Introduction to Hubbard Model

S. A. JafariDepartment of Physics, Isfahan Univ. of Tech.

Isfahan 8415683111, IRAN

Tackling the Hubbard Model

• Exact diagonalization for small clusters (Lect. 1)• Various Mean Field Methods (Lect. 2)• Dynamical Mean Field Theory (D 1,Lect. 3, practical)• Bethe Ansatz (D=1)• Quantum Monte Carlo Methods• Diagramatic perturbation theories• Combinations of the above methods• Effective theories:

1- Luttinger Liquids (D=1)

2- t-J model (Lect. 4)

Lecture 1

• What is the Hubbard Model?

• What do we need it for?

• What is the simplest way of solving it?

Band InsulatorsEven no. of e’s per unit cell

Even no. of e’s per unit cell +band overlap

Odd no. of e’s per unit cell

CCa, Sr

Na, K

According to band theory, odd no. of e’s per unit cell ) Metal

Failure of Band Theory

Co: 3d74s2

O : 2s22p4

Total no. of electrons = 9+6 = 15

Band theory predicts CoO to be metal, while it is the toughest insulator known

Failure of band theory ) Failure of single particle picture ) importance of interaction effects (Correlation)

Gedankenexperiment: Mott insulator

Imagin a linear lattice of Na atoms:Na: [1s2 2s2 2p6] 3s1

- Band is half-filled - At small lattice constants overlapand hence the band width is large ) Large gain in kinetic energy ) Metallic behavior - for larger “a”, charge fluctuationsare supressed:

Coulomb energy dominates:

) cost of charge fluctuations increases ) Insulator at half filling

A Simple Model

At (U3s/t3s)cr=4® Coulomb energy cost starts to dominate the gain in the charge fluctuations ) |FSi becomes unstable) Insulating states becomes stabilized

Hubbard Model

Metal-Insulator Trans. (MIT)(1) Band Limit (U=0):

(2) Atomic Limit (UÀ t):

• For t=0, two isolated atomic levels ²at and ²at+U• Small non-zero t¿ U broadens theatomic levels into Hubbard sub-bands• Further increasing t, decreases theband gap and continuously closes the gap(Second order MIT)

Symmetries of Hubbard Modelparticle-hole symmetry For L sites with N e’s, the transformation

At half-filling, N=L, H(L) H(L)

Symmetries of Hubbard ModelSU(2) symmetry

When Hubbard Model is Relevant?

• Long ragne part of the interaction is ignored ) Screening must be strong

• Long range interaction is important, but we are addressing spin physics.

Two-site Hubbard ModelN and Sz are good quantum numbers. Example: N=2, Sz=0 for L=2 sites

Exact Diagonalization

Ground state

Excited states

Excitation Spectrum

Low-energy physics

Low-energy physics of the Hubbard model at half-filling and large U is a spin model!

Energy scale forsinglet-triplet transitions

Why Spin Fluctuations?

U

In the large U limit,double occupancy (d)is expensive: each (d)has energy cost UÀ t

Hopping changes thedouble occupancy

Tackling the Hubbard Model

• Exact diagonalization for small clusters (Lect. 1)• Various Mean Field Methods (Lect. 2)• Dynamical Mean Field Theory (D 1,Lect. 3, practical)• Bethe Ansatz (D=1)• Quantum Monte Carlo Methods• Diagramatic perturbation theories• Combinations of the above methods• Effective theories:

1- Luttinger Liquids (D=1)

2- t-J model (Lect. 4)

Questions and commentsare welcome

Lecture 2Mean Field Theories

• Stoner Model

• Spin Density Wave Mean Field

• Slave Boson Mean Field

Mean Field Phase Diagram

Metal

insulator

Broken Symmetry: Ordering

• Mean field states break a symmetry• hAi, hBi are order parameter

Hartree: Diagonal cy

c

Hartree-Fock

Stoner Criterion

Metallic Ferromagnetism

³

°

43

22=3

Exercise

Generalized Stoner: SDW

For half filled bands with perfect nesting property,

arbitrarily small U>0 causes a transition to an antiferromagnetic (AF)

state

Formation of SDW state

Math of SDW state

Double occupancy of the SDW ansatz vs. exact resutls from the Bethe ansatz in 1D

Lecture 3Dynamical Mean Field Theory

Limit Of Infinite Dimensions

Hubbard Model:

• Purely onsite U remains unchanged

Scaling in large coordination limit:Spin Models:

Simplifications in Infinite Dim.

,

( ) ( , ; )2

i j

ikin i j i j

R R

dE t R R e G R R

i

| |/ 2R Ri jd

Dimension dependence of Green’s functions:

| |/ 2R Ri jd

| |R Ri jdL d

Number of n.n. hoppings to jump a distance Rji

The Green functions decay at large distances as a power of dimension of space

Real Space Collapse:

Luttinger-Ward free energy (AGD, 1965)

Above HF, more than 3 independent lines connect all vertices )

Example of non-skeleton diagram that cant be collapsed ! momentum conservation hold from, say j to l vortices

Site Diagon

al

Real Space Collapse: For nearest neighbors skeleton Sij involves at least 3 transfer matrices

No. of n.n. transfers » d ) total Sij/ d-1/2

For general distance RI and Rj :

Number of such n.n. transfers is »

Perturbation Theory in d=1 is

purely :

Effective Local TheoryOriginal Hubbard model

In any dimension

Diagram CollapseIn Infinite

Dimension

“”t

DMFT Equations

St

e

“Dynamical” Mean Field

Generic Impurity ModelAnderson impurity model:

Integrate out conduction degrees of freedom:

A solvable limit:

Lorentzian DOS

Start with

Iterated Perturbation Theory

SOPT

FFT

Projection

Update

FFTConvergence

Yes

No

Miracle Of SOPTAtomic Limit:

• Height of Kondo peak at Fermi surface is constant

• Width of Kondo peak exponentially narrows with increasing U

• DMFT (IPT) captures both sides: Insulating and Metallic

• DMFT clarifies the nature of MIT transition

IPT for +i

Laplace transform

P-h bubble

SOPT diagram

G

)

)

Optical Conductivity

Ward Identity I

: Arbitrary quantum amplitude

Ward Identity II

Corrections to two photon vertex

One-photon vertex corrections

Odd parts of current vertex is projected! ) Only remaining even part of G is 1 ) vertex corrections=0

• In nonlinear optics we have more phonons attached to bubble

• Above argument works also in nonlinear optics

(3)() In D=1

Lehman Representation

General structure:

Questions and commentsare welcome

Lecture 4

t-J model $ Hubbard model How spin physics arises

fromStrong Electron Correlations?

Projected HoppingLocal basis:

Projection operators:

Ensure there is + at j

Ensure there is " at i

Perform the hopping

Ensure site j is |di

Ensure site i is |0i

projected hopping

Classifying HoppingsEnsure there is no + at i

Perform Hopping

Ensure there is a + at j

Double occupancy increaded: D D+1

Correlated Hopping

Mind the local

correlationsDon’t care

the correlations

Questions and commentsare welcome