Transcript 8 - 1
Introduction to Hubbard Model S. A. Jafari Department 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 Insulators Even no. of e’s per unit cell C Even no. of e’s per unit cell + band overlap Odd no. of e’s per unit cell Ca, Sr According to band theory, odd no. of e’s per unit cell ) Metal Na, K Failure of Band Theory Co : 3d7 4s2 O : 2s2 2p4 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 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 (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 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 Sz are good quantum numbers. Example: N=2, Sz=0 for L=2 sites Exact Diagonalization Excited states Ground state 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 UÀ t Hopping changes the double occupancy U 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 • hAi, hBi are order parameter Hartree: Diagonal cy c Hartree-Fock Stoner Criterion Metallic Ferromagnetism ° 22=3 4 3 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 3 Dynamical Mean Field Theory Limit Of Infinite Dimensions Spin Models: Hubbard Model: • Purely onsite U remains unchanged Scaling in large coordination limit: Simplifications in Infinite Dim. Dimension dependence of Green’s functions: Ekin d i e G ( Ri , R j ; ) 2 i t ( Ri R j ) Ri , R j |Ri R j |/ 2 d |Ri R j | L d d Number of n.n. hoppings to jump a distance Rji d |Ri R j |/ 2 The Green functions decay at large distances as a power of dimension of space Real Space Collapse: F Luttinger-Ward free energy (AGD, 1965) Above HF, more than 3 independent lines connect all vertices ) Site Diagon al Example of non-skeleton diagram that cant be collapsed ! momentum conservation hold from, say j to l vortices Real Space Collapse: S 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 » Total contribution to self energy: Perturbation Theory in d=1 is purely local: Effective Local Theory Original Hubbard model In any dimension Diagram Collapse In Infinite Dimension Dynamical (t) Local Field A. Georges, et.al, Rev. Mod. Pys. 1996 DMFT Equations Imurity solver gets tis Self consistency condition S(t) is obtained from solution of a quantum imurity roblem Te only ay lattice enters is via D(e) to roject Gji onto site o Tere are many metods to solve imurity roblem, e.g. QMC, Conformal Field Teory, Perturbation Teory, etc. “Dynamical” Mean Field A. Georges, et.al, Rev. Mod. Pys. 1996 Generic Impurity Model Anderson impurity model: Integrate out conduction degrees of freedom: A solvable limit: Metallic Pase Lorentzian DOS A. Georges, et.al, Pys. Rev. B 45, 6479 (1992) Iterated Perturbation Theory Start with SOPT FFT Projection No Yes Update Convergence FFT Miracle Of SOPT Atomic Limit: XY Zang et al., PRL 70, 1666 (1993) IPT interolates beteen eak and strong couling limits • 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 +ig P-h bubble SOPT diagram Laplace transform Optical Conductivity T. Pruscke et.al, Pys. Rev. B 47,3553 (1993) Vertex factor: By oer counting argument, G collases and becomes local Momentum conservation becomes irrelevant ) momentum sums factorize as odd arity ) vertex corrections 0 T. Pruscke et.al, Pys. Rev. B 47,3553 (1993) Ward Identity I M. E. Peskin et al., Introduction to QFT : Arbitrary quantum amplitude Ward Identity II 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 Corrections to two photon vertex (3) c (n) 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: projected hopping Ensure there is + at j Ensure there is " at i Perform the hopping Ensure site j is |di Ensure site i is |0i 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