×5),

j ‖(x

),and

hole

den

sity

h(x

)in

a2×

32

ladder

with

δ=

1/16,J/t=

0.5

and

φ/2π

=0.1

25.

⊗H

s0.0

10.0

1

FIG

.9:

(Col

oron

line)

Loc

alex

pect

atio

nva

lues

with

the

con

vent

ions

ofF

ig.8

onth

en=

δ=

1/2

line

(qua

rter

-filli

ng)

inth

ebo

nd-d

ensi

ty-w

ave

phas

eat

low

flux.

✻✛✲

❄ ✲✛

✻✛✲

❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻ ✲

✛❄ ✲✛

✻✛✲

❄ ✲✛

✻✛✲

❄✇

❞♣

❜❵

❛❵

❛❵

❵❵

❵❵

❵❵

❵❵

❵❵

❵❵

❵❵

❵❛

❵❛

❵❜

♣❞

✇

✇❞

♣❜

❵❛

❵❛

❵❵

❵❵

❵❵

❵❵

❵❵

❵❵

❵❵

❵❵

❛❵

❛❵

❜♣

❞✇

Plo

toflo

calcu

rren

tsj ⊥

(x),

j ‖(x

),and

hole

den

sity

h(x

)in

a2×

32

ladder

with

δ=

0.5

,J/t=

0.2

and

φ/2π

=0.0

625.

⊗H

✉0.0

20.0

1

7

Ref. 11 was the absence of algebraic transverse current corre-lations in a C1S0 phase because of the strong spin fluctuationsassociated with the spin gap. Here, although the spin-gappedC1S0 phase survives at low flux, the situation is quite differ-

ent because the chain exchange symmetry is explicitly brokenby the magnetic field. Indeed, Eq. (11) can be rewritten in thebasis of thed, u bands, which leads in the continuum limit toterms with different wave-vectors:

j⊥(x) = it⊥

[

(a2u − b2u)[e−2ikuxψ†

u,R,σψu,L,σ − e2ikuxψ†u,u,σψu,R,σ] + (a2d − b2d)[e

−2ikdxψ†d,R,σψd,L,σ − e2ikdxψ†

d,L,σψd,R,σ]

+(buad + aubd)[e−i(kd+ku)x(ψ†

u,R,σψd,L,σ − ψ†d,R,σψu,L,σ) + ei(kd+ku)x(ψ†

u,L,σψd,R,σ − ψ†L,d,σψu,R,σ)]

+ (bubd + auad)[e−i(ku−kd)x(ψ†

u,R,σψd,R,σ − ψ†d,L,σψu,L,σ) + ei(ku−kd)x(ψ†

u,L,σψd,L,σ − ψ†d,R,σψu,R,σ)]

]

(13)

where the coefficientsad/u, bd/u are defined in Appendix C.We now turn to the bosonization representation of the opera-tors appearing in Eq. (13). Terms with the lowest wave-vectorkd − ku contain operators of the form

ψ†u,R,σψd,R,σ ∼ ei[−φc−+θc−−σ(φs−−θs−)] (14)

with σ = (↑, ↓) = ±. In the C1S0 phase, their correlationfunctions decay exponentially as they involve the dual fieldsθs− andφc− which are disordered. For the terms with thewave-vectorkd + ku, we find similarly that

ψ†u,R,σψd,L,σ ∼ ei[φc++θc−+σ(φs++θs−)] (15)

which also have short ranged correlation functions becauseof the presence of the disordered fieldθs− in the bosonizedrepresentation Eq. (15). This is exactly the same result asin Ref.11 albeit extended to nonzero flux. Without magneticfield, DMRG calculations11 showed that the dominant wave-vector in the exponential signal wask0+kπ rather thank0−kπprobably because the bosonized expression of the correspond-ing Fourier component contains two strongly fluctuating fieldfor the latter term (see Eq. (14) , but only one (see Eq. (15)) forthe former term. In the presence of a magnetic field, we seethat the terms in Eq. (13) corresponding to the wavevectors2ku and2kd have a magnitude

(bp)2 − (ap)

2 ∝ φ

at small fluxφ, i.e. they exactly cancel forφ = 0 but arepresent once the magnetic field is turned on. These new termare allowed by the symmetry reduction induced by the mag-netic field. Furthermore, they have the bosonized expression:

ψ†p,R,σψp,L,σ ∼ ei[φc++pφc−+σ(φs++pφs−)] , (16)

in which the spin contribution has long range order, but thecharge contribution contains the dual fieldφc− leading to ex-ponential decay of the associated correlation function. There-fore, all the “2kF ” contributions in the transverse current cor-relation functions display an exponential decay.

In the expression (13), the higher “4kF ” harmonics of thecurrent are not taken into account. The reason for this is thatin the full Hilbert space, the4kF component ofj⊥ is simply

proportional toc†3kFc−kF . However, in bosonization, a mo-

mentum cutoff is introduced, and the high energy states whichinvolve the creation or annihilation of fermions with momen-tum farther from the Fermi momenta than the cutoff are elim-inated. Thus, the expression of the current at lowest order inthe interactionU/t in Eq. (13) cannot contain any4kF contri-butions. However, virtual processes in which a fermion is cre-ated and annihilated far from the Fermi points give contribu-tions of higher order inU/t to the4kF components that onlyinvolve fermion operators close to the Fermi points. Such con-tributions can be derived in perturbation theory followingtheapproach in Refs.42,43. In the case of the transverse current,an interaction of the formUc†kF

c†kFc−kF c3kF yields a pertur-

bative contribution proportional toU/t c†kFc†kF

c−kF c−kF tothe4kF component which involve only operators belonging tothe low energy subspace and thus cannot be neglected. Suchcorrections can be viewed as a pair hopping or a correlatedhopping between the chains. Thus, we expect to find a4kFcontribution to the transverse current of the form:

ψ†d,R,σψd,L,σψ

†u,R,σψu,L,σ ∼ ei2φc+ , (17)

1 10x

10−8

10−6

10−4

φ = 0.25π1/x

2

|J(x)|

0 10 20 30 40 50 60x

−5×10−4

0

5×10−4

x2J(x)

δ−1

J/t = 0.5 and δ = 1/16

(a) (b)

FIG. 10: (a) Transverse current correlations become algebraic in theC1S0 phase once the magnetic field is turned on (same parameters asin Fig. 8). (b) Demodulation of the signal enables to extract clearlythe wave-vector2πδ of the correlations.

8

TABLE I: Summary of the bosonization result for operators inthelow field C1S0 phase (see Appendix C for notations). We have2kF = πn. If not short-range (”exp.” notation), we give the decayexponent of the associated correlations. Numerically,〈n(x)n(0)〉and〈j⊥(x)j⊥(0)〉 are algebraic because they pick up the 4kF terms.

in the C1S0 phase

Operator exponent wave-vector

Sz(x) exp. 2kF

∆(x) 1/(2Kc+) 0

n2kF (x) exp. 2kF

n4kF (x) 2Kc+ 4kF

j⊥,2kF (x) exp. 2kF

j⊥,4kF (x) 2Kc+ 4kF

associated with the wave-vector2(kd + ku) = 2π(1 − δ).The〈j⊥,4kF (x)j⊥,4kF (0)〉 correlations have a power-law de-cay with exponent2Kc+. This result is very similar to theCDW fluctuations associated with the correlations〈n(x)n(0)〉which, while being short-range at2kF , also possess a4kFpower-law decay.6 Note that these CDW correlations con-tain terms analogous to (14, 15, 16) but with different pref-actors. This is in good agreement with numerical results forwhich we found algebraic correlations with wave-vector2πδ.The wave-length of the correlations is again associated to thelength scaleδ−1 of the local hole and transverse current pat-terns. We found numerically a larger Luttinger exponent fromthe current correlationsKc+ ∼ 1 while superconducting cor-relations rather giveKc+ ∼ 0.6 for the same parameters. Thisdifference, also found for charge correlations,2 could be at-tributed to the need for virtual high energy processes to createthe4kF correlations, leading to somehow low and noisy sig-nals. The behavior of the bosonized operators is summarizedin table I.

B. Zero-field susceptibility and the commensurate phases

Since only very small flux per plaquetteφ can be achievedexperimentally, we now focus on theφ → 0 limit of the or-bital susceptibilityχ0 ≡ χorb(0) which is calculated numeri-cally from Eq. (5). For the non-interacting system, this quan-tity is finite and positive at half-filling and then increaseswithdoping (see Fig. 16 of Appendix A). Once interactions areturned on, this susceptibility is zero at half-filling because ofthe Mott insulating state (see Fig. 11 and Appendix A for ageneral discussion of the susceptibility). When doped, thesystem acquires a susceptibility roughly proportional to den-sity of charge carriers withχ0 ∼ δ. The proportionality coef-ficient decreases withJ/t which is reminiscent of the fact thatlargeJs reduce the mobility of holes (see insert of Fig. 11).Compared with the non-interacting result, the susceptibility isthus strongly reduced by the interaction. When the ratioJ/tis lowered, a strong reduction ofχ0 is clearly visible for thehole commensurate dopingδ = 1/4 up to finite size effectsdiscussed in Appendix B. This drop of the susceptibility in-

0 0.25 0.5δ

−1

0

1

2

3

χ0

J/t = 0.4J/t = 0.25

0.25 0.5 0.75 1

J/t

3

4

5

6

7

8

dχ0/dδ|δ=0

strong finitesize effects

FIG. 11: (Color online) Zero field susceptibilityχ0 as a function ofdopingδ and interactions parametersJ/t for a ladder withL = 48.A strong reduction at lowJ/t and commensurabilitiesδ = 1/4 and1/2 are clearly visible. The computed susceptibility is nearlyzero atδ = 1/4 but subject ot finite size effects (see Appendix B). On thecontrary, theδ = 1/2 phase has a finite susceptibility.Insert: thederivativedχ0/dδ|δ=0 as a function ofJ/t.

creases continuously asJ/t is lowered (data not shown, forlargerJ/t, the convergence is better) so that we are confidentthat the observation is not an artifact of the finite size effects.For δ = 1/2, a discontinuity of the slope is found but thesusceptibility remains finite in this phase (there, the finite sizeeffects are smaller). The occurrence of insulating CDWs waspreviously studied6 in this part of the(δ, J/t) phase diagramof the t-J model. However, only theδ = 1/4 CDW phase wasdiscussed and Fig. 11 suggests that theδ = 1/2 phase is of adifferent nature.

0 0.25 0.5δ

0

1

2

∆ 2p/J

J/t = 0.75; φ=0J/t = 0.75; φ=4π/LJ/t = 0.4 ; φ=0J/t = 0.4 ; φ=4π/LJ/t = 0.25; φ=0J/t = 0.25; φ=4π/L

L = 48

FIG. 12: (Color online) Two-particle charge gap as a function ofdoping showing the emergence of insulating phases at commensura-bilities δ = 1/2 andδ = 1/4 at lowJ/t. The smallest flux we haveaccess to is enough to destroy theδ = 1/4 CDW phase (see textfor details), while theδ = 0.5 BOW phase is more stable. Data arecomputed with an extraJ = +0.3 at the two extremal rungs.

9

To study more precisely the occurrence of these phases, wehave computed the two-particle charge gap

∆2p = E0(nh + 2) + E0(nh − 2)− 2E0(nh) (18)

as a function of doping and interaction at zero magnetic fieldand for the lowest flux4π/L we have access to. A systemwith pairs always has a finite one-particle charge gap (or pair-ing energy), but is insulating if the two-particle charge gap isalso finite. The results on a ladder of finite lengthL = 48are given on Fig. 12. On the one hand, a strong disconti-nuity at δ = 1/2 is found even for rather largeJ/t and thecharge gap of this phase survives to nonzero flux (away fromthe commensurability,∆2p is much smaller but finite becauseof the finite length of the system). On the other hand, theδ = 1/4 discontinuity is only found at smallJ/t and is de-stroyed for the lowest flux we can use. No other discontinuityof the two-particle charge gap is found for the range of doping0 ≤ δ ≤ 0.6. The system withδ = 1/4 has edge effects andwe have added an extraJ⊥ = +0.3 on the two extremal rungsto control the spinons at the edges as it was done in Ref.6. Thisphase is thus difficult to study under magnetic flux but it wasstudied previously and it was proposed to be a four-fold de-generate CDW phase with pairing and a small spin gap on thebasis of the behavior of the Friedel oscillations. These gapswere found to be numerically very small.6 The observed sen-sitivity to the flux is consistent with small gaps. Indeed, sucha four-fold degenerate phase is difficult to stabilize. Qualita-tively, if pairs of holes are well-formed on rungs, it is hardtogenerate an effective long range repulsion between these pairsto stabilize a crystal of hole pairs. On the contrary, if pairs arespread over a few rungs, they can repel each other more easilybut will have a smaller spin gap and pairing energy. This latestpicture of a pair of holes delocalized on a plaquette every twoplaquettes seems to be more suited to describe this phase.

In the insulating phase withδ = 1/2 (quarter-filling),instead of the pronounced Friedel oscillations obtained forδ = 1/4, a uniform electronic density (see Fig. 9) is found.However, if one computes the bond order parameterstν(x)along the bonds at zero magnetic field by using the definitions

t1,‖(x) = t‖〈c†x+1,1,σcx,1,σ + c†x,1,σcx+1,1,σ〉t2,‖(x) = t‖〈c†x+1,2,σcx,2,σ + c†x,2,σcx+1,2,σ〉t⊥(x) = t⊥〈c†x,1,σcx,2,σ + c†x,2,σcx,1,σ〉 ,

one finds strong oscillations with a period of two lattice sitesfor the ‖ bonds making this phase an insulating bond or-der wave (BOW) phase (see Fig. 13). This is confirmedby the current pattern under magnetic field found in Fig. 9which has well-defined orbits around plaquettes but smallcurrents between plaquettes. The local transverse currentisstaggered while the transerve bond density wave order pa-rameter is uniform. Such local orbits allow a finite orbitalsusceptibility even if the system remains insulating because〈j⊥(x)j⊥(0)〉 ≃ 〈j⊥(1)j⊥(0)〉 in Eq. (D2) which is simplythe local response on a plaquette.

0 6 12 18 24

x

0.4

0.5

0.6

t2(x)

t1(x)

tperp

(x)

δ = 0.5J/t = 0.5

FIG. 13: (Color online) Local kinetic bondstν(x) in the δ = 1/2bond order wave (BOW) phase computed at zero magnetic field. Theparallel bond orders are strongly oscillating at wave-vector π whilethe transverse bond and the transverse kinetic bond and electronicdensities are uniform (see Fig. 9).

IV. CONCLUSION

We have studied the effect of a magnetic flux through adoped two-leg ladder by means of bosonization and DMRGcalculations. As a function of the flux, a rich phase diagram isobserved with an intermediate Luttinger liquid phase and thereentrance of the Luther-Emery phase at high flux. Both in theweak- and in the strong-coupling limits, the phase diagram isgoverned by the evolution of the band structure. Focusing onthe small field physics of the Luther-Emery phase, we observethat local currents develop in the ladder inside the hole pairsregions. Their typical length-scaleδ−1 is controlled by holedoping δ. The transverse current correlations also developas soon as the magnetic field is turned on with an algebraicbehavior contrary to what was found without magnetic field.Lastly, we have computed numerically the zero-field suscepti-bility of the system as a function of the interaction parameterJ/t and of hole doping. We found that insulating commen-surate phases at lowJ/t exists only at dopingsδ = 1/4 and1/2 and that the two phases have different responses undermagnetic field. The contribution of the conduction electronsto the orbital susceptibility might thus be useful to probe thesephases experimentally. Results on theδ = 1/4 phase are con-sistent with a four-fold degenerate ground state with a smallpairing and spin-gap, making it very sensitive to the flux. Onthe contrary, theδ = 1/2 phase appears to be a robust bond or-der wave phase with a two-fold degenerate ground state. De-spite its insulating nature, this phase has a finite susceptibilitydue to local orbits of electrons around plaquettes.

The ladder compound Sr14−xCaxCu24O41 (SCCO) wasthe first non-square cuprate compounds showing super-conductivity under high pressure.44 The presence of aspin gap in its superconducting phase has been addressedexperimentally45,46,47,48 but no consensus has risen on theactual nature of superconductivity in this material. An-other exciting feature of SCCO is the occurrence of charge

10

density waves at ambient pressure.49,50,51,52,53,54,55,56,57,58

Experiments57,58 have suggested that CDW could appear atwave-vectors 1/3 and 1/5 which was discussed theoreticallyusing a multiband charge transfer model solved by Hartree-Fock approximation.59 Here, we have showed that the t-Jmodel on a single ladder only displays 1/4 and 1/2 commensu-rabilities, as proposed in Ref. 6, and that orbital susceptibilitycould help to understand the nature of these commensuratephases.

We would like to point out that an interesting realization ofquasi-one dimensional systems in which magnetic flux can af-fect the band structure is provided by carbon nanotubes.60 Fol-lowing the theoretical prediction,60 experiments on multiwallnanotubes (where notable fluxes can be achieved due to thelarge diameter of the outmost shell) have shown that the bandstructure of these systems was indeed sensitive to magneticfluxes.61,62,63 In the case of gapped zig-zag single wall nan-otubes, although the experimentally accessible fluxes throughthe tubes were small, an effect on the conductance oscillationsin the Fabry-Perot regime could nevertheless be evidenced asa result of the lifting of the degeneracy between two subbands.As there exists some evidence for strong electronic correla-tions in carbon nanotubes,64,65,66,67,68and as carbon nanotubespossess some analogies with ladders,69 an interesting exten-sion of the theoretical results developed in the present paperwould be to study quasi-one dimensional models mimickingmore closely carbon nanotubes. It would be particularly in-teresting to compute the behavior of the Luttinger exponentcontrolling the Zero Bias Anomaly as a function of the ap-plied field.

Acknowledgments

GR would like to thank IDRIS (Orsay, France) andCALMIP (Toulouse, France) for use of supercomputer facil-ities. GR and DP thank Agence Nationale de la Recherche(France) for support. GR and SRW acknowledge the supportof the NSF under grant DMR-0605444.

APPENDIX A: FORMULAS FOR THE NON-INTERACTINGSYSTEM

Part of these results were first given in Refs. 22,23. We re-produce them for clarity and notation conventions (which aredifferent) and extend them when necessary. In what followsα = t⊥/t‖.

1. Band structure

At zero flux, the interchain coupling lifts the degeneracy be-tween chains giving birth to a bonding bandky = 0 and anti-bonding bandky = π. The flux breaks the reflection symme-try between chains and couples these0 andπ modes. We calldown and up (with labelsd/u) the two bands in presence ofa flux. It is straightforward to diagonalize the non-interacting

Hamiltonian by taking the Fourier transform, which gives theenergies

Ed/u(k, φ) = −2t‖

{

cos k cosφ

2±√

sin2 k sin2φ

2+(α

2

)2}

.

(A1)The basis transformation can be written using coherence fac-torsak, bk > 0

(

ck,1ck,2

)

=

(

ak bkbk −ak

)(

ck,dck,u

)

, (A2)

with

a2k =1

2

1− sink sin φ2

√

sin2 k sin2 φ2 +

(

α2

)2

(A3)

b2k =1

2

1 +sink sin φ

2√

sin2 k sin2 φ2 +

(

α2

)2

. (A4)

We havea−k = bk and the factors depend on the wave-vectork while at zero fluxak = bk = 1/

√2. For any finite flux and

k > 0 we haveak < bk with

b2k − a2k =sink sin φ

2√

sin2 k sin2 φ2 +

(

α2

)2. (A5)

Below are a few considerations on the band structure (A1)which are summarized in Fig. 2:

(i) The condition to have a band gap is thatmax Ed =Ed(π, φ) < min Eu = Eu(0, φ) which givesα

2 = cos φ2 . This

condition can be reformulated asφ > φ0 with

sinφ02

=

√

1−(α

2

)2

. (A6)

(ii) The condition to have a double well in theEd bandcomes from the sign of the second derivative atk = 0 and

d

u

FIG. 14: Depending on the filling, the number of Fermi points can beeither 2 or 4 (sketched on the bands of the B phase of Fig. 2). Notethat at low and high filling, the two Fermi velocities have oppositesigns while at intermediate filling, they have the same sign.

11

is α2 cos φ

2 = sin2 φ2 . This condition can be reformulated as

φ > φc with

sinφc2

=

[√α4 + 16α2 − α2

8

]1/2

. (A7)

In this case, the wave-vectors corresponding to the minimumof the down band and the maximum of the up band read

kdmin(φ) = arcsin

√

sin2φ

2−(α

2

)2

cot2φ

2(A8)

kumax(φ) = π − kdmin(φ) (A9)

which appears with a finite value.(iii) The two curves intersection isα =

√2 for which we

haveφ0/c = π/2. If α < αc we haveφc < φ0, elseφ0 < φc.

(iv) The condition to empty theu band isµ = α − 2 cos φ2

and is the same as the condition for which thed band is com-pletely filled.

(v) We can show that only situations with two or four Fermipoints can occur: for monotonous bands (φ < φc), this isobvious. For non-monotonousdispersion (φ > φc), if φ > φ0,the two bands are not overlapping so we have either two orfour Fermi points. For any fluxφ ∈ [φc, φ0], we can convinceourselves that sinceEu(k, φ) > Ed(k, φ) (for anyk) and sincethe up band has a unique maximum (atkumax(φ)) while thedown band has a unique minimum (atkdmin(φ)), it necessarilyimplies that only two or four Fermi points are allowed (seeFig. 14).

2. Filling the bands : finding Fermi points

Fermi pointskd/uF are deduced from their relation to thechemical potentialµ from Eq. (A1). If needed, this equationcan be inverted into

cos kd/uF (µ, φ) = − µ

2t‖cos

φ

2

± sd/u

√

√

√

√

[

1−(

µ

2t‖

)2]

sin2φ

2+(α

2

)2

.

(A10)

This latest is useful when working at fixedµ. Note that, de-pending onφ andµ, the above equation can have two rootslabelled bysd/u = ±1 for eachsectord, u (see Fig. 14). Ifthere are two Fermi points in the same bandp, we use thenotationkpF,1/2. The Fermi velocities can be evaluated from

vd/uF (k, φ) = 2t‖ sin k

cosφ

2∓ cos k sin2 φ

2√

sin2 k sin2 φ2 +

(

α2

)2

.

(A11)Note that one of the two Fermi velocities is negative in caseof non-monotonous bands.

Working at fixed electronic densityn, we have to relate theFermi points directly ton using Eq. (A1) and the Luttingersum rule (see Fig. 14 for sketch of all possible situations).Ifthe system has only two Fermi points, then we have eitherkdF = πn or kuF = π(n − 1), where both relations do notdepend on the flux. When there are four Fermi points andlimiting the discussion ton ≤ 1, we have either that:(i) the bands are overlapping:

kuF + kdF = πn

sin

(

kuF − kdF2

)

=

[

(α2 )2 cos2 φ

2

sin2(πn2 )− sin2 φ2

+ sin2φ

2

]1/2

.

(ii) there is a non-monotonous dispersion, for the down band:

kdF,1 − kdF,2 = πn

sin

(

kdF,1 + kdF,2

2

)

=

[

(α2 )2 cos2 φ

2

sin2(πn2 )− sin2 φ2

+ sin2φ

2

]1/2

.

These equations reproduce the correct result in theφ = 0 andα = 0 limits. It is also straightforward to compute the 4 crit-ical densities at which the number of Fermi points changesfrom 2 to 4 and 4 to 2 (see Fig. 3). By arranging them accord-ing tonc1 < nc2 < nc3 < nc4, we have

nc1 = arccos(cosφ+ α cos(φ/2))/π if φ ∈ [φc, π]

nc2 = arccos(cosφ− α cos(φ/2))/π if φ ∈ [0, φ0]

nc3 = 2− nc2 if φ ∈ [0, φ0]

nc4 = 2− nc1 if φ ∈ [φc, π] (A12)

These equations can be inverted to give the critical fluxφ2↔4

at which the transitions from 2↔ 4 Fermi points occur:

cosφ2↔4

2=

1

2

[

±α2+

√

(α

2

)2

+ 4 cos2(πn

2

)

]

(A13)

this last expression is useful to check that the change of signof the susceptibility is associated with these transitions.

3. Orbital susceptibility

Knowing the location of the Fermi points , we can com-pute quantities integrated over the bands such as the totalenergyE0(φ), the screening currentj‖(φ) and the associ-ated orbital susceptibility from Eqs. (5). One contribution tothis current from electrons of momentumk is jd/u‖ (k, φ) =

∂Ed/u(k, φ)/∂φ, which reads:

jd/u‖ (k, φ) = −t‖

cos k sinφ

2∓ 1

2

sin2 k sinφ√

sin2 k sin2 φ2 +

(

α2

)2

,

(A14)

12

0 π/2 π

0

PSfrag replacements

φ

φ4→2

φ2→4

t⊥/t‖ = 1

χorb

n = 0.7

FIG. 15: Susceptibility as a probe to the number of Fermi points for0.5 < 1− δ < 1.0 andt⊥ = t‖. φ2↔4 indicate the transitions from4 to 2 and 2 to 4 Fermi points.

and similarly, the corresponding contribution to the suscepti-bility reads

χorbd/u(k, φ) = − t‖

2

cos k cosφ

2∓ sin2 k cosφ[

sin2 k sin2 φ2 +

(

α2

)2]1/2

±1

4

sin4 k sin2 φ[

sin2 k sin2 φ2 +

(

α2

)2]3/2

.

(A15)

It is important to remark that additional contributions comefrom the derivatives∂kp(φ)/∂φ and∂2kp(φ)/∂φ2 because,in the case of four Fermi points, the location of the Fermipoints depends on the flux. Thus, Eqs. (5) can be computedeither analytically or numerically. A typical plot of the inte-gratedχorb(φ) is given on Fig. 15 which shows that its dis-continuities are associated with the2 ↔ 4 Fermi points tran-sitions of Eqs. (A13). We have checked that the latest resultis valid only for0.5 < n = 1 − δ < 1.0 whenα = 1 (seezero-field susceptibility below).

Lastly, the behavior of the zero-field susceptibilityχorb0 can

be computed as a function of the electronic densityn. With afactor two for the spins, we have:if α > 1− cosπn (two Fermi points),

χorb0 (n) = −t‖

[

sinπn

(

1 +1

αcosπn

)

− π

αn

]

, (A16)

else, ifα < 1− cosπn (four Fermi points), we have

χorb0 (n) = −t‖

[√

sin2(πn

2

)

−(α

2

)2(

2 +cosπn

sin2(πn/2)

)

− 2

αarcsin

(

α

2

1

sin(πn/2)

)]

.

(A17)

0 0.5 1

−0.5

−0.25

0

0.25

0.5

PSfrag replacements

t⊥/t‖ = 1

χorb0

n

FIG. 16: Zero field orbital susceptibility of the non-interacting sys-tem for t⊥ = t‖ from Eqs. (A16-A17). The singularity atn = 0.5marks the transition from 2 to 4 Fermi points. Whenn < 0.5, thesusceptibility changes sign forn ≃ 0.3975 . . . while the system al-ways has two Fermi points.

The curve forα = 1 is plotted on Fig. 16 which shows thatfor n < 0.5, even when there are only two Fermi points, thesusceptibility can be either positive or negative.

APPENDIX B: FLUX QUANTIZATION AND FINITE SIZEEFFECTS

In this section, we discuss the quantization of the flux on afinite size ladder. First, Hamiltonian (1) clearly givesE(φ) =E(2π−φ) so that we can restrict ourselves to the windowφ ∈[0, π]. Similarly, from Eqs. (5) we havej(φ) = −j(2π − φ)which implies thatj(0) = j(π) = 0. What is quantizationof the flux φ on a finite system ? Using periodic boundaryconditions with the gauge of Fig. 1, the integrated flux alongaleg is±φ/2× L so that there is no remanent flux through thecylinder holeof the periodic ladder if

φ = m4π/L (B1)

with m an integer. Actually, this can also be simply under-stood from momentum quantizationk = 2πm/L and lookingat the dispersions−2t‖ cos(k±φ/2) on each leg whent⊥ = 0.This quantization can be checked numerically with exact diag-onalization. Furthermore, another possible gauge which givesthe same flux per plaquette is to takeφ⊥(x) = φx, with φ⊥the flux along a rung, and no flux along the legs. We can showthat the two gauges are strictly equivalent on a finite systemwith periodic boundary conditions only if (B1) is satisfied.

With DMRG, we are using open boundary conditions forwhich we expect similar effects due to momentum quantiza-tion. For most quantities and parametersJ/t, the points withφ = 2πm/L interpolates nicely with the ones using (B1) sowe can relax the constraint (see Fig. 4 for instance). However,when taking the derivatives such as in (5) at lowJ/t wheremore finite size effects are present, one must strictly obey (B1)

13

0 2π/L 4π/L 6π/L 8π/L

φ

−62.70

−62.65

−62.60

χ0 ~ −1

χ0 ~ 0

E0(φ)

0 0.25 0.5

δ

−1

0

1

2

3J/t = 0.25 dφ = 2π/LJ/t = 0.25 dφ = 4π/LJ/t = 0.40 dφ = 2π/LJ/t = 0.40 dφ = 4π/L

χ0(δ)

J/t = 0.25δ = 0.25

(a) (b)

FIG. 17: (Color online) Finite size effects on the calculation of thezero-field susceptibility at lowJ/t. (a) energies as a function of flux.(b) χ0 as a function of the dopingδ from Eq. (B2) for two differentdφ.

to have correct estimates (this is necessary in Fig. 11 for in-stance). On Fig. 17, because we know thatj‖(0) = 0, the zerofield susceptibility is

χ0 = (E0(2dφ)− E0(0))/(2dφ2) . (B2)

Takingdφ = 2π/L gives a negative susceptibility whiledφ =4π/L givesχ0 ∼ 0. Finite size effects are smaller away fromtheδ = 0.25 commensurability and also at largerJ/t and .

The DMRG simulations were performed using the standardfinite system algorithm on systems ranging fromL = 32 toL = 64, with minor modifications to treat complex wavefunc-tions. The density matrix at each step was the sum of the den-sity matrices constructed using the real and imaginary partsseparately. Typically, we keptm = 1400 states per block,giving a typical truncation error of10−6. The presence of themagnetic field did not increase the truncation error notably.Correlation functions are computed by averaging two-pointcorrelations of equal distance. Correlations with one pointtoo close to one of the edges are removed. Even if there isno translation symmetry with open boundary conditions, thismethod gives comparable results with the one using one pointfixed at the middle (but this method gives access to a largerdistancex).

APPENDIX C: BOSONIZATION CONVENTIONS

We use the same conventions as in Ref. 70. The bosoniza-tion procedure starts from the linearization of the band dis-persion in the vicinity the Fermi points. When there are fourFermi points, two of then in the up band and the two otherin the down band down bands (corresponding to the low-fieldC1S0 phase), we use:

(

c1c2

)

=1√L

∑

k

eikx

(

ak bkbk −ak

)(

ck,dck,u

)

,

with implicit spin index if not explicitly required. We denoteby ψR/L,d/u the bosonized right and left movers inside eachbands. Note that we have different Fermi levelskF,d ≡ kd 6=kF,u ≡ ku. From (A3) and (A4), we deduce thata−kd/u

=bkd/u

≡ bd/u andb−kd/u= akd/u

≡ ad/u. The bosonizedversion of the local fermion operators depends on how manyFermi points we have and which bands are filled. If we havefour Fermi points and overlapping bands, we use

c1(x)/√a→ ade

ikdxψd,R(x) + bde−ikdxψd,L(x)

+ bueikuxψu,R(x) + aue

−ikuxψu,L(x)

c2(x)/√a→ bde

ikdxψd,R(x) + ade−ikdxψd,L(x)

− aueikuxψu,R(x)− bue

−ikuxψu,L(x)

where the right and left moving Fermi fields have thebosonized representation

ψp,r,σ =ηrp,σ√2πα

eiǫrφr,p,σ

with α a cutoff (nott⊥/t‖) and r = R/L, p = d/u andǫR/L = ∓1. ηrp,σ are Klein factors than ensure anticommu-tation of fermion operators having different spin or band in-dex. The fieldsφr,p,σ are chiral boson fields. The non-chiralbosons fields are defined by:

φp,σ = [φL,p,σ + φR,p,σ]/2, (C1)

θp,σ = [φL,p,σ − φR,p,σ]/2, (C2)

and they satisfy commutation relations[φp,σ(x), θp′,σ′(x′)] =iδpp′δσ,σ′δ(x−x′). As usual in the framework of two coupledchains, we also introduce the following combinations of theφandθ fields: the charge and spin modes in each bandsp are

φc,p = [φp,↑ + φp,↓]/√2 (C3)

φs,p = [φp,↑ − φp,↓]/√2 (C4)

and similar transformations for theθ. And lastly, the± com-binations

φc/s,± = [φc/s,d ± φc/s,u]/√2 (C5)

The Luttinger parameters associated with these bosons areKc± for the charge sectors andKs± for the spin sectors.

In the case of two Fermi points (intermediate flux C1S1phase) andn < 1, only the down band is filled and we can usethe results of a single chain but using

c1(x)/√a → ade

ikdxψd,R(x) + bde−ikdxψd,L(x)

c2(x)/√a → bde

ikdxψd,R(x) + ade−ikdxψd,L(x) .

We simply denote byKc andKs the Luttinger parameterscorresponding to the charge and spin modes.

In the high-flux C1S0 phase, four points are present in thedown band. With the notationkF,1,d ≡ k1 6= kF,2,d ≡ k2, wehave after linearizing the band structure:

c1(x)/√a→ a1e

ik1xψ1,R(x) + b1e−ik1xψ1,L(x)

+ a2eik2xψ2,L(x) + b2e

−ik2xψ2,R(x)

c2(x)/√a→ b1e

ik1xψ1,R(x) + a1e−ik1xψ1,L(x)

+ b2eik2xψ2,L(x) + a2e

−ik2xψ2,R(x) ,

14

and similar expressions for the Fermi operatorsψp,r(x).

APPENDIX D: DIAMAGNETIC SUSCEPTIBILITY

Let us consider the Hamiltonian (3) or (4) in the limitφ→0. By expanding to second order, we have:

H = H(φ = 0)− φ

2L(j1 − j2)−

φ2

8L(K1 +K2),

wherej1,2 are the densities of current operators along chains1 and2 andK1,2 represent the densities of kinetic energy inchains1 and2. Note that these operators are taken atφ =0. We obtain the density of screening current operatorj‖ =

− 1L∂H/∂φ as:

j‖ =1

2(j1 − j2) +

φ

4(K1 +K2) .

Using linear response theory, we obtain the expectation valueof this current in this limit as:

j‖(φ) = 〈j‖〉0 =φ

4[〈〈(j1 − j2); (j1 − j2)〉〉0 + 〈K1 +K2〉0] ,

where〈〈;〉〉0 represents the retarded response function and〈〉0is the expectation value in the ground state without magneticfield. In the absence of interchain hopping, the cross responsefunction〈〈j1; j2〉〉0 would vanish and〈j‖〉0 would be simplythe sum of Drude weights of each chain. The expression of〈j‖〉0 can be rearranged by noting that:

j1 − j2 = 2

∫ x

j⊥ (D1)

as a consequence of Kirchhoff’s law. So we have that:

χ0 =

∫

dx〈j⊥(x)j⊥(0)〉0 +1

4〈K1 +K2〉0 . (D2)

In the case of negligible transverse current correlations,thisterm reduces to the expectation value of the kinetic energy.Inan insulator, this yields a vanishing diamagnetic susceptibility.

∗ Electronic address: roux@irsamc.ups-tlse.fr1 E. Dagotto, J. Riera, and D. Scalapino, Phys. Rev. B45, 5744

(1992).2 C. A. Hayward, D. Poilblanc, R. M. Noack, D. J. Scalapino, and

W. Hanke, Phys. Rev. Lett.75, 926 (1995).3 M. Troyer, H. Tsunetsugu, and T. M. Rice, Phys. Rev. B53, 251

(1996).4 L. Balents and M. P. A. Fisher, Phys. Rev. B53, 12133 (1996).5 H. J. Schulz, Phys. Rev. B53, R2959 (1996).6 S. R. White, I. Affleck, and D. J. Scalapino, Phys. Rev. B65,

165122 (2002).7 P. W. Anderson, Science235, 1196 (1987).8 N. Nagaosa, Sol. State Comm.94, 495 (1995).9 S. T. Carr and A. M. Tsvelik, Phys. Rev. B65, 195121 (2002).

10 A. A. Nersesyan, Phys. Lett. A153, 49 (1991).11 D. J. Scalapino, S. R. White, and I. Affleck, Phys. Rev. B64,

100506(R) (2001).12 K. Tsutsui, D. Poilblanc, and S. Capponi, Phys. Rev. B65,

020406(R) (2001).13 U. Schollwock, S. Chakravarty, J. O. Fjærestad, J. B. Marston,

and M. Troyer, Phys. Rev. Lett.90, 186401 (2003).14 J. O. Fjærestad, J. B. Marston, and U. Schollwock, Ann. Phys.

321, 894 (2006).15 T. Kjeldaas and W. Kohn, Phys. Rev.105, 806 (1957).16 J. E. Hebborn and E. H. Sondheimer, Phys. Rev. Lett.2, 150

(1959).17 W. Kohn, Phys. Rev.115, 1460 (1959).18 D. C. Cabra, A. De Martino, P. Pujol, and P. Simon, Europhys.

Lett. 57, 402 (2002).19 G. Roux, S. R. White, S. Capponi, and D. Poilblanc, Phys. Rev.

Lett. 97, 087207 (2006).20 G. Roux, E. Orignac, P. Pujol, and D. Poilblanc, Phys. Rev. B75,

eid245119 (2007).21 A. F. Albuquerque and G. B. Martins, J. Phys.: Condens. Matter

17, 2419 (2005).22 B. N. Narozhny, S. T. Carr, and A. A. Nersesyan, Phys. Rev. B71,

161101(R) (2005).23 S. T. Carr, B. N. Narozhny, and A. A. Nersesyan, Phys. Rev. B73,

195114 (2006).24 E. Orignac and T. Giamarchi, Phys. Rev. B64, 144515 (2001).25 S. R. White, Phys. Rev. Lett.69, 2863 (1992).26 S. R. White, Phys. Rev. B48, 10345 (1993).27 U. Schollwock, Rev. Mod. Phys.77, 259 (2005).28 R. E. Peierls, Z. Phys.80, 763 (1933).29 W. Kohn, Phys. Rev.133, A171 (1964).30 P. Gagliardini, S. Haas, and T. M. Rice, Phys. Rev. B58, 9603

(1998).31 C. M. Varma and A. Zawadowski, Phys. Rev. B32, 7399 (1985).32 K. Penc and J. Solyom, Phys. Rev. B41, 704 (1990).33 M. Fabrizio, Phys. Rev. B48, 15838 (1993).34 H. H. Lin, L. Balents, and M. P. A. Fisher, Phys. Rev. B56, 6569

(1997).35 M. Fabrizio, Phys. Rev. B54, 10054 (1996).36 S. Daul and R. M. Noack, Phys. Rev. B58, 2635 (1998).37 C. Bourbonnais and L. G. Caron, Physica (Utrecht)143B, 450

(1986).38 C. Bourbonnais and L. G. Caron, Int. J. Mod. Phys. B5, 1033

(1991).39 G. Roux, S. R. White, S. Capponi, A. Lauchli, and D. Poilblanc,

Phys. Rev. B72, 014523 (2005).40 D. Poilblanc, O. Chiappa, J. Riera, S. R. White, and D. J.

Scalapino, Phys. Rev. B62, R14633 (2000).41 D. Poilblanc, E. Orignac, S. R. White, and S. Capponi, Phys. Rev.

B 69, 220406(R) (2004).42 K. Penc and J. Solyom, Phys. Rev. B44, 12690 (1991).43 M. Tsuchiizu, H. Yoshioka, and Y. Suzumura, J. Phys. Soc. Jpn.

70, 1460 (2001).44 M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Mori, and

15

K. Kinoshita, J. Phys. Soc. Jpn.65, 2764 (1996).45 H. Mayaffre, P. Auban-Senzier, M. Nardone, D. Jerome, D. Poil-

blanc, C. Bourbonnais, U. Ammerahl, G. Dhalenne, andA. Revcolevschi, Science279, 345 (1998).

46 D. Jerome, P. Auban-Sauzier, and Y. Piskunov, inHigh Mag-netic Fields, edited by C. Berthier, L. P. Levy, and G. Martinez(Springer, 2002), vol. 595 ofLecture notes in physics.

47 N. Fujiwara, N. Mori, Y. Uwatoko, T. Matsumoto, N. Motoyama,and S. Uchida, Phys. Rev. Lett.90, 137001 (2003).

48 N. Fujiwara, N. Mori, Y. Uwatoko, T. Matsumoto, N. Motoyama,and S. Uchida, J. of Phys.: Condens. Matter17, S929 (2005).

49 T. Osafune, N. Motoyama, H. Eisaki, S. Uchida, and S. Tajima,Phys. Rev. Lett.82, 1313 (1999).

50 B. Gorshunov, P. Haas, T. Room, M. Dressel, T. Vuletic, B.Korin-Hamzic, S. Tomic, J. Akimitsu, and T. Nagata, Phys. Rev. B66,060508(R) (2002).

51 G. Blumberg, P. Littlewood, A. Gozar, B. S. Dennis, N. Mo-toyama, H. Eisaki, and S. Uchida, Science297, 584 (2002).

52 T. Vuletic, B. Korin-Hamzic, S. Tomic, B. Gorshunov, P. Haas,T. Room, M. Dressel, J. Akimitsu, T. Sasaki, and T. Nagata,Phys.Rev. Lett.90, 257002 (2003).

53 P. Abbamonte, G. Blumberg, A. Rusydi, A. Gozar, P. G. Evans,T. Siegrist, L. Venema, H. Eisaki, E. D. Isaacs, and G. A.Sawatzky, Nature431, 1078 (2004).

54 A. Gozar and G. Blumberg,Collective Spin and Charge Exci-tations in (Sr,La)14 − xCaxCu24O41 Quantum Spin Ladders(Spinger-Verlag, 2005), p. 653, Frontiers in Magnetic Materials,cond-mat/0510193.

55 T. Vuletic, T. Ivek, B. Korin-Hamzic, S. Tomic, B. Gorshunov,P. Haas, M. Dressel, J. Akimitsu, T. Sasaki, and T. Nagata, Phys.Rev. B71, 012508 (2005).

56 K.-Y. Choi, M. Grove, P. Lemmens, M. Fischer, G. Guntherodt,U. Ammerahl, B. Buchner, G. Dhalenne, A. Revcolevschi, andJ. Akimitsu, Phys. Rev. B73, 104428 (2006).

57 A. Rusydi, P. Abbamonte, H. Eisaki, Y. Fujimaki, G. Blumberg,

S. Uchida, and G. A. Sawatzky, Phys. Rev. Lett.97, 016403(2006).

58 A. Rusydi, M. Berciu, P. Abbamonte, S. Smadici, H. Eisaki, Y.Fu-jimaki, S. Uchida, M. Rubhausen, and G. A. Sawatzky, Phys. Rev.B 75, 104510 (2007).

59 K. Wohlfeld, A. M. Oles, and G. A. Sawatzky, Phys. Rev. B75,180501(R) (2007).

60 H. Ajiki and T. Ando, J. Phys. Soc. Jpn.62, 1255 (1993).61 S. Zaric, G. N. Ostojic, J. Kono, J. Shaver, V. C. Moore, M. S.

Strano, R. H. Hauge, R. E. Smalley, and X. Wei, Science304,1129 (2004).

62 U. C. Coskun, T.-C. Wei, S. Vishveshwara, P. M. Goldbart, andA. Bezryadin, Science304, 1132 (2004).

63 B. Lassagne, J.-P. Cleuziou, S. Nanot, W. Escoffier, R. Avriller,S. Roche, L. Forro, B. Raquet, and J.-M. Broto, Phys. Rev. Lett.98, 176802 (2007).

64 M. Bockrath, D. H. Cobden, J. Lu, A. G. Rinzler, R. E. Smalley,L. Balents, and P. L. Mceuen, Nature397, 598 (1999).

65 R. Egger, A. Bachtold, M. S. Fuhrer, M. Bockrath, D. H. Cobden,and P. L. McEuen, inInteracting Electrons in Nanostructures,edited by R. Haug and H. Schoeller (Springer, Berlin, 2001),vol.579 ofLecture Notes in Physics, p. 125.

66 H. Ishii, H. Kataura, H. Shiozawa, H. Yoshioka, H. Otsubo,Y. Takayama, T. Miyahara, S. Suzuki, Y. Achiba, M. Nakatake,T. Narimura, M. Higashiguchi, K. Shimada, H. Namatame andM. Taniguchi, Nature426, 540 (2003).

67 B. Gao, A. Komnik, R. Egger, D. C. Glattli, and A. Bachtold,Phys. Rev. Lett.92, 216804 (2004).

68 J. Lee, S. Eggert, H. Kim, S.-J. Kahng, H. Shinohara, and Y. Kuk,Phys. Rev. Lett.93, 166403 (2004).

69 H.-H. Lin, Phys. Rev. B58, 4963 (1998).70 T. Giamarchi,Quantum Physics in one Dimension, vol. 121 of

International series of monographs on physics(Oxford UniversityPress, Oxford, UK, 2004).