Introduction to Hubbard Model S. A. Jafari Department of Physics, Isfahan Univ. of Tech. Isfahan , IRAN TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

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 Insulators Even 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 C Ca, Sr Na, K According to band theory, odd no. of e’s per unit cell ) Metal

Failure of Band Theory 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: [1s 2 2s 2 2p 6 ] 3s 1 - Band is half-filled - At small lattice constants overlap and hence the band width is large ) Large gain in kinetic energy ) Metallic behavior - for larger “a”, charge fluctuations are supressed: Coulomb energy dominates: ) cost of charge fluctuations increases ) Insulator at half filling

A Simple Model At (U 3s /t 3s ) cr =4 ® Coulomb energy cost starts to dominate the gain in the charge fluctuations ) |FS i becomes unstable ) Insulating states becomes stabilized

Hubbard Model

Metal-Insulator Trans. (MIT) (1) Band Limit (U=0): (2) Atomic Limit (U À t): For t=0, two isolated atomic levels ² at and ² at +U Small non-zero t ¿ U broadens the atomic levels into Hubbard sub-bands Further increasing t, decreases the band gap and continuously closes the gap (Second order MIT)

Symmetries of Hubbard Model particle-hole symmetry For L sites with N e’s, the transformation At half-filling, N=L, H(L)  H(L)

Symmetries of Hubbard Model SU(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 Model N and S z are good quantum numbers. Example: N=2, S z =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 for singlet-triplet transitions

Why Spin Fluctuations? In the large U limit, double occupancy (d) is expensive: each (d) has energy cost U À t Hopping changes the double 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 comments are welcome

Lecture 2 Mean 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 h A i, h B i are order parameter Hartree: Diagonal c y c

Hartree-Fock

Stoner Criterion

Metallic Ferromagnetism

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 3 Dynamical 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. Dimension dependence of Green’s functions: Number of n.n. hoppings to jump a distance R ji 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 S ij involves at least 3 transfer matrices No. of n.n. transfers » d ) total S ij / d -1/2 For general distance R I and R j : Number of such n.n. transfers is »   Perturbation Theory in d= 1 is purely  :

Effective Local Theory Original Hubbard model In any dimension Diagram Collapse In Infinite Dimension   “  ”  t  

DMFT Equations        S  t   e      

 “Dynamical” Mean Field

Generic Impurity Model Anderson impurity model: Integrate out conduction degrees of freedom:   A solvable limit: Lorentzian DOS  

Start with Iterated Perturbation Theory SOPT FFT Projection Update FFT Convergence Yes No

Miracle Of SOPT Atomic 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 comments are welcome

Lecture 4 t-J model $ Hubbard model How spin physics arises from Strong Electron Correlations?

Projected Hopping Local basis: Projection operators: Ensure there is + at j Ensure there is " at i Perform the hopping Ensure site j is |d i Ensure site i is |0 i projected hopping

Classifying Hoppings Ensure there is no + at i Perform Hopping Ensure there is a + at j Double occupancy increaded: D  D+1

Correlated Hopping Mind the local correlations Don’t care the correlations

Questions and comments are welcome