introduction to hubbard model s. a. jafari department of physics, isfahan univ. of tech. isfahan...
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Introduction to Hubbard Model
S. A. JafariDepartment of Physics, Isfahan Univ. of Tech.
Isfahan 8415683111, IRAN
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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)
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Lecture 1
• What is the Hubbard Model?
• What do we need it for?
• What is the simplest way of solving it?
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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
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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)
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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
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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
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Hubbard Model
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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)
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Symmetries of Hubbard Modelparticle-hole symmetry For L sites with N e’s, the transformation
At half-filling, N=L, H(L) H(L)
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Symmetries of Hubbard ModelSU(2) symmetry
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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.
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Two-site Hubbard ModelN and Sz are good quantum numbers. Example: N=2, Sz=0 for L=2 sites
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Exact Diagonalization
Ground state
Excited states
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Excitation Spectrum
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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
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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
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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)
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Questions and commentsare welcome
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Lecture 2Mean Field Theories
• Stoner Model
• Spin Density Wave Mean Field
• Slave Boson Mean Field
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Mean Field Phase Diagram
Metal
insulator
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Broken Symmetry: Ordering
• Mean field states break a symmetry• hAi, hBi are order parameter
Hartree: Diagonal cy
c
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Hartree-Fock
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Stoner Criterion
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Metallic Ferromagnetism
³
°
43
22=3
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Exercise
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Generalized Stoner: SDW
For half filled bands with perfect nesting property,
arbitrarily small U>0 causes a transition to an antiferromagnetic (AF)
state
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Formation of SDW state
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Math of SDW state
Double occupancy of the SDW ansatz vs. exact resutls from the Bethe ansatz in 1D
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Lecture 3Dynamical Mean Field Theory
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Limit Of Infinite Dimensions
Hubbard Model:
• Purely onsite U remains unchanged
Scaling in large coordination limit:Spin Models:
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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
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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
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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 :
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Effective Local TheoryOriginal Hubbard model
In any dimension
Diagram CollapseIn Infinite
Dimension
“”t
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DMFT Equations
St
e
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“Dynamical” Mean Field
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Generic Impurity ModelAnderson impurity model:
Integrate out conduction degrees of freedom:
A solvable limit:
Lorentzian DOS
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Start with
Iterated Perturbation Theory
SOPT
FFT
Projection
Update
FFTConvergence
Yes
No
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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
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IPT for +i
Laplace transform
P-h bubble
SOPT diagram
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G
)
)
Optical Conductivity
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Ward Identity I
: Arbitrary quantum amplitude
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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
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(3)() In D=1
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Lehman Representation
General structure:
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Questions and commentsare welcome
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Lecture 4
t-J model $ Hubbard model How spin physics arises
fromStrong Electron Correlations?
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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
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Classifying HoppingsEnsure there is no + at i
Perform Hopping
Ensure there is a + at j
Double occupancy increaded: D D+1
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Correlated Hopping
Mind the local
correlationsDon’t care
the correlations
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Questions and commentsare welcome