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= -1
Perfect diamagnetism (Shielding of magnetic field)
(Meissner effect)
Dynamic variational principle and the phase diagram of high-temperature superconductors
André-Marie Tremblay
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CuO2 planes
YBa2Cu3O7-
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Experimental phase diagram
n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)
++-
-
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Ut
Simplest microscopic model for Cu O2 planes.
t’t’’
LCO
H ij t i,j c i c j c j
c i Uinini
The Hubbard model
No mean-field factorization for d-wave superconductivity
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T
UU
A(kF,)A(kF,)
Weak vs strong coupling, n=1
Mott transitionU ~ 1.5W (W= 8t)
LHB UHB
tEffective model, Heisenberg: J = 4t2 /U
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Theoretical difficulties
• Low dimension – (quantum and thermal fluctuations)
• Large residual interactions – (Potential ~ Kinetic) – Expansion parameter? – Particle-wave?
• By now we should be as quantitative as possible!
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Methodology
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Heuristics
M. Potthoff et al. PRL 91, 206402 (2003).
C-DMFTV-
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Dynamical “variational” principle
tG G TrG0t 1 G 1 G Tr ln G
GG
tG
G G0t
1 G 1 0
G 1G0t
1
Luttinger and Ward 1960, Baym and Kadanoff (1961)
G + + + ….
H.F. if approximate by first order
FLEX higher order
Universality
Then is grand potentialRelated to dynamics (cf. Ritz)
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Another way to look at this (Potthoff)
t G Tr G Tr ln G0t 1
GG
tG G TrG0t 1 G 1 G Tr ln G
M. Potthoff, Eur. Phys. J. B 32, 429 (2003).
t F Tr ln G0t 1
Still stationary (chain rule)
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SFT : Self-energy Functional Theory
M. Potthoff, Eur. Phys. J. B 32, 429 (2003).
t t Tr ln G0 1 1 Tr ln G0
1 1.
With F Legendre transform of Luttinger-Ward funct.
t F Tr ln G0 1 1 #
For given interaction, F is a universal functional of no explicit dependence on H0(t). Hence, use solvable cluster H0(t’) to find F
Vary with respect to parameters of the cluster (including Weiss fields)
Variation of the self-energy, through parameters in H0(t’)
is stationary with respect to and equal to grand potential there
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The T = 0 ordered phases
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Different clusters
L = 6
L = 8
L = 10The mean-fields decrease with system size
David Sénéchal
V-CPT
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Order parameters for competing d-SC and AF
HM M a 1ana na
HD a,b
abcacb H.c.
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Experimental Phase diagram
Optimal dopingOptimal doping
n, electron density
Hole dopingElectron doping
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
Stripes
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AF order parameter, size dependence
U = 8tt’ = -0.3tt’’ = 0.2t
e-dopedh-doped
e h
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AF order parameter, size dependence
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AF order parameter, size dependence
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dSC order parameter, size dependence
Lichtenstein et al. 2000
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dSC order parameter, size dependence
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dSC order parameter, size dependence
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AF and dSC order parameters, U = 8t, for various sizes
dSC
AF
Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005
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AF and dSC order parameters, L=8, for various U
dSC
AF
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Other numerical results
• Th. Maier et al. PRL 85, 1524 (2000)
• A. Paramekanti et al. PRL 87 (2001).
• S. Sorella et al. PRL 88, 2002
• D. Poilblanc et al. cond-mat/0202180 (2002)
• Th. Maier et al. cond-mat/0504529
• Review: Th. Maier, RMP (2005)
Here AFM and d-SC for both hole and electron doping
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The ARPES spectrum
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Fermi surface plots, U = 8t, L = 8
15%
10%
4%
MDC at the Fermi energy
Wave-particle
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Single-particle (ARPES) spectrum, e-dopedn = 1.1
For coexisting AF and d-SC
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Comparison with mean-field n = 1.1
AF
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Single-particle (ARPES) spectrum, e-doped n = 1.1
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Armitage et al. PRL 87, 147003; 88, 257001
H.Kusunose et al., PRL 91, 186407 (2003)
Observation of in-gap states
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Single-particle (ARPES) spectrum, e-dopedn = 1.1
Mott gap
AF
Atomic limit
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Dispersion, U = 8t, L = 8, n =1.1
d-wave gap
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e-doped
h-doped
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Conclusion
• With V-CPT • t-t’-t’’-U Hubbard model
has AF and d-SC where expected from experiment on both h-, e-doped cuprates. (fixed t-t’-t’’)
• ARPES in agreement with experiment (40 meV)
• Homogeneous coexistence of AF + dSC is stable on e-doped side.
• D. Sénéchal, Lavertu, Marois AMST PRL 2005
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Conclusion
• The Physics of High-temperature superconductors is in the Hubbard model (with a very high probability).
• We are beginning to know how to squeeze it out of the model!
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Mammouth, série
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C’est fini…
enfin
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Alexis Gagné-Lebrun
Bumsoo Kyung
A-M.T.
Sébastien Roy
Alexandre Blais
D. Sénéchal
C. Bourbonnais
R. Côté
K. LeHur
Vasyl Hankevych
Sarma Kancharla
Maxim Mar’enko
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Steve Allen
François Lemay
David Poulin Hugo Touchette
Yury VilkLiang Chen
Samuel Moukouri
J.-S. Landry M. Boissonnault
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Theory without small parameter: How should we proceed?
• Identify important physical principles and laws to constrain non-perturbative approximation schemes– From weak coupling (kinetic)– From strong coupling (potential)
• Benchmark against “exact” (numerical) results.
• Check that weak and strong coupling approaches agree at intermediate coupling.
• Compare with experiment
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Starting from weak coupling, U << 8t
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Theory difficult even at weak to intermediate coupling!
• RPA (OK with conservation laws)– Mermin-Wagner– Pauli
• Moryia (Conjugate variables HS <– Adjustable parameters: c and Ueff
– Pauli
• FLEX– No pseudogap– Pauli
• Renormalization Group– 2 loops
XXX
Rohe and Metzner (2004)Katanin and Kampf (2004)X
= +1
32 2
13 3
12
2
3
4
5
-
Vide
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• General philosophy– Drop diagrams– Impose constraints and sum rules
• Conservation laws
• Pauli principle ( <n> = <n > )
• Local moment and local density sum-rules
• Get for free: • Mermin-Wagner theorem
• Kanamori-Brückner screening
• Consistency between one- and two-particle G = U<n n->
Two-Particle Self-Consistent Approach (U < 8t)- How it works
Vilk, AMT J. Phys. I France, 7, 1309 (1997); Allen et al.in Theoretical methods for strongly correlated electrons also cond-mat/0110130
(Mahan, third edition)
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TPSC approach: two steps
I: Two-particle self consistency
1. Functional derivative formalism (conservation laws)
(a) spin vertex:
(b) analog of the Bethe-Salpeter equation:
(c) self-energy:
2. Factorization
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TPSC…
II: Improved self-energy
3. The F.D. theorem and Pauli principle
Kanamori-Brückner screening
Insert the first step resultsinto exact equation:
n n 2 n n 2nn
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Benchmark for TPSC :Quantum Monte Carlo
• Advantages of QMC– Sizes much larger than exact diagonalizations– As accurate as needed
• Disadvantages of QMC– Cannot go to very low temperature in certain
doping ranges, yet low enough in certain cases to discard existing theories.
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-5.00 0.00 5.00 -5.00 0.00 5.00/ t
(0,0)
(0, )
( , )2
2
( , ) 4 4
( , ) 4 2
(0,0) (0, )
( ,0)
Monte Carlo Many-Body
-5.00 0.00 5.00
Flex
U = + 4
Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).
Proofs...
TPSC
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Moving on to experiment : pseudogap
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Fermi surface, electron-doped caseArmitage et al. PRL 87, 147003; 88, 257001
15%
10%
4%
15%10%4%Pseudogap at hot spots
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15% doping: EDCs along the Fermi surfaceTPSC
Exp
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004
Umin< U< Umax
Umax also from CPT
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Hot spots from AFM quasi-static scattering
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AFM correlation length (neutron)
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 Expt: P. K. Mang et al., cond-mat/0307093, Matsuda (1992).
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Pseudogap temperature and QCP
ΔPG≈10kBT* comparable with optical measurements
Prediction QCPmay be masked by 3D transitions
Hankevych, Kyung, A.-M.S.T., PRL 2004 : Expt: Y. Onose et al., PRL (2001).
Prediction
ξ≈ξth at PG temperature T*, and ξ>ξth for T<T*
supports further AFM fluctuations origin of PG
Prediction
Matsui et al. PRL (2005)Verified theo.T* at x=0.13with ARPES
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Observation
Matsui et al. PRL 94, 047005 (2005)
Reduced, x=0.13AFM 110 K, SC 20 K
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Precursor of SDW state(dynamic symmetry breaking)
• Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769-1771 (1995).
• Y. M. Vilk, Phys. Rev. B 55, 3870 (1997).
• J. Schmalian, et al. Phys. Rev. B 60, 667 (1999).
• B.Kyung et al.,PRB 68, 174502 (2003).• Hankevych, Kyung, A.-M.S.T., PRL, sept 2004
• R. S. Markiewicz, cond-mat/0308469.
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What about d-wave superconductivity?
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1.00.8
0.60.4
0.2
0.0
43
21
0.100.150.200.250.300.350.400.450.500.55
0.60
0.65
0.70
0.75
U = 0 U = 4 U = 6 U = 8 U = 10
dx2-y2-wave susceptibility for 6x6 lattice
d x2 -y2
0.0 0.2 0.4 0.6 0.8 1.00.1
0.2
0.3
0.4
0.5
0.6
0.7 L = 6 U = 0, = 4
U = 4
d
Doping
= 4 = 3 = 2 = 1
QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)
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QMC: symbols. Solid lines analytical Kyung, Landry, A.-M.S.T., Phys. Rev. B (2003)
1.00.8
0.60.4
0.2
0.0
43
21
0.100.150.200.250.300.350.400.450.500.55
0.60
0.65
0.70
0.75
U = 0 U = 4 U = 6 U = 8 U = 10
dx2-y2-wave susceptibility for 6x6 lattice
d x2 -y2
0.0 0.2 0.4 0.6 0.8 1.00.1
0.2
0.3
0.4
0.5
0.6
0.7 L = 6 U = 0, = 4
U = 4
d
Doping
= 4 = 3 = 2 = 1
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QMC: symbols. Solid lines analytical. Kyung, Landry, A.-M.S.T. PRB (2003)
1.00.8
0.60.4
0.2
0.0
43
21
0.100.150.200.250.300.350.400.450.500.55
0.60
0.65
0.70
0.75
U = 0 U = 4 U = 6 U = 8 U = 10
dx2-y2-wave susceptibility for 6x6 lattice
d x2 -y2
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7L = 6 U = 0, = 4
U = 4
d
Doping
= 4 = 3 = 2 = 1
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0.00 0.05 0.10 0.15 0.20 0.250.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
L = 256
Tc
Doping
U = 4 U = 6
Kyung, Landry, A.-M.S.T. PRB (2003)
DCA Maier et al. cond-mat/0504529
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What about the pseudogap in large U?
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Vary cluster
shape and size
D. Sénéchal et al., PRL. 84, 522 (2000); PRB 66, 075129 (2002), Gross, Valenti, PRB 48, 418 (1993).
W. Metzner, PRB 43, 8549 (1991).Pairault, Sénéchal, AMST, PRL 80, 5389 (1998).
Cluster perturbation theory (CPT)
David Sénéchal
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Fermi surface, hole-doped case 10%
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Hole-doped (17%)
t’ = -0.3tt”= 0.2t
= 0.12t= 0.4t
Sénéchal, AMT, PRL 92, 126401 (2004).
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Hole doped, 75%, U = 16 t
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Hole-doped 17%, U=8t
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Electron-doped (17%)
t’ = -0.3tt”= 0.2t
= 0.12t= 0.4t
Sénéchal, AMT, PRL in press
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4x4
Electron-doped 12.5%, U=8t
15%
10%
4%
12.5%
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Electron-doped, 17%, U=4t
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Strong coupling pseudogap (U > 8t)
• Different from Mott gap that is local (all k) not tied to .
• Pseudogap tied to and only in regions nearly connected by (e and h),
• Pseudogap is independent of cluster shape (and size) in CPT.
• Not caused by AFM LRO– No LRO, few lattice spacings.– Not very sensitive to t’– Scales like t.
Sénéchal, AMT, PRL 92, 126401 (2004).
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Weak-coupling pseudogap
• In CPT – is mostly a depression in
weight – depends on system size
and shape.– located precisely at
intersection with AFM Brillouin zone
• Coupling weaker because better screened U(n) ~ d/dn U=4
n
Sénéchal, AMT, PRL 92, 126401 (2004).
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Conclusion
• Ground state of CuO2 planes (h-, e-doped)
– V-CPT, (C-DMFT) give overall ground state phase diagram with U at intermediate coupling.
– TPSC reconciles QMC with existence of d-wave ground state.
0.0 0.2 0.4 0.6 0.8 1.00.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7L = 6 U = 0, = 4
U = 4
d
Doping
= 4 = 3 = 2 = 1
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Conclusion
• Normal state (pseudogap in ARPES)– Strong and weak
coupling mechanism for pseudogap.
– CPT, TPSC, slave bosons suggests U ~ 6t near optimal doping for e-doped with slight variations of U with doping.
U=5.75
U=5.75
U=6.25
U=6.25