Download Optical Response in Infinite Dimensions

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. Pys.
1996
DMFT Equations
Imurity
solver gets
tis
Self
consistency
condition
 S(t) is obtained from solution of a quantum imurity roblem
 Te only ay lattice enters is via D(e) to roject Gji onto
site o
 Tere are many metods to solve imurity roblem, e.g. QMC,
Conformal Field Teory, Perturbation Teory, etc.
“Dynamical” Mean Field
A. Georges, et.al, Rev. Mod. Pys. 1996
Generic Impurity Model
Anderson impurity model:
Integrate out conduction degrees of freedom:
A solvable limit:
Metallic
Pase
Lorentzian
DOS
A. Georges, et.al, Pys. Rev. B 45, 6479
(1992)
Iterated Perturbation Theory
Start with
SOPT
FFT
Projection
No
Yes
Update
Convergence
FFT
Miracle Of SOPT
Atomic Limit:
XY Zang et al., PRL 70, 1666
(1993)
IPT interolates beteen eak and strong
couling 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. Pruscke et.al, Pys. Rev. B 47,3553
(1993)
Vertex
factor:
 By oer counting argument, G collases and becomes local
 Momentum conservation becomes irrelevant ) momentum sums
factorize

as odd arity ) vertex corrections  0
T. Pruscke et.al, Pys. 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