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2357-3
Innovations in Strongly Correlated Electronic Systems: School and Workshop
Andre-Marie TREMBLAY
6 - 17 August 2012
Departement de Physique and RQMP Universite de Sherbrooke, QC J1K 281
CANADA
Mott transition, Hubbard model and superconductivity: an introduction
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Mott transition, Hubbard model andsuperconductivity: an introduction
A.-M. TremblayG. Sordi, K. Haule, D. Sénéchal, P. Sémon, B. Kyung, G. Kotliar
Trieste, 6 August, 2012
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How to make a metal
Courtesy, S. Julian
ω
r
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Superconductivity
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Attraction mechanism in the metallic state
p
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Attraction mechanism in the metallic state
p’
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Attraction mechanism in the metallic state
-p
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Attraction mechanism in the metallic state
-p’
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Attraction mechanism in the metallic state
p -p
p’
-p’
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#1 Cooper pair, #2 Phase coherence
EP ! !p,p!Up!p!!p!,!p"!p!
!,!p!
"
"
EP!!p,p!
Up!p! !!p!,!p" "!p!
!,!p!
"
"
" !p!,!p" !p!
!,!p!
"
"
|BCS!!"# !"#eiN! |N# # ei!N!2"! |N # 2# #" .
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Half-filled band is metallic?
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Half-filled band: Not always a metal
NiO, Boer and Verway
Peierls, 1937 Mott, 1949
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« Conventional » Mott transition
T
p
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tUµ
t’t’’
H ! !!!ij"! t i,j ci!
! cj! " cj!! ci! " U! i ni#ni$H ! !!
!ij"! t i,j ci!! cj! " cj!
! ci! " U! i ni#ni$
Hubbard model
1931-1980
tEffective model, Heisenberg: J = 4t2 /U
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Superconductivity and attraction?
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Bare Mott critical point in organics
40
30
20
10
0
T (
K)
600 400 200 0
P (bar)
AF
U-SC AF/SC
PI
M
F. Kagawa, K. Miyagawa, + K. Kanoda PRB 69 (2004) +Nature 436 (2005)
Phase diagram (X=Cu[N(CN)2]Cl)S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)
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High-temperature superconductors
• Competing order– Current loops: Varma, PRB
81, 064515 (2010)– Stripes or nematic:
Kivelson et al. RMP 75 1201(2003); J.C.Davis
– d-density wave : Chakravarty, Nayak, Phys. Rev. B 63, 094503 (2001); Affleck et al. flux phase
– SDW: Sachdev PRB 80, 155129 (2009) ...
• Or Mott Physics?– RVB: P.A. Lee Rep. Prog.
Phys. 71, 012501 (2008)
Armitage, Fournier, Greene, RMP (2009)
• n = 1 Mott• Non-FL (Pseudogap)• QCP• non-BCS (phonons? d-wave)
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Perspective
U/t
δ
t’/t
t
tt’
t
tt’
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« Big things » induced by correlations
• Metal to insulator, heavy fermion behavior, high temperature superconductivity, colossal magnetoresistance, giantthermolectricity …………..
• The Kohn Sham approach cannot possibly describe spectroscopic properties of correlated materials, – because these retain atomic physics aspects
(Motness, e.g. multiplets, transfer or spectral weight, high Tc’s, ) which are not perturbative
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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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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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Theoretical Methods
Example of some that have been usedto search for d-wave
superconductivity in Hubbard model
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d-wave superconductivity• Weak coupling
– C. J. Halboth and W. Metzner, Phys. Rev. Lett. 85, 5162 (2000). Functional Renormalization Group– B. Kyung, J.-S. Landry, and A. M. S. Tremblay, Phys. Rev. B 68, 174502 (2003). TPSC– C. Bourbonnais and A. Sedeki, Physical Review B 80, 085105 (2009). Functional RG– D. J. Scalapino, Physica C: Superconductivity 470, Supplement 1, S1 (2010), ISSN 0921-4534, FLEX
proceedings of the 9th International Conference on Materials and Mech anisms of Superconductivity.– A. Abanov, A. V. Chubukov, and J. Schmalian, Adv. Phys. 52, 119 (2003). Feynman diagrams
• Renormalized Mean-Field Theory (Gutzwiller)– P. W. Anderson, P. A. Lee, M. Randeria, T. M. Rice, N. Trivedi, and F. C. Zhang, Journal of Physics:
Condensed Matter 16, R755 (2004).– K.-Y. Yang, T. M. Rice, and F.-C. Zhang, Phys. Rev. B 73, 174501 (2006).
• Slave particles (Gauge theories)– G. Kotliar, Liu, P.R.B (1988).– P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006).
– M. Imada, Y. Yamaji, S. Sakai, and Y. Motome, Annalen der Physik 523, 629 (2011)
• Variational approaches– T. Giamarchi and C. Lhuillier, Phys. Rev. B 43, 12943 (1991).– A. Paramekanti, M. Randeria, and N. Trivedi, Phys. Rev. B 70, 054504 (2004).
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d-wave superconductivity
• Quantum cluster methods– T. Maier, M. Jarrell, T. Pruschke, and J. Keller, Phys. Rev. Lett. 85, 1524 (2000).– T. A. Maier, M. Jarrell, T. C. Schulthess, P. R. C. Kent, and J. B. White, Phys. Rev. Lett. 95, 237001
(2005).– K. Haule and G. Kotliar, Phys. Rev. B 76, 104509 (2007).– + More in this talk
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But…
Aimi and Imada, J. Phys. Soc. Jpn (2007)
QMC constrained pathS. Zhang, Carlson, Gubernatis Phys. Rev. Lett. 78, 4486 (1997)
Refined variational approach: no
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Outline
• More on the model• Method DMFT
– Validity– Impurity solvers
• Finite T phase diagram– Normal state
• First order transition• Widom line and pseudogap
• T=0 phase diagram– The « glue »
• Superconductivity T finite
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The correct model?
Cuprates as doped Mott insulators
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Spectral weight transfer
Meinders et al. PRB 48, 3916 (1993)
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Experiment: X-Ray absorption
Chen et al. PRL 66, 104 (1991)
Peets et al. PRL 103, (2009), Phillips, Jarrell PRL , vol. 105, 199701 (2010)
Number of low energy states above ω = 0 scales as 2x +Not as 1+x as in Fermi liquid
Meinders et al. PRB 48, 3916 (1993)
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Charge transfer insulator
Eskes, Meinders, Sawatzky, PRL 67, 1991
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Outline
• More on the model• Method DMFT
– Validity– Impurity solvers
• Finite T phase diagram– Normal state
• First order transition• Widom line and pseudogap
• T=0 phase diagram– The « glue »
• Superconductivity T finite
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Method
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Mott transition and Dynamical Mean-Field Theory. The beginnings in d = infinity
• Compute scatteringrate (self-energy) of impurity problem.
• Use that self-energy(ω dependent) for lattice.
• Project lattice on single-site and adjustbath so that single-site DOS obtained bothways be equal.
W. Metzner and D. Vollhardt, PRL (1989)A. Georges and G. Kotliar, PRB (1992)
M. Jarrell PRB (1992)
Bath
DMFT, (d = 3)
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DCA
2d Hubbard: Quantum cluster method
C-DMFTV-
DCA
Hettler …Jarrell…Krishnamurty PRB 58 (1998)Kotliar et al. PRL 87 (2001)M. Potthoff et al. PRL 91, 206402 (2003).Maier, Jarrell et al., Rev. Mod. Phys. 77, 1027 (2005)
REVIEWSMaier, Jarrell et al., RMP. (2005) Kotliar et al. RMP (2006)AMST et al. LTP (2006)
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Self-consistency
G!
imp!i!n"
!1! G!
0!imp!i!n"
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!!i!n"
Nc !dd!k"2!#d
1G!
k!0"i"n #
!1!!
!"i"n #
! G#
imp!i!n"
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Methods of derivation
• Cavity method• Local nature of perturbation theory in
infinite dimensions• Expansion around the atomic limit• Effective medium theory• Potthoff self-energy functional
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DMFT as a stationnary point
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When is cluster DMFT OK?Example: The Mott transition
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Local moment and Mott transition
T
U
n = 1, unfrustrated d = 3 cubic lattice
J = 4t2 /UAFM
Slater Heisenberg
Mott
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Local moment and Mott transition
T
U
n = 1, d = 2 square lattice
M I
Understanding finite temperature phase from a mean-field theory down to T = 0
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Size dependence
Gull, Parcollet, MillisarXiv:1207.2490v1 Kancharla et al. PRB 77, 184516 (2008)
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Size dependence near FS
2 4 8 16
Sakai et al. arXiv:1112.3227
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Understanding finite temperature phase from a mean-field theory down to T = 0
• Fermi liquid– Start from Fermi sea– Self-energy analytical– One to one
correspondence of elementary excitations
– Landau parameters– Long-wavelength
collective modes canbecome unstable
• Mott insulator– Hubbard model– Atomic limit– Self-energy singular– DMFT– How many sites in the
cluster determines how low in temperatureyour description of the normal state is valid.
– Long-wavelengthcollective modes canbecome unstable
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+ and -
• Long range order:– Allow symmetry breaking in the bath (mean-field)
• Included:– Short-range dynamical and spatial correlations
• Missing: – Long wavelength p-h and p-p fluctuations
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Some many-body theory for the Hubbard model
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H ! !!!ij"! t i,j ci!
! cj! " cj!! ci! " U! i ni#ni$H ! !!
!ij"! t i,j ci!! cj! " cj!
! ci! " U! i ni#ni$
U = 0
ci! !
1N!k eik!rick!
H ! !k,! !kck!! ck!
|!! " !
k,!
ck!! |0!
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tij = 0
H ! !!!ij"! t i,j ci!
! cj! " cj!! ci! " U! i ni#ni$H ! !!
!ij"! t i,j ci!! cj! " cj!
! ci! " U! i ni#ni$
U
U 2N
|!! " !
ic i!!
!
jc j"! |0!
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Green’s function: free electrons, atomic limit
H ! !!!ij"! t i,j ci!
! cj! " cj!! ci! " U! i ni#ni$H ! !!
!ij"! t i,j ci!! cj! " cj!
! ci! " U! i ni#ni$
Gk!!i!n" !1
i"n!!#k!$"
H ! !!!ij"! t i,j ci!
! cj! " cj!! ci! " U! i ni#ni$H ! !!
!ij"! t i,j ci!! cj! " cj!
! ci! " U! i ni#ni$
!n" ! 1G!!i!n" !
1/2i"n!
U2
"
1/2i"n!
U2
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Impurity solvers
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C-DMFT
K
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CDMFT + ED
Caffarel and Krauth, PRL (1994)
++
-
-
++
-
-
Sarma Kancharla
No Weiss field on the cluster!
Marcello Civelli
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Competition AFM-dSC
See also, Capone and Kotliar, Phys. Rev. B 74, 054513 (2006), Macridin, Maier, Jarrell, Sawatzky, Phys. Rev. B 71, 134527 (2005).
++
-
-
++
-
-
David Sénéchal
B. KyungS. Kancharla
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Monte Carlo method
Gull, Millis, Lichtenstein, Rubtsov, Troyer, Werner, Rev.Mod.Phys. 83, 349 (2011)
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Monte Carlo: Markov chain
• Ergodicity• Detailed balance
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Reminder on perturbation theory
exp!!!!Ha ! Hb"" " exp!!!Ha "U!!"
"U!!"
"!
" !Hb!!"U!!"
U!!" " 1 ! !0
!
d"Hb!"" ! !0
!
d" !0
"
d" !Hb!""Hb!"!
" !#
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Partition function as sum over configurations
Z !Tr!exp"Ha " Hb#$
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Updates
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Solving cluster in a bath problem
• Continuous-time Quantum Monte Carlo calculations to sum all diagrams generatedfrom expansion in powers of hybridization.
– P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97, 076405 (2006).
– K. Haule, Phys. Rev. B 75, 155113 (2007).
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Expansion in powers of the hybridization
Pm !
!m |e!!Hlocdjk "!k #djk!
"
"!k!#!dj1"!1 #dj1
!
"
"!1!#|m$
!
n
!n|e!!Hlocdjk "!k #djk!
"
"!k!#!dj1"!1 #dj1
!
"
"!1!#|n$
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Sign problem
P. Sémon, A.-M.S. Tremblay, (unpub.)
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Outline
• More on the model• Method DMFT
– Validity– Impurity solvers
• Finite T phase diagram– Pseudogap normal state
• First order transition• Widom line and pseudogap
• T=0 phase diagram– The « glue »
• Superconductivity T finite
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The normal state pseudogap
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High-temperature superconductors
• Competing order– Current loops: Varma, PRB
81, 064515 (2010)– Stripes or nematic:
Kivelson et al. RMP 75 1201(2003); J.C.Davis
– d-density wave : Chakravarty, Nayak, Phys. Rev. B 63, 094503 (2001); Affleck et al. flux phase
– SDW: Sachdev PRB 80, 155129 (2009) ...
• Or Mott Physics?– RVB: P.A. Lee Rep. Prog.
Phys. 71, 012501 (2008)
Armitage, Fournier, Greene, RMP (2009)
What is under the dome?Mott Physics away from n = 1
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Three broad classes of mechanisms for pseudogap
• Rounded first ordertransition
• Precursor to a lowertemperature brokensymmetry phase
• Mott physics
• Competing order– Current loops: Varma, PRB
81, 064515 (2010)– Stripes or nematic: Kivelson
et al. RMP 75 1201(2003);J.C.Davis
– d-density wave : Chakravarty, Nayak, Phys. Rev. B 63, 094503 (2001); Affleck et al. flux phase
– SDW: Sachdev PRB 80, 155129 (2009) ...
• Or Mott Physics?– RVB: P.A. Lee Rep. Prog.
Phys. 71, 012501 (2008)
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Normal state of high-temperaturesuperconductors
• e-doped more weakly coupled– Sénéchal, AMST, PRL 92,
126401 (2004)– Weber et al. Nature Phys. 6,
574 (2010)
• e-doped T* fromprecursors of AFM– Kyung et al. PRL 93,
147004 (2004).– Motoyama et al. Nature 445,
186 (2007).
Armitage, Fournier, Greene, RMP (2009)
(hvF / kBT*)~ξth(T*) ∼ ξAFM(T*)
- Vilk, AMST J. Phys. France 7, 1309 (1997)
!" # kBT # $vF!k
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d = 2 precursors, e-doped
Motoyama et al. Nature 445, 186 (2007)
Vilk, A.-M.S.T (1997)
Kyung, Hankevych, A.-M.S.T., PRL, sept. 2004
Semi-quantitative fits of both ARPES and
neutron
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Fermi surface plots
be not too large
increase for smaller doping
Hubbard repulsion U has to…
U=5.75
U=5.75
U=6.25
U=6.25
B.Kyung et al.,PRB 68, 174502 (2003)Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004
15%
10%
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Hot spots from AFM quasi-static scattering
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Hole-doped case: Competing phases?
Leboeuf, Doiron-Leyraud et al. PRB 83, 054506 (2011)
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Pseudogap from Mott physics
G. Sordi, et al. Scientific Reports 2, 547 (2012)Competing order is a consequence of the pseudogap, not its cause:
Parker et al. Nature 468, 677 (2010)
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Doping-induced Mott transition (t’=0)
Kristjan HauleKristjan HauleKristjan Haule
Giovanni SordiGiovanni SordiGiovanni Sordi
G. Sordi, K. Haule, A.-M.S.TPRL, 104, 226402 (2010)
andPhys. Rev. B. 84, 075161 (2011)
T
U
µµµµ Not just adding new piece: Lesson from DMFT, first order transition + critical
point governs phase diagram
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C-DMFT
K
Continuous-time Quantum Monte Carlo calculations to sum all diagrams generated
from expansion in powers of hybridization.
P. Werner, A. Comanac, L. de’ Medici, M. Troyer, and A. J. Millis, Phys. Rev. Lett. 97,
076405 (2006).
K. Haule, Phys. Rev. B 75, 155113 (2007).
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Doping driven Mott transition, t’ = 0Method t’ Orbital
selectiveU Critical
pointRef.
D+C+H 8 7 Werner et al. cond-mat (2009)
D+C+H 4 Gull et al. EPL (2008)-0.3 10,6 Liebsch, Merino… (2008)
Ferrero et al. PRB (2009)D+C+H 8 7 Gull, et al. PRB (2009)
K. Haule, G. Kotliar, PRB (2008) Vildhyadhiraja, PRL (2009)
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Doping driven Mott transition
Gull, Werner, Millis, (2009)E. Gull, M. Ferrero, O. Parcollet, A. Georges, and A. J. Millis (2010)
T = 0.25 t
Gull, Parcollet, MillisarXiv:1207.2490v1
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Link to Mott transition up to optimal doping
Doping dependence of critical point as a function of U
Correlated metal
PG
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First order transition at finite doping
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Link to Mott transition up to optimal doping
Doping dependence of critical point as a function of U
H. Park, K. Haule, and G. Kotliar PRL 101, 186403 (2008)
Smaller D and S
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Density of states
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Density of states
Khosaka et al. Science 315, 1380 (2007);
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Density of states
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Spin susceptibility
Underdoped Hg1223Julien et al. PRL 76, 4238 (1996)
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Pseudogap T* along the Widom line
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The Widom line
Giovanni SordiGiovanni SordiGiovanni SordiKristjan HauleKristjan HauleKristjan Haule
Patrick SémonPatrick Sémon
G. Sordi, et al. Scientific Reports 2, 547 (2012)
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What is the Widom line?
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Pseudogap T* along the Widom line
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Summary: normal state
• Mott physics extends waybeyond half-filling
• Pseudogap is a phase• Pseudogap T* is a Widom line• High compressibility (stripes?)
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Outline
• More on the model• Method DMFT
– Validity– Impurity solvers
• Finite T phase diagram– Normal state
• First order transition• Widom line and pseudogap
• Superconductivity T=0 phase diagram– The « glue »
• Superconductivity T finite
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A bit of physics: superconductivity and repulsion
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Cartoon « BCS » weak-coupling picture
! p " !
12V ! p !
U!p ! p !
"
" p !
E p !1 ! 2n E p !
Exchange of spin waves?Kohn-Luttinger
p
p’
Tc with pressure
D. J. Scalapino, E. Loh, Jr., and J. E. HirschP.R. B 34, 8190-8192 (1986).
Béal–Monod, Bourbonnais, EmeryP.R. B. 34, 7716 (1986).
Kohn, Luttinger, P.R.L. 15, 524 (1965).P.W. Anderson Science 317, 1705 (2007)
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A cartoon strong coupling picture
H M F ! !
k! ,!
"!k#"ck! ,!" c k! ,! ! 4Jmm$ ! Jd!d$ % d$ "
" % F 0
J !!i,j "
S i ! S j ! J !!i,j "
12 c i
!
!"c i !
12 c j
!
!"c j
d ! !d" " ! 1/N !
k!#cos k x ! cos k y $!c k! ,"c !k! ,# "
Pitaevskii Brückner: Pair state orthogonal to repulsive core of Coulomb interaction
Kotliar and Liu, P.R. B 38, 5142 (1988)Miyake, Schmitt–Rink, and Varma
P.R. B 34, 6554-6556 (1986)
P.W. Anderson Science 317, 1705 (2007)
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T = 0 phase diagram n = 1
Phase diagramExact diagonalization as impurity
solver (T=0).
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Theoretical phase diagram BEDT
Kyung, A.-M.S.T. PRL 97, 046402 (2006)
40
30
20
10
0
T (
K)
600 400 200 0
P (bar)
AF
U-SC AF/SC
PI
M
t’ = 0.6t
Y. Kurisaki, et al. Phys. Rev. Lett. 95, 177001(2005) Y. Shimizu, et al. Phys. Rev. Lett. 91, (2003)
X= Cu2(CN)
3 (t’~ t)
t
tt’
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T = 0 phase diagram: cuprates
Phase diagramExact diagonalization as impurity
solver (T=0).
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Theory: Tc down vs Mott
S. Kancharla et al. Phys. Rev. B (2008)
1.00.90.80.7
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Dome vs Mott (CDMFT)
Kancharla, Kyung, Civelli, Sénéchal, Kotliar AMST
Phys. Rev. B (2008)
0.7 0.8 0.9 1.0
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CDMFT global phase diagram
Kancharla, Kyung, Civelli, Sénéchal, Kotliar AMST
Phys. Rev. B (2008)AND Capone, Kotliar PRL (2006)
Armitage, Fournier, Greene, RMP (2009)
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Homogeneous coexistence (experimental)
• H. Mukuda, M. Abe, Y. Araki, Y. Kitaoka, K. Tokiwa, T. Watanabe, A. Iyo, H. Kito, and Y. Tanaka, Phys. Rev. Lett. 96, 087001 (2006).
• Pengcheng Dai, H. J. Kang, H. A. Mook, M. Matsuura, J. W. Lynn, Y. Kurita, Seiki Komiya, and YoichiAndo, Phys. Rev. B 71, 100502 R (2005).
• Robert J. Birgeneau, Chris Stock, John M. Tranquada and KazuyoshiYamada, J. Phys. Soc. Japan, 75, 111003 (2006).
• Chang, … Mesot PRB 78, 104525 (2008).
PCCO
La2CuO4.11
La1.9Sr0.1CuO4Applied field
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Consistent with following experimentsH. Mukuda, Y. Yamaguchi, S. Shimizu, … A. Iyo JPSJ 77, 124706 (2008)
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Magnetic phase diagram of YBCO
Haug, … Keimer, New J. Phys. 12, 105006 (2010)
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Materials dependent properties
C. Weber, C.-H. Yee, K. Haule, and G. Kotliar, ArXiv e-prints (2011), 1108.3028.
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T = 0 phase diagram
The glue
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Im Σan and electron-phonon in Pb
Maier, Poilblanc, Scalapino, PRL (2008)
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The glue
Wakimoto … BirgeneauPRL (2004)
Kyung, Sénéchal, Tremblay, Phys. Rev. B 80, 205109 (2009)
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The glue and neutrons
Wakimoto … Birgeneau PRL (2007);PRL (2004)
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Outline
• More on the model• Method DMFT
– Validity– Impurity solvers
• Finite T phase diagram– Normal state
• First order transition• Widom line and pseudogap
• T=0 phase diagram– The « glue »
• Superconductivity T finite
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Finite T phase diagram
Superconductivity
Giovanni SordiGiovanni SordiGiovanni SordiKristjan HauleKristjan HauleKristjan Haule
Patrick SémonPatrick Sémon
Sordi et al. PRL 108, 216401 (2012)
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Unified phase diagram
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Cuprates (doping driven transition)
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Cuprates (doping driven transition)
Pseudogap vs pairF. Rullier-Albenque, H. Alloul, and G.Rikken,
Phys. Rev. B 84, 014522 (2011).
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Larger clusters
• Is there a minimal size cluster where Tcvanishes before half-filling?
• Learn something from small clusters as well• Local pairs in underdoped
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Meaning of Tcd
• Local pair formation
K. K. Gomes, A. N. Pasupathy, A. Pushp, S. Ono, Y. Ando, and A. Yazdani,
Nature 447, 569 (2007)
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Fluctuating region
Dubroka et al. 106, 047006 (2011)
Infrared response
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Tpair
Kondo, Takeshi, et al. Kaminski Nature Physics 2011, 7, 21-25
ARPESBi2212
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T2
Magnetoresistance, LSCOFluctuating vortices
Patrick M. Rourke, et al. Hussey Nature Physics 7, 455–458 (2011)
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Giant proximity effect
Morenzoni et al., Nature Comms. 2 (2011)
Tc = 32 KTc < 5K
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Actual Tc in underdoped
• Quantum and classical phase fluctuations– V. J. Emery and S. A. Kivelson, Phys. Rev. Lett. 74, 3253 (1995).– V. J. Emery and S. A. Kivelson, Nature 374, 474 (1995).– D. Podolsky, S. Raghu, and A. Vishwanath, Phys. Rev. Lett. 99, 117004 (2007).– Z. Tesanovic, Nat Phys 4, 408 (2008).
• Magnitude fluctuations– I. Ussishkin, S. L. Sondhi, and D. A. Huse, Phys. Rev. Lett. 89, 287001 (2002).
• Competing order– E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. Eisenstein, and A. P. Mackenzie, Annual
Review of Condensed Matter Physics 1, 153 (2010).
• Disorder– F. Rullier-Albenque, H. Alloul, F. Balakirev, and C. Proust, EPL (Europhysics Letters) 81,
37008 (2008).– H. Alloul, J. Bobro, M. Gabay, and P. J. Hirschfeld, Rev. Mod. Phys. 81, 45 (2009).
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Gaussian amplitude fluctuations in Eu-LSCO
Chang, Doiron-Leyraud et al.
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Phase fluctuations and disorder?
A. T. Bollinger et al. & I. Božovi!, Nature 472, 458–460
Monolayer LSCO, field doped
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Effect of disorder
F. Rullier-Albenque, H. Alloul, and G.Rikken, Phys. Rev. B 84, 014522 (2011).
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Superconductivity in underdoped vs BCS
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First-order transition leaves its mark
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Summary• Below the dome finite T
critical point (not QCP) controls normal state
• First-order transition destroyed but traces in the dynamics
• Pseudogap differentfrom pairing.
• Actual Tc in underdoped– Competing order– Long wavelength
fluctuations (see O.P.)– Disorder
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André-Marie Tremblay
Sponsors:
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Mammouth
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C’est fini…Merci
Thank you