sponsors€¦ · 36 - slightly overdoped high-tc superconductor tlsr2cacu2o6.8 guo-qing zheng et...
TRANSCRIPT
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André-Marie Tremblay
Sponsors:
CENTRE DE RECHERCHE SUR LES PROPRIÉTÉS ÉLECTRONIQUES
DE MATÉRIAUX AVANCÉS
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I. All is not well with the theory of solids
II. A microscopic model
III. Inching our way up the weak-coupling regime.
IV. What was the competition up to?
V. Conclusion
Mathematics, Physics, and Computers : strongly correlated electrons intwo dimensions as a case study.
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H = Kinetic + Coulomb
- Many new ideas and concepts needed for progress(Born-Oppenheimer, H-F, Bands...)
- Successful program - Semiconductors, metals and superconductors- Magnets
- Is there anything left to do?- Unexplained materials: High Tc, Organics...- Strong correlations:
strong interactions, low dimensionstrong fluctuations
Theory of solids
I. All is not well with the theory of solids
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4
Quasiparticles, Fermi surface and Fermi liquids- LDA (Nobel prize 1998)
L.F. Mattheiss, Phys. Rev. Lett. 58, 1028 (1987).
La2CuO4
The standard approaches :
-
ePhoton
= E + ω + µµµµ - W k2m
2
k
ph
Angle-Resolved Photoemission (ARPES)
Quasi 2-d material
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6
T. Valla, A. V. Fedorov, P. D. Johnson, and S. L. HulbertP.R.L. 83, 2085 (1999).
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n = 1, Metal according to bandAFM insulator in reality
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8
Optimally doped BISCCONormal Superconductor
A. V. Fedorov, T. Valla, P. D. Johnson et al. P.R.L. 82, 2179 (1999)
- d=2 partial vanishing actof the Fermi surface awayfrom n = 1.
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κ−(BEDT)2X
II. A microscopic model
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Ut
•Compute [ ][ ]
[ ][ ]Tr Oe
Tr e
H k T
H k T
B
B
−
−
/
/
•Size of Hilbert space : 4 N (N = 16)
µ
Simplest microscopic model for Cu Oplanes.
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11
H = ¡ X¾
ti;j³cyi¾cj¾ + c
yj¾ci¾
´+ U
Xi
ni"ni#
Hubbard model (Kanamori, Gutzwiller, 1963) :
- Screened interaction U- U, T, n- a = 1, t = 1, h = 1
- 2001 vs 1963: Numerical solutions to check analytical approaches
t
t
Ut
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12
A(k,ω)
ω- U/2 U/2
ω
k
π/a−π/a
ω
k
π/a−π/a
ω
A(k,ω)
- 4 t
A(k,ω)
+ 4 t
U = 0 t = 0
U
-U/2
+U/2
µ
ω
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13
T
U
A(kF,ω)
A(kF,ω)A(kF,ω)
A(kF,ω)
A(kF,ω)
∆∆
U U
U
D. Sénéchal
ω
ωω
ω
ω
Weak vs strong coupling
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•QMC•André Reid, Christian Boily, Hugues Nélisse
L. Chen, C. Bourbonnais, T. Li, and A.–M. S. TremblayPhys. Rev. Lett. 66, 369-372 (1991)
Liang Chen
Claude Bourbonnais
III. Inching our way up the weak coupling regime
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•Problems:
•Cannot compute charge structure factor with satisfactory accuracy
•Predicts a finite T antiferromagnetic phase transition in d = 2
•Contradicts Mermin-Wagner theorem
( )∇ → → → ∞∫θ θ θ θ α2 2 2 2 2q k T d q k Tqq q BB
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•Yury Vilk, 1993
•Forget diagrams
•Keep RPA form since satisfies conservation laws
•Determine renormalized interaction from sum-rule (Singwi)
•(Double-occupancy determined self-consistently)
•Get the charge fluctuations from Pauli principle
•Mermin-Wagner theorem automatically satisfied
Yury Vilk
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17
A non-perturbative approach for both U > 0 and U < 0
q0.0
0.4
(0,0) (π,π)(π,0) (0,0) q0
20.0
0.4
0
2
n S c
h(q)
n S s
p(q)
U = 8
U = 8
U = 4
U = 4n = 0.26
n = 0.94n = 0.60
n = 0.45n = 0.20
n = 0.80n = 0.33n = 0.19
n = 1.0n = 0.45n = 0.20
(a)
(b) (c)
(d)
U > 0
QMC + cal.: Vilk et al. P.R. B 49, 13267 (1994)
Notes:-F.L.parameters
-Self alsoFermi-liquid
- Proofs that it works
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0.50 0.75 1.00T
0.0
0.5
1.0
1.5
2.0U=4.0n=0.87
Monte Carlo
8x8 (BWS)
8*8 (BW)
This work
FLEX no AL
FLEX
π,π
χ(
,0)
sp
U = + 4
QMC: Bulut, Scalapino, White, P.R. B 50, 9623 (1994).Calc.: Vilk, et al. J. Phys. I France, 7, 1309 (1997).
Proofs...
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0.0 0.2 0.4 0.6 0.8 1.0T
0
4
8
12
16S s
p(π
,π)
Monte Carlo
4x4
6x6
8x8
10x10
12x12
ξ = 5
U=4n=1
TX Tmf
Calc.: Vilk et al. P.R. B 49, 13267 (1994) QMC: S. R. White, et al. Phys. Rev. 40, 506 (1989).
)( ∞=NO A.-M. Daré, Y.M. Vilk and A.-M.S.T Phys. Rev. B 53, 14236 (1996)
( )ξ ~ exp ( ) /C T T
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20
= +
1
32 2
1
33
12
2
3
4
5
=Σ1 2 +
-
2
45
2
1
1
21
What about single-particle properties? (Ruckenstein)
Y.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995).Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996);
N.B.: No Migdal theorem
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Quantitative agreement with QMC
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Qualitatively new result: effect of critical fluctuations on particles (RC regime)
!ωsf Bk Tξ ξ ξ
ξ ξ ξ π> ≡th th F Bv k T( / )!
∆ ∆ ∆ε ε≈ ∇ ⋅ ≈ =k F Bk v k k T!
Σ" , ln /R
F thUk 0 02a f a f d i∝ − ξ ξ ξ
TcY.M. Vilk and A.-M.S. Tremblay, J. Phys. Chem. Solids 56, 1769 (1995).Y.M. Vilk and A.-M.S. Tremblay, Europhys. Lett. 33, 159 (1996);
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M. Vekic and S. R. White, Phys. Rev. B 47, 1160–1163 (1993)
J. J. Deisz, D. W. Hess, and J. W. SerenePhys. Rev. Lett. 76, 1312-1315 (1996)
FLEXQMC
IV. What was the competition up to?
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State of the art analytical tools
Φ [G] = +
Σ [G] = δΦ [G] / δG = +
12
+ ...
Γ [G] = δΣΣΣΣ [G] / δG = +
+ ...
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- Thermodynamically consistent:dF/dµ = Tr[G]
- Satisfies Luttinger theorem (Volume of Fermi surface at T = 0 preserved)
- Satisfies Ward identities (conservation laws):G2(1,1;2,3) appropriately related to G(1,2)
Gordon Baym, Phys. Rev. 127, 1391 (1962).N.E. Bickers and D.J. Scalapino, Annals of Physics, 193, 206 (1989)
Advantages
FLEX
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- Integration over coupling constant of potential energydoes not give back the starting Free energy.
- The Pauli principle in its simplest form is not satisfied(It is used in defining the Hubbard model in the first place)
- There is an infinite number of conserving approximations(How do we pick up the diagrams?)
- Inconsistency:Strongly frequency-dependent self-energy, constant vertex
No Migdal theorem, sovertex corrections should be included
Disdvantages
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-1 0 10.0
0.5
1.0
A(ω)
-1 0 1 -1 0 1
Monte Carlo Many-Body Σ(s) FLEX
-0.5
0.0
G( τ
)
0 β
a) b) c)L=4
L=6
L=8
L=10
β/2
U = + 4
Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).
D. PoulinH. Touchette
Back to us
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-5.00 0.00 5.00 -5.00 0.00 5.00ω/t
(0,0)
(0, )π
( , )π2 π2( , )π π4 4( , )π π4 2
(0,0) (0, )π
( ,0)π
Monte Carlo Many-Body
-5.00 0.00 5.00
Flex
U = + 4
Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).
S. Moukouri F. Lemay S. Allen Y. Vilk B. Kyung
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29
I
II ωωωω
kF
A(ωωωω)
E
k
A(ωωωω)
ωωωω
kF
III
99-aps/Explication...
E
k
A(ωωωω)
ωωωω
kF
TX
c
kF
kF
2∆2∆
T = 0n
T
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§(1)¾
³1; 1
´G(1)¾
³1; 2
´= AG
(1)¡¾
³1; 1+
´G(1)¾ (1; 2)
where A depends on external ¯eld and is chosen such that the exact result
§¾³1; 1
´G¾
³1; 1+
´= U
Dn"n#
Eis satis¯ed. One ¯nds
A = U
Dn"n#
EDn"E Dn#E
Functional derivative ofDn"n#
E=³Dn"E Dn#E´drops out of spin vertex
Usp = A = U
Dn"n#
EDn"E Dn#E
First step: Two-Particle Self-ConsistentHow it works...
S. Allen
Y. Vilk
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To close the system of equations, while satisfying conservation laws and
the Pauli principle¿³n" ¡ n#
´2À=
Dn"E+Dn#E¡ 2
Dn"n#
ET
N
Xeq
Â0(q)
1¡ 12UspÂ0(q)= n¡ 2hn"n#i (1)
Recall
Usp = U
Dn"n#
EDn"E Dn#E (2)
To have charge °uctuations that satisfy Pauli principle as well,
T
N
Xq
Â0(q)
1 + 12UchÂ0(q)= n+ 2hn"n#i ¡ n2 (3)
(Bonus: Mermin-Wagner theorem)
How it works...
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§¾³1; 1
´G¾
³1; 2
´= ¡U
DÃy¡¾
³1+´Ã¡¾ (1)þ (1)Ãy¾ (2)
EÁ
§¾³1; 1
´G¾
³1; 2
´= ¡U
"±G¾ (1; 2)
±Á¡¾ (1+; 1)¡G¡¾
³1; 1+
´G¾ (1; 2)
#
Last term is Hartree Fock (lim! ! 1). Multiply by G¡1, replace lowerenergy part results of TPSC
§(2)¾ (1; 2) = UG
(1)¡¾
³1; 1+
´± (1¡ 2)¡ UG(1)
"±§(1)
±G(1)±G(1)
±Á
#Transverse+longitudinal for crossing-symmetry
§(2)¾ (k) = Un¡¾ +
U
8
T
N
Xq
�3UspÂ
(1)sp (q) + UchÂ
(1)ch (q)
¸G(1)¾ (k+ q): (4)
Second step: improved self-energy
[ ]
How it works...
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-4.0 0.0 4.0ω
0.0
0.2
0.4T = 0.25, n = 0.8, U = -4
QMCMany-Body
-4.0 0.0 4.0ω
0.0
0.4
0.8
1.2N( )ω A(k, )ω
(a) (b)
(c) (d)
-4.0 0.0 4.00.0
0.1
0.2
0.3T = 0.2, n=0.8, U = -4
Many-BodyQMC
-4.0 0.0 4.00.0
0.2
0.4
0.6
τ-0.6
-0.4
-0.2
0.0
T = 0.25, n = 0.8, U = -4QMC
Many Body
τ-0.6
-0.4
-0.2
0.0G(k, )τ G(k, )τ
(a) (b)
(c) (d)
-0.6
-0.4
-0.2
0.0
T = 0.2, n=0.8, U = -4
Many-Body
QMC
-0.6
-0.4
-0.2
0.0
Σk
U = - 4
Kyung et al. cond-mat/0010001
Proof that generalization for U < 0 works
S. Allen
B. Kyung
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34
0.00 0.50 1.00 1.50 2.00ω
T=1/3
T=1/4
T=1/5
Even part of the pair susceptibility at q = 0, for different temperatures
U = - 4
Allen, et al. P.R. L 83, 4128 (1999)
Mechanism for pseudogap formation in the attractive model:
d = 2 is crucial
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U > 0- Renormalized classical regime
for spin fluctuations in pseudogap regime?
Philippe Bourges cond-mat/0009373
T
δ
pseudogap
AF
SC QCP ?
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36
- Slightly Overdoped High-Tc Superconductor TlSr2CaCu2O6.8Guo-qing Zheng et al., P. R. L. 85, 405 (2000) - Pseudogap in Knight shift and NMR relaxation strongly
H dependent, contrary to underdoped (up to 23 T).
- Underdoped in a range ∆Τ ∼ 15 Κ near Tc see evidence for renormalized classical regime (KT behavior).
Corson et al.Nature, 398, 221 (1999).
- Higher symmetry group creates large range of T where there is a pseudogap.
n = 1 n = 0
SO(2)
KT
Pseudogap
T
SO(3)Allen et al. P.R.L. 83, 4128 (1999)
U < 0 Pairing-fluctuation induced pseudogap
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V. Conclusion•What is happening now?
•Methods for thermodynamics and for crossed channels.
•Computers : •Small sizes, not all relevant parameter regimes are accessible
•Analytical studies :•No small parameter, no perfect approximation
•Limiting cases, physical intuition•Not always reliable
-
Steve Allen
François Lemay
David Poulin Hugo Touchette
Yury VilkLiang Chen
Samuel MoukouriBumsoo Kyung
-
David Sénéchal A.-M.T.
Michel Barrette
Alain Veilleux
Mehdi Bozzo-Rey
Elix
-
Jean-Séastien Landry
Bumsoo Kyung
A-M.T.
Sébastien Roy Alexandre Blais
D. Sénéchal
Claude Bourbonnais
R. Côté
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41
- How can we understand electronic systems that showboth localized and extended character?
- Why do both organic and high-temperature superconductors show broken-symmetry states where mean-field-like quasipar-ticles seem to reappear?
- Why is the condensate fraction in this case smaller than what would be expected fromthe shape of the would-be Fermi surface in the normal state?
- Are there new elementary excitations that couldsummarize and explain in a simple way the anomalousproperties of these systems?
- Do quantum critical points play an important role in the Physics of these systems?
- Are there new types of broken symmetries? - How do we build a theoretical approach that can include bothstrong-coupling and d = 2 fluctuation effects?
- What is the origin of d-wave superconductivity in the high-temperature superconductors?