Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
KITP, August 2002
1
Assa Auerbach,
Technion, Haifa, Israel
Collaborators:
Ehud Altman, Erez Berg, Technion
Outline
1. Renormalization and Effective Hamiltonians
2. Contractor Renormalization
3. 2-D Hubbard => Plaquette Boson-Fermion Model
4. Quantum Frustration: Checkerboard and Pyrochlore lattices
References:
Cuprates:
E. Altman and A. Auerbach, Phys. Rev. B65, 104508, 2002.
Pyrochlores: E. Berg, E. Altman and A. Auerbach, cond-mat/0206384;
PRL submitted.
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
1
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
The Cuprate Problem
T
Hubbard model
t-J model
Effective Hamiltonian
which includes:
T∗
Heff
1. Tightly bound d-wave pairs
AF
d-SC
2. Antiferromagnetic correlations
x
Can we get this from a strong repulsively interacting electrons
model ?
Problem: Quantum Frustration
2
+ const
H = J ∑ Si ⋅ S j = J ∑ Stet
ij
tet
Checkerboard
Pyrochlore
Extensive number (N/2)
degrees of freedom in
classical GS manifold
Spinwave theory is poorly controlled
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
2
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Highly frustrated magnets
E
H = J ∑Si ⋅ S j
ij
J
0
Checkerboard
Heff = ?
Pyrochlore
spin configuration
Strong Quantum fluctuations (spin-½) ?
Spin-½ Checkerboard Antiferromagnet
E
Exact diagonalization
(Palmer and Chalker, 2001)
J
How to describe ground state and low energy singlets ?
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
3
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
k-Space Renormalization (Shankar)
K
K'
Eliminate high momenta single particle states
(
Z micro = ∫ D 2 z exp − ∫ dτz ∂ τ z − H [ z , z ]
)
= ∫ D 2 z k < K' exp(− S [ z ( τ' ), τ ] )
S K' ≈
∑ (iω − ε
k
k < K'
− Σ( k , ω )) zk zk − f zk z k' z k'' zk +k' − k'' + ...
micro
"Effective Hamiltonian"
Renormalization group: S(K)--> S(K')
Couplings flow:
f(K),
f
K''
K'
Σ(Κ)
Provision: S(K') should be similar to S(K)
Σ
Renormalization by Cannonical Transformation
H = −t ∑cis c js +U ∑ni↑ni↓
+
ij
Schrieffer-Wolff ->
Advantages
i
U
4t 2
H ≈ ∑Si ⋅ S j + O( t U , EU )
U ij
eff
Problems:
1. Reduced Hilbert Space
+
s
S = c σ ss' cs' (1) Pertubative limit of (t/U)<<1
2. Bosonic degrees of freedom
1
2
3. Better mean field approximations
(2) H(E) is not really a Hamiltonian
for Heisenberg than for the Hubbard Model
(large S, Large N)
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
4
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Direct Numerical Correlations
L > ξcorrelation
Need
L
ξcorrelation
<S>
L 4x4
1/ ξ
1/L
Exact diagonalization computation time is
( )
≈ exp L d
If ξ is large, it is futile to try to extract thermodynamic correlations
What are numerics good for?
Emerging Low Energy Degres of Freedom
GeV
quarks&leptons
MeV
eV
nucleons
atoms
0.1 eV
chemical bonds
OH
C C
C
The Captain's Weight
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
5
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Numerical Renormalization
Numerics
ξcorrelation
Heffective
Need only L > ξcoherence !
L
ξ coherence
ξ coherence= size of “atoms”
L
How do we identify “atoms” and calculate
effective couplings??
Contractor Renormalization (CORE)
Morningstar-Weinstein, PRD (1996)
Step I: Divide lattice to disjoint blocks
Truncate:
“atoms”
{φ }
M
N
M lowest states per block
i
i=1
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
6
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Step II: Hren for a particular cluster of N blocks
α = φi1 ,φ i 2 ,φi N
Reduced Hilbert space:
( dim= MN )
ground
state
up.
(Gramm-Schmidt)
1. Orthonormalize
2.
3.
Diagonalize
Project
onfrom
reduced
H on
the
Hilbert
connected
space
cluster.
εn , n ,
H
ren
(1 , , N
∑P
α
)
≡
M
ψn
n
N
∑ ε n ψ~n ψ~ n
ren
h ( 1 ,... N ) ≡ H (1 ,
,N
n =1
)−
conn . subclus .
∑h
i1 ,... i L
i1 ,... i L
Step III: Cluster Expansion
2. CORE Exact Identity:
ren
∑hijk h+ sub∑−cluster
,i n h
H effhi1=, ,∑
hijk ...
(i1 ,∑
)−
in =hH
i +
ij + ∑
i
ij
+
connected
ijk
sub-clusters
+
+
ijk ..
+
3. Truncate at finite range
truncation error?
A short emerging length scale
ξ coherence controls expansion
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
7
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Test: 1D Heisenberg
S=1/2 Heisenberg chain
E0 / N = ∑r h0r
JSi ⋅ S i +1
C.J. Morningstar&M. Weinstein hep-lat/000202
Heisenberg Ladders: Piekarewicz and Shepard,(1997)
Altman & A.A., (2001), Poilblanc & Capponi, preprint.
Hubbard Plaquette States
tα
b
π,π) Triplet
(π
D-wave hole pair
Ω ~
+
= RVB vacuum
AFmagnet and Superconductor degrees of freedom !
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
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Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Coupling Plaquettes (CORE)
hole fermion
hole pair
magnon
computed interplaquette
hopping & interactions
"Plaquette Boson Fermion Model of Cuprates"
E. Altman and A.A., Phys. Rev. B65, 104508 2002.
Coupling plaquettes:
Failure of perturbative approach
t’
Energy of 2 holes on the cluster
-8.8
t′
JC ∝
∆b
2
Pair hopping:
Fails for
t' > ∆b
-9
-9.2
E0
-9.4
2nd order pert.
-9.6
-9.8
-10
0
0.0 5
0.1
0.1 5
0.2
t’/t
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
9
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Good convergence of the cluster expansion!
H (approx
1, 2 ,3 )
hijk
−1
hij ~ 10 − 10
−2
h1
h12
h2
h23
H (1, 2,3)
h3
2 holes
⇓
Short coherence length!
Half filling
filling
Half
E/t
S.H. Pan etal (PRL 00)
Pairs keep their integrity on the full
lattice!!
Pair Integrity
Pair correlations in t-J model by
DMRG (White&Scalapino)
6x8
Hole pairs stay tightly
bound in larger clusters
short pair
coherence length
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
10
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Four Boson Model
Projected SO(5) Theory
S-C Zhang, J.P. Hu,
E. Arrigoni, W. Hanke,
A. Auerbach, PRB60, (99)
Numerics:
Dorneich, Hanke,Arrigoni,
Troyer, Zhang,(02)
Higher order interactions:
Projected SO(5) point
Similar scale for pair
and magnon hopping!
Small superfluid density
T
H =
1
2
((
))
ρ c ∫ d 2 x ∇ + iA ϕ
2
Ginzburg-Landau
T∗
ρ
c
∝ 1 λ2
Uemura’s Plot (89)
AF
d-SC
x
BCS
Tc
2
ρc ≈ ne
≈ ε ≈1eV
2m F
Tc ≈ ωDe−1/ λ ≈10−3eV
ne ≈ 1a2
Tc ,ρc , ne unrelated
ρ c ∝ 1 − ne
BEC of real-space pairs
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
11
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Plaquettized Lattice?
Experiment
Howland etal (02)
x=0.125
i
Q
r
condensate wave br ∝ e
Modulation of period 4 in
Q= ( π 2 ,0 ) + ( 0, π2 )
tunneling conductance
Stripes,
or Plaquettized order parameter?
Back to the Pyrochlores:
back
Goal: Heff in terms of lowest tetrahedron states
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
12
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Tetrahedron eigenstates
E
S=2
H(
) = JStot2 + c
S=1
S=1
S=1
2J
S=0
S=0
1
Pseudospin Variable:
3
1
2
Effective Hamiltonian
Heff =
∑h
∑h
ijkl
ijk
ij
(
∑h
∑h
ij
i
i
ijkl
ijk
)(
)
ˆ S ⋅Ω
ˆ − h ∑Sz
⇒ H eff = − J I ∑ S i ⋅ Ω
ij
j
ij
I
i
ij
ˆ
Ω
Verticle
=
i
JI ≅ J 2
hI ≅ J 4
ˆ
Ω
Horizontal =
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
13
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Mean Field solution
0.15
ene rgy pe r pla que tte
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
-1
-0.5
0
θ/π
0.5
1
Direction of pseudospins
Thermodynamics
Crossed plaquettes are
symmetric with respect to the
degenerate ground states!
Singlet Excitations
Ising Domain walls
Palmer and Chalker (2001)
Heff
Spin flip excitation energy ~ J/2
Number of low singlet excitation
grows linearly with size
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
14
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Quantum Pyrochlore Antiferromagnet
No MF order down to
zero temperature!
Villain (79);
Moessner and Chalker (98);
Spectrum of a finite cluster of the spin-½ Pyrochlore
-6.2
3
E [J ]
-6.4
3
3 3
-6.6
3
-6.8
3
1
2
3
1
2
2
1
3
3
-7
2
0
2
6 S(S+1)
First CORE Step
First stage: Tetrahedral clustering
Pseudospins defined on
a FCC lattice
Effective 3-body hamiltonian
1
 1
 1

H eff ≈ J 3 ∑  + Si ⋅ eijk  + S j ⋅ eijk  + S k ⋅ eijk 
 2
 2

ij  2
Perturbative Expansion: Harris, Berlinsky,Bruder (92)
MF level: four sublattice “Order”
Remaining macroscopic degeneracy!
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
15
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Second CORE step
Second stage: “Super Tetrahedral” clustering
Basic block:
“supertetrahedron”
Spectrum of a
single block:
Cubic lattice
Pseudospin - ½
Mean Field Solution
ij
(
J b ∑ Si ⋅ eij
)(S ⋅ e )+
j
ij
)(S ⋅ e )+ ...
j
z
(
H eff = J a ∑ Si ⋅ eij
ij
ij
Mean field ground state:
y
x
Supertetrahedron pseudospins
• Ground state: 6 fold degeneracy, rotational and
translational symmetry breaking
•Coherence length: ~ single supertetrahedron
•Domain wall excitations
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
16
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Energy Scale of the effective couplings
E/J
Effective model
pyrochlore
1
Fcc
10-1
Cubic
z
10-2
y
x
Summary
1. CORE renormalizes a microscopic Hamiltonian to an effective
Hamiltonian, non perturbatively.
2. The truncation error can be controlled by a short
coherence length (``atom size'').
3. Applying CORE to the Hubbard Model yields
an effective Plaquette Boson Fermion Model.
4. Applying CORE to the Pyrochlore yields a lattice symmetry
breaking ground state and very low lying singlet excitations.
Future:
Development of larger scale CORE computations (S. Capponi)
Explore the PBFM.
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
17
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Pair Binding on a
Plaquette
∆ ≡ E2 −2E3 + E4 < 0
| ∆|
Pair binding was found for Hubbard
and t-J clusters close to half filling!
(Hirsch et.al., Fye et.al 89.)
Pair binding
∆/t
Is this the Pairing Mechanism? U/t
Test: Tight Binding models
2D
-4cos(k)
ground state
k
dimer zone edge
1D
k
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
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Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Fermion holes
24 x 24
εk ≈ t′(cosk x + cos k y )2 + t′′(cos kx − cos k y )2
4t’
4t”
Plaquette
fermions
EQ
(0,π )
(π ,π )
(π,0)
(0,0)
ZQ
(0,π )
(0,0)
(π,π)
(π,0)
Rajiv1
dnk / dT
h
e
∇nk
h
e
Non Interacting, Tight Binding
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
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Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
dnk / dT ( T = 0.5 J )
dnk / dT ( T = 0.3J )
Rajiv2e
e
h
h
e
e
t-J Model
Plaquette Boson – Fermion Model
H PBFM = H bosons + H fermions + H bf
STM
ε
Boson DOS
Hole fermion DOS
Pseudogap
∆ ≅ Ek sp − µ (x, T )
pg
Bosons
condense
out of vortex
at q = 0
µ (x ) − µ (0 ) =
in vortex
In vortex
x
b +κ f )
Superconducting gap
out of
vortex
S.H. Pan etal (PRL 00)
(2κ
∆sck = g bd k b
Hole fermions at
q ≈ (± π2 , ± π2 )
Ek =
(ε k − µ )2 + (∆SCk )2
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
20
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Numerical support
4X4 cluster
Holes in pockets
E(N)
(π/2,π/2)
(π/2,π/2)
(π/2,π/2)
(0,0)
(0,0)
(0,0)
(0,0)
N(holes)
3 pairs in a
condensate
Pair condensate
The problem of Two Gaps
Pseudogap decreases with doping
(measures chemical potential)
µ (x ) − µ (0 ) =
x
(2κ b + κ f )
∆pg ≅ Ek sp − µ (x, T )
Coherence gap increases with doping
(measures hole pairs order parameter)
Tunneling:
pseudogap
∆sck = g b d k b
Prediction:
∆ SC ∝ TC
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
21
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
Short coherence length
T
ξ ∼ 20 A ~ 5a
T∗
AF
d-SC
100 nm
x
Antiferromagnetic resonance
in neutron scattering
Small vortex cores in
STM experiments
Pan etal (PRL 00)
Phase diagram (Monte Carlo)
Dispersion of the (π
π,π)
triplet
Experiment
Uemura' splot
ρc ∝ x
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
22
Contractor Renormlization for the 2D Hubbard and Frustrated Heisenberg Models
1
 1
 1

hijk ≈ J 3  + S j ⋅ eijk  + S j ⋅ eijk  + S k ⋅ eijk 
2
 2
 2

i
j
k
Dr. Assa Auerbach, Technion (KITP Correlated Electron Materials 8/23/02)
23