part iii€¦ · "plaquette boson fermion model of cuprates" e. altman and a.auerbach,...
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![Page 1: Part III€¦ · "Plaquette Boson Fermion Model of Cuprates" E. Altman and A.Auerbach, Phys. Rev. B (in press), also cond-mat/0108087 Part III Derivation of Plaquette Boson-Fermion](https://reader036.vdocuments.us/reader036/viewer/2022071101/5fda1e087ca62a439303d195/html5/thumbnails/1.jpg)
References1. ``1nteracting Electrons and Quantum Magnetism'',
A.Auerbach, Springer-Verlag (NY).2. ``Quantum Magnetism Approaches to SCES'' ,
A. Auerbach, F. Berutto and L. Capriotti"Field Theories in Low .. Eds. G. Morandi et. al. Springer-verlag (00), also cond-mat/9801294)
3. “Projected SO(5) Hamiltonian for Cuprates and its Applications'' A.Auerbach and E. Altman,IJMPB 15, 2509 (01), also cond-mat/0009421.
4. "Plaquette Boson Fermion Model of Cuprates"E. Altman and A.Auerbach, Phys. Rev. B (in
press), also cond-mat/0108087
Part IIIDerivation of Plaquette Boson-Fermion
Model for Cuprates, using COntractor REnormalization
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732 OCuYBa42 CuOSrLa xx−
Tc = 400 Tc = 900
Bednorz &Muller ‘86 Wu et.al. ‘87
Cuprate (High Tc)Superconductors
What is the Mechanism? (Non BCS?)
Review 1
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“Hi Tc”spin fluctuationsd-wave superconductivityLow Superfluid DensityShort Coherence Length
Abnormal “Normal State”Mysteries of ARPESWeird TransportPseudogap temperatureT* versus doping
High Tc Phemomenology: Summary of Problems
unco
nven
tiona
l SC
Non
Fer
mi L
iuid
Review 2
IN NEED OF A MODEL & ITS SOLUTION
Hubbard Model:
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2D Hubbard Model : Previous Status
t-J Model: spins and holes are strongly entangled on lattice scale.
We need a lower effective H!
UndopedUndopedHeisenberg model, Neel phase.
Single hole:Single hole:ground state momentum (π/2,π/2)π/2,π/2)π/2,π/2)π/2,π/2), ε(ε(ε(ε(k)))) .
::::Two holesTwo holesPair binding on small clusters (numerical). Will they bind in large lattices?
Many holes:Many holes:Phase separation/stripes? Superconductivity?What destroys Neel order? Conflicting mean field theories…..
Review 3
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What is the effective Hamiltonian for the Hubbard Model in this regime?
Can it describe the experimental phase diagram?
Energy scales
5% 15% xunderdoped overderdoped
AF d-SC
∗T
Hubbard modelt ~ 0.5 eV
t << U
HeffTSC<12meV
TNeel ~30meVt-J model t>J
J ~ 0.1eV
spins-holes entanglement
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CORE
COntractor REnormalizationC.J. Morningstar & M. Weinstein PRD 54, ‘96
Discrete Real Space Renormalization
ijH efflkh: Full H.S. : Reduced H.S.
2D
chains
1 Step CORE: Microscopic d.o.f. (electrons, spins)
−−−−> > > > local collective d.o.f. (hole pairs, singlets, triplets)
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Erez Berg, Ehud Altman,AA, preprint.
Heff
Checkerboard
Pyrochlore
Deriving the low spectrum in local eigenstates representation
E
J
∑ ⋅=ij
jiJH SS
spin configuration
0
1 & 2 Step CORE
Highly frustrated magnets
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1. Truncate each plaquette to M states.
2. The Reduced Hilbert has size MΛΛΛΛ = |α_1,..α_Ν > << Λ= |α_1,..α_Ν > << Λ= |α_1,..α_Ν > << Λ= |α_1,..α_Ν > << Λ
N
Choose a connected cluster on N plaquettes (N=3)
3. Diagonalize H on the cluster and obtain n=1,,M ε , ψ ε , ψ ε , ψ ε , ψ
n nN
4. Gram-Schmidt Orthogonalize
The Scheme
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5. Determine renormalized Hamiltonian
6. Connected Renormalized Hamiltonian
conn. subclus.
7. Cluster Expansion of Full Lattice
Range 1 Range 2 Range 3
6. Truncate at finite range interactions OR
Pade approximate to infinite range.
The Scheme—cont’d
A similar cluster expansion exists for all observables
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Test: Tight Binding models
-4cos(k)
dimer zone edge
1D
2D
k k
ground state
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1. Quantum Hamiltonian Renormalization (no time retardation)
2. Non perturbative (in parameters of H)3. Estimable truncation error4. Efficient in memory usage:
stores interactions, not wavefunctions.
Test of CORE
1+⋅ ii SJS ∑=r
rhNE0
/0
C.J. Morningstar&M. Weinstein hep-lat/000202
S=1/2 Heisenberg chain
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The Hubbard Dimer
)0,0(),1(3 π×
),0( π
)0,0(
)0,(2 21×
),(2 21 π×
)0,0(
ε
Ut /4 2
U
+
n=2 n=0n=1),( kS
00|!f
0|!b
0|!αt
vacuum hole fermion
hole pair
magnon
21
24 SSUt rr
⋅
No pair binding, no d-wave hole pairs.
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The secret of High Tc Superconductivity hides in the Hubbard Plaquette!
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The Hubbard Plaquette
The Vacuum (RVB)
The (antiferro)magnon
The d-wave Hole Pair
- = + 1/3t ts s
|Ω> ∼|Ω> ∼|Ω> ∼|Ω> ∼
t |Ω> ∼|Ω> ∼|Ω> ∼|Ω> ∼s t t s
+
b |Ω> ∼|Ω> ∼|Ω> ∼|Ω> ∼ + + +
hole pair-magnon repulsionHeisenberg model
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hole pair magnon
hole fermion
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Pair binding
U/t
∆∆∆∆/t
432 2 EEE +−≡∆
Pair binding survives weak interplaquette coupling ∆<<'t
0 0 .0 5 0 .1 0 .1 5 0 .2-1 0
-9 . 8
-9 . 6
-9 . 4
-9 . 2
-9
-8 . 8
2nd order pert.
Exact
E0
t’/t
Ene
rgy
of 2
hol
es o
n th
e cl
uste
r
.Will it survive non perturbative coupling?
Pair binding on the plaquette
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N(holes)
E(N)
(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)(0,0)
(0,0)(0,0)(0,0)(0,0)
(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)
(0,0)(0,0)(0,0)(0,0)
(π/2,π/2)(π/2,π/2)(π/2,π/2)(π/2,π/2)
3 pairs in a condensate
Fermion holes in nodal pockets
Pair binding
Pair correlations in t-J model by DMRG (White&Scalapino)
short pair coherence length
44×
6x8
2 hole pairing
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CORE: The 4 Boson Model
Parameters table: cond-mat/01018087
hole pair
magnons
1=+αα α∑ iiii bbtt !!Hard core bosons:
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∑∑
∑∑
αααα=α
αα ++−ε+
+−µ−ε=
ijjjii
t
iiis
ijjib
iiib
pSO
ttttJtt
cHbbJbbH
))((2
).()2(
3,2,1
)5(
!!!
!!
Hole Pair Hopping
S-C Zhang, J.P. Hu, E. Arrigoni, W. Hanke, AA, PRB60, (99)
Projected SO(5) point
Projected SO(5) Theory
[ (
( )) ]∑∫∫
α ατ
ααµα
φ+ψ+ψ∇ρ+ψµ−∂ψ
φ−φ∂ρτ−ψφ=
22221*
222
12
||||)2(
)(exp
U
axddDDZ
cn
sm
b
s
continuum:
+
S.Sachdev
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h12 h12h 1 2 1h h
12 site Hubbard model
Hubbard Model CORE: Range-2
E. Altman
Coherence length for hole pairs and magnons isof order 1 plaquette
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The Pairing Mechanism
Hole Pair coherence length is of order one plaquette
1. Rapid decay of CORE interactions from range 2 to 3.2. Pair correlations in t-J model (White&Scalapino)
Kinematic (QED) hole pairing in a quantum spin liquid
Weigmann, Lee, Wen:
Pair correlations are short even without plaquette pair binding U/t > 4.5
Superconductivity stabilized by large value of pair hopping J ~ J b t
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Short Coherence Length
Tunneling Microscopy
Howald et.al (Stanford) Pan et.al
(Berkeley)
Unconventional Superconductivity cont’d
Hoogenboom et.al
020 A≈ξ
30 A
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Variational Theory
d-wave SC ground state
Neel State
Phase diagram
Results (U/t=8)
Magnon resonance
First order
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Experiments
underdoped
21λρ ∝c
xTcc ∝≈ρ Uemura’s Plot (89)
Antiferromagnetic resonance
resonance energy decreasing with doping
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Role of Pair Hopping
1. - J ΣΣΣΣ b b determines superfluid density, andT .
2. Competes with charge ordering.
b
c
3. Destroys antiferromagnetic order
Ordered Moment
Heisenberg model
Spin liquid
Pair hopping energy
Variational MC:Havilio and Auerbach,PRL'99
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Single hole dispersionSemiclassics and numerics yield same sublattice hopping
24 x 24
t’’ ~ 0.1 t’t’ ~ J
Quantum Monte-Carlo
ground state
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Hole Fermions
4t’’
4t’24 x 24t’’ ~ 0.1 t’t’ ~ J
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Plaquette Boson Fermion Model
Andreev coupling
1. Chemical equilibrium:
2.Induced fermion gap (vanishes at Tc)
nodal points
Fermion-magnon interaction
Im ΣΣΣΣ
ωωωω
Dip-hump spectral feature
Eschrig&Norman, Abanov&Chubukov
3. Dirac Fermions+QED Franz&Tesanovic
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Boson fermion theormodynamics
Boson density of states Hole fermion density of states
pseudogap
Hole entropy
Pair evaporation
chemical pot.
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Experimental Predictions1. Hole spectral weight in Luttinger theorem violating
momenta
(0,π)(0,π)(0,π)(0,π) (π,π)(π,π)(π,π)(π,π)
Luttinger FS
Forbiddenarea
Forbiddenshadow peaks
Hole pocket
Nodal direction
(0,π)(0,π)(0,π)(0,π) (π,π)(π,π)(π,π)(π,π)
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Predictions cont’d
Injecting holes
2. Asymmetry in tunneling conductance
~ x
Injectingelectrons
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Predictions cont’d
Nodal transverse velocity
v ~ Tc ~ ρρρρsc
Pairing gap
pseudogap
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Vacuum Crossing
underdoped is plaquettizedSpeculation:
Plaquette bosons
(0,π)(0,π)(0,π)(0,π) (π,π)(π,π)(π,π)(π,π)
Hole fermions
overdoped has full lattice symmetry
|Ω>|Ω>|Ω>|Ω>Hole pair
|Ω>|Ω>|Ω>|Ω>Hole pair
Increase µµµµ
new vacuumnew boson
Is there a phase transition?
(0,π)(0,π)(0,π)(0,π) (π,π)(π,π)(π,π)(π,π)
electrons
underdopedoverderdoped
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Summary
Hubbard model """" PBFM using CORE
Underdoped phase:
d-wave superconductivity
pSO(5) phase diagram
Pseudogap in fermion spectrum Future:
CORE with fermions (4x4clusters)
(with S. Capponni, Toulouse)
PBFM: Transport and Thermodynamics, vortex cores, surface states, impurity states, etc.
Underdoped-overdoped transition
Extended Hubbard models