Mott –Hubbard Transition & Thermodynamic Properties in

Nanoscale Clusters.

Armen Kocharian

(California State University, Northridge, CA)

Gayanath Fernando (University of Connecticut, Storrs, CT)Jim Davenport (Brookhaven National Laboratories, Upton, NY)Kalum Palandage (University of Connecticut, Storrs, CT)

Outline

• Motivation• Small Hubbard clusters (2-site, 4-site)• Ground state properties• Exact Thermodynamics

– Charge dos and Mott-Hubbard crossover – Spin dos and AF Nee l crossover– Phase diagrams

• QMC calculations in small clusters• Conclusions

Quantum Monte Carlo

• Exact analytical results and QMC

h=0

MotivationElectron Correlations

- Large Thermodynamic System: • Interplay between charge and spin degrees • Mott-Hubbard Transition• AFM-PM (Nee l Transition)• Magnetic and transport properties

-Nanoscale Clusters: • Mott-Hubbard crossover?• Charge and spin degrees?• AFM-PM crossover?

Finite size Hubbard model• Simplest lattice model to include correlations:

Tight binding with one orbital per site Repulsion: on-site only Nearest neighbor hopping onlyMagnetic field

• Bethe ansatz solution [Lieb & Wu. (’67)] Ground state but not correlation functions

Finite size Hubbard cluster

Lieb & Wu. (’67)

Thermodynamics (T≠0)

Ground state (T=0)• Weak correlations in 1d systems:

power law decay (Schulz ’91, Korepin & Frahm ’90)

• Long range order in finite clusters: saturated ferromagnetism (Nagaoka’65)

• Signature of short range correlations: weak magnetization (Aizenman & Lieb’90)correlations decay faster than power law like (Koma & Tasaki ’92)

• No long range correlations:no magnetic order in 1d (Mermin & Wagner. ’66, Ghosh ’71)

Large Clusters: • Bethe-ansatz calculations • Lanczos • Monte Carlo• Numerical diagonalization• DMFT

Small Clusters:

• Exact analytical diagonalization • Charge and spin gaps (T=0)• Pseudogaps (T≠0)

Lieb & Wu. (’67)

Dagotto et al. (’ 84)

Canio et al. (’96)

Jarrell et al. (’70)

Kotliar. et al. (’97)

Mott-Hubbard transition: • Temperature• Magnetic field

AF-PM Transition: • Exchange• Susceptibility

HTSC superconductivity:• Pseudogap formation• Chemical potential (n≠1)

Kotliar (’67)

Schrieffer et al. (’ 90)

Canio et al. (’96)

Neel Magnetic Phase TN

Mott-Hubbard Phase TMH

Two phase transitions in Hubbard Model

From D. Mattis et al. (’69)

M. Cyrot et al. (’70)

J. R. Schrieffer et al. (’70)

Approaching to TMH from metallic state: U↑, T↓

TN consequence of Mott-Hubbard phase

Mott Hubbard and AF transitions

Brinkman et al. (’70)

Anderson (’97)

Slater (’51)TMH consequence of Neel anti-ferromagnetism Approaching to MH phase from insulator: T↑,U↓

Hubbard (’64)

Evolution of dos and pseudogaps, TMH and TN for 2 and 4 site clusters at arbitrary U, T and h

Thermodynamics of small clusters

From Shiba et al., (‘72)

Specific heatof finite chains N=2, 3, 4, 5

Low temperature peak – AFM-PM

High temperature peak – MH transition

Focus on 2 and 4-site clusters

Mott-Hubbard Transition

AFM-PM Transition

Driven by h and T

A single hydrogen molecule acting as a nanowire

Shumann (’02)Shiba et al. (’70)

Harris et al. (’72) Kocharian et al. (’ 96)

Exact ground state properties

Exact mapping of 2-site Hubbard and Heisenberg ground states at half filling (A. Kocharian et al. ’91, 96):

e.g., hC=J(U)

hC - critical field of ferromagnetic saturation

Ground state charge gap (N=2)

e.g., h<hC

e.g., h≥hC

Half filling

• Gap is monotonic versus U and non monotonic versus h

Ground state charge gap (N=1, 3)

Quarter and three quarter fillings

e.g., h<hC

e.g., h≥hC

• Charge gap versus h and Uis monotonic everywhere

Exact thermodynamics (T≠0)

• Number of particles Nat h=0 versus µ and T

• Sharp step like behavior only in the limit T 0

h=0

2 sites: n 24

4 sites: n 44

h=0

N versus chemical potential (T/t=0.01)

Real plateaus exist only T=0

(not shown)

h=0

Chemical potential in magnetic field

• Number of particles Nat h/t=2 versus µ and T

• Sharp step like behavior only in limit T 0

h/t=2.0, U/t=5.0

• Plateaus at N=1 and N=3 increases with h

• Plateau at N=2 decreases with h

Magnetic susceptibility χ at half filling

• Susceptibility versush at T=.05

• As temperature T 0peaks of χ closely tracks U dependence of hC (U)

hc(U)/t

U/t

4

Number of electrons vs. μ clusters

h=0

• Plateaus at integer N exist only at T=0(not shown in figure)

Charge pseudogap at infinitesimal T≠0

h=0

Charge and spin dos in 2-site cluster

• Charge dos for general N has four peaks

• Spin dos at half fillinghas two peaks

U=6 and h=2

Thermodynamic charge dos and pseudogap

• Charge dos for general U≠0 has four peaks

U=0 and h=0

• Charge pseudogap disappears at TMH

Two peaks merge in one peak saddle point

Saddle point

U=5 and h=0

TMH

Charge dos and pseudogap

• Charge dos for general N has four peaks

Charge dos for general N has four peaks

σ

h=0 h=2t

Spin dos and pseudogap

• Spin dos at half filling has two peaks

U=6

• spin pseudogap at TN disappears (saddle point)

Saddle point

Thermodynamic charge and spin dos

• Charge dos for general N has four peaks

• Spin dos at half fillinghas two peaks

σ

Weak singularity in charge dos

• Infinitesimal temperature smears ρ(μC)≠0 and results in pseudo gap

• At TMH, ρ(μC)≠0 and ρ′(μC)=0 ρ″(μC)>0 . It is a saddle point

MH Transition at half-filling (N=2)

• True gap at μC=U/2 exists only at T=0

n 1• Forth order MH

phase transition

Weak singularity in spin dos

• Infinitesimal temperature smears σ(0)≠0 at h=0 and results in pseudo gap

• At TN, σ(0)≠0 and σ′(0)=0 σ ″(0)>0. . It is a saddle point

Neel Transition at N=2

• True gap exists only at T=0

n 1• Forth order Nee l

phase transition

Weak singularity in charge dos

• Distance between charge peak positions versus temperature

N=2

TMH versus μ MH crossover

Bifurcations atμ=U/2 &μ≠U/2

Weak singularity in spin dos

• Distance between spin peak positions versus temperature

N=2

TN versus h crossover

Spin magnetization

• Magnetization at quarter filling (no spin gap)

• Magnetization at half filling (spin gap)

Magnetization versus h

h=0

• No spin gap at N=1 and 3

Zero field spin susceptibility (N=2)

• TN from maximum susceptibility

• TN from peaks distance

TN temperature versus U

• TN versus U (AF gap)

• TN from maximumof spin susceptibility

• TN from spin dos peaks

• TF versus U• (Ferro gap)

Zero field magnetic susceptibility χ

h=0

• At large U magnetic susceptibility ~T

At U/t»1 χ increases linearly

N=2

Spin susceptibility

• Susceptibility at quarter filling (no spin gap)

• Susceptibility at half filling (UC/t=6)

h/t=2 N=1N=2

Phase diagram, TMH versus U

• At t=0 TMFMH =U/2

result at t=0. D.Mattis’69• At t=0 TMH = U/2ln2

and ρ(µC )=2ln2/5U

h=0

Phase diagram, TMH versus U

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

TMH

Phase Diagram at half filling

• TMH & TN versus U, at which pseudogap disappears

T/t

U/t

4

MH + AF

MH

N TMH

TN

Mott-Hubbard crossover

• TMH versus h and U

σ

N=2

4-site clusters

h=0

2 216U t

2 216U t

322

32 2

4 3arccos

16

t U

U t

0zs s

4-site clusters

2 3

32 2

36arccos

48

t U U

U t

1,0, 1zs 1s

4-site clusters

h=0

2 216U t

2 216U t1,0, 1zs 1s

h=0

• Plateaus at integer N exist only at T=0(not shown in figure)

N versus μ in 4 site cluster

No gap atU=0 and N=4

T/t=0.01U/t=4.0

h=0

Bifurcations at N = 2, 4 and 6

Weak singularity in charge dos

h=0

Bifurcations at N = 1, 2, 4, 6 and 7

Weak singularity in charge dos

Thermodynamic dos for 4-site cluster

Analytical calculations

h=0

DMFT calculations

Quantum Monte Carlo

• Exact analytical results and QMC

h=0

• Magnetism and MH crossovers in rings and pyramids

h=0

Quantum Monte Carlo studies

QMC studies of small clusters

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

5 sites pyramids

h=0

• Magnetization versus h

5 sites pyramids

h=0

• Staggered magnetization versus n

14 sites pyramids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

14 sites pyramids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

Conclusions

• Exact mapping Hex~HU in the ground state• True spin and charge gaps exist only at T=0

ECGap(U)≠ES

Gap(U ) at U≠0

• Pseudogaps appear at infinitesimal T• Charge dos - MH crossover (TMH >TN )• Spin dos - AFM-PM crossover (TN )• Temperature driven bifurcation – generic feature • 1d Hubbard model, UC=0 and true gap in ρ(μC)=0 exists

only at T=0 and n=1• 2 and 4 site Hubbard clusters reproduces main features of

small and large system• Evolution of pseudogap versus μ in HTSC

Rigid spin dynamics

h=0

QMC studies of small clusters

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

14 sites pyrmids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

QMC studies of small clusters

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

14 sites pyrmids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

14 sites pyrmids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

• Staggered magnetization (spin at site i)

14 sites pyrmids

h=0

• Staggered magnetization i (-1)x+y+z (spin at site i)

4-site clusters

h=0

1,0, 1zs 0zs s

1s 1,0, 1zs

Ground state charge gap

• Gap is monotonic versus U and non monotonic versus h