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Transcript of = -1 Perfect diamagnetism (Shielding of magnetic field) (Meissner effect) Dynamic variational...
= -1
Perfect diamagnetism (Shielding of magnetic field)
(Meissner effect)
Dynamic variational principle and the phase diagram of high-temperature superconductors
André-Marie Tremblay
CuO2 planes
YBa2Cu3O7-
Experimental phase diagram
n, electron density Damascelli, Shen, Hussain, RMP 75, 473 (2003)
++-
-
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
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
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!
Methodology
Heuristics
M. Potthoff et al. PRL 91, 206402 (2003).
C-DMFTV-
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)
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)
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
The T = 0 ordered phases
Different clusters
L = 6
L = 8
L = 10The mean-fields decrease with system size
David Sénéchal
V-CPT
Order parameters for competing d-SC and AF
HM M a 1ana na
HD a,b
abcacb H.c.
Experimental Phase diagram
Optimal dopingOptimal doping
n, electron density
Hole dopingElectron doping
Damascelli, Shen, Hussain, RMP 75, 473 (2003)
Stripes
AF order parameter, size dependence
U = 8tt’ = -0.3tt’’ = 0.2t
e-dopedh-doped
e h
AF order parameter, size dependence
AF order parameter, size dependence
dSC order parameter, size dependence
Lichtenstein et al. 2000
dSC order parameter, size dependence
dSC order parameter, size dependence
AF and dSC order parameters, U = 8t, for various sizes
dSC
AF
Sénéchal, Lavertu, Marois, A.-M.S.T., PRL, 2005
AF and dSC order parameters, L=8, for various U
dSC
AF
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
The ARPES spectrum
Fermi surface plots, U = 8t, L = 8
15%
10%
4%
MDC at the Fermi energy
Wave-particle
Single-particle (ARPES) spectrum, e-dopedn = 1.1
For coexisting AF and d-SC
Comparison with mean-field n = 1.1
AF
Single-particle (ARPES) spectrum, e-doped n = 1.1
Armitage et al. PRL 87, 147003; 88, 257001
H.Kusunose et al., PRL 91, 186407 (2003)
Observation of in-gap states
Single-particle (ARPES) spectrum, e-dopedn = 1.1
Mott gap
AF
Atomic limit
Dispersion, U = 8t, L = 8, n =1.1
d-wave gap
e-doped
h-doped
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
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!
Mammouth, série
C’est fini…
enfin
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
Steve Allen
François Lemay
David Poulin Hugo Touchette
Yury VilkLiang Chen
Samuel Moukouri
J.-S. Landry M. Boissonnault
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
Starting from weak coupling, U << 8t
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
• 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)
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
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
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.
-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
Moving on to experiment : pseudogap
Fermi surface, electron-doped caseArmitage et al. PRL 87, 147003; 88, 257001
15%
10%
4%
15%10%4%Pseudogap at hot spots
15% doping: EDCs along the Fermi surfaceTPSC
Exp
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004
Umin< U< Umax
Umax also from CPT
Hot spots from AFM quasi-static scattering
AFM correlation length (neutron)
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004 Expt: P. K. Mang et al., cond-mat/0307093, Matsuda (1992).
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
Observation
Matsui et al. PRL 94, 047005 (2005)
Reduced, x=0.13AFM 110 K, SC 20 K
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.
What about d-wave superconductivity?
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)
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
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
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
What about the pseudogap in large U?
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
Fermi surface, hole-doped case 10%
Hole-doped (17%)
t’ = -0.3tt”= 0.2t
= 0.12t= 0.4t
Sénéchal, AMT, PRL 92, 126401 (2004).
Hole doped, 75%, U = 16 t
Hole-doped 17%, U=8t
Electron-doped (17%)
t’ = -0.3tt”= 0.2t
= 0.12t= 0.4t
Sénéchal, AMT, PRL in press
4x4
Electron-doped 12.5%, U=8t
15%
10%
4%
12.5%
Electron-doped, 17%, U=4t
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).
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).
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
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