Pseudogaps and their relation tosuperconductivity*A.-M.S. Tremblay, S. Arya, D. Bergeron,
M. Charlebois, L. Fratino,
P. Sémon, G. Sordi, and A. Venne
CORPES2017 Hiroshima

Thèse de Francis Laliberté,
Université de Sherbrooke

Our road map

CDWHubbard model
t’t’’


Ut

1931-1980

H  − ∑ ij t i,j c i c j  c j c i  U ∑ i n i↑ n i↓t = 1, kB = 1, h = 1
Attn: Charge transfer insulator

Weak vs Strong correlationsn = 1, unfrustrated d = 3 cubic lattice
Mottt
Heisenberg

Slater
AFM/t

Method 1
Dynamical Mean Field Theory (+ clusters)
Concept: atomic-like localized correlations
consistent with delocalized aspect
REVIEWS
Maier, Jarrell et al., RMP. (2005)
Kotliar et al. RMP (2006)
AMST et al. LTP (2006)

Hettler et al, PRB 1998
Lichtenstein et al.,PRB 2000
Kotliar et al., PRB 2000
M. Potthoff, EJP 2003

Cellular DMFT + CT-QMC
00Z   Dd , d exp −S c −  d  d ∑′′ K ′ ′′dΔ,d i  ,iii iSome groups using these methods for cuprates• Europe:– Georges, Parcollet, Ferrero, Civelli, (Paris)– Lichtenstein, Potthoff, (Hamburg) Aichhorn (Graz),Liebsch (Jülich) de Medici (Grenoble) Capone (Italy)• USA:– Gull (Michigan) Millis (Columbia)– Kotliar, Haule (Rutgers)– Jarrell (Louisiana)– Maier, Okamoto (Oakridge)• Japan– Imada (Tokyo) Sakai, Tsunetsugu, Motome

Outline• The model• The method• Part I: Weakly vs strongly correlated AFM• Part II: Strong correlations h-doped cuprates• Part III: Weaker correlations e-doped cuprates

Part I

Weakly vs strongly correlated AFM
CDMFT 2x2

Giovanni Sordi

Lorenzo Fratino

Maxime Charlebois

Patrick Sémon

Mott transition as an organizing principle
Influence of the underlying normal
state on the ordered stateAFM phase diagram d=2, t’=0

L. Fratino, P. Sémon, M. Charlebois, G. Sordi, AMT Phys. Rev. B 95, 235109 (2017)

Double occupancy at weak and strong
interactions: benchmarks

K = -2V

Underlying Mott transitionn = 1, d = 2 square lattice
Critical point visible inV2 O3 ,BEDT organics

Understanding finite temperature phase from a mean-field theory down
to T = 0Change in mechanism for stability of the AFM

L. F. Tocchio, F. Becca, and S. Sorella, Phys. Rev. B 94, 195126 (2016).

Part II

Strong correlations :CDMFT h-doped

Giovanni Sordi

Kristjan Haule

Influence of the Mott transition
away from half-filling
Pseudogap in the normal state
Sordi et al., PRL 104, 226402 (2010)
Sordi et al., PRB 84, 075161 (2011)
Fratino et al., PRB 93, 245147 (2016) [Emery model]
Sordi et al., Sci. Rep. 2 547 (2012);
Sordi et al., PRB 87, 041101(R) (2013)
Fratino et al., PRB 93, 245147 (2016) [Emery model]

Influence of Mott transition
away from half-fillingn = 1, d = 2 square latticeInfluence of Mott transition
away from half-fillingn = 1, d = 2 square latticeSpin susceptibility

G. Sordi, et al. Scientific Reports 2, 547 (2012)

Spin susceptibility

G. Sordi, et al. Scientific Reports 2, 547 (2012)

Spin susceptibility

Underdoped Hg1223
Julien et al. PRL 76, 4238 (1996)

Two crossover lines
Giovanni Sordi

Patrick Sémon

G. Sordi et al. Phys. Rev. Lett. 108, 216401/1-6 (2012)
P. Sémon, G. Sordi, A.-M.S.T., Phys. Rev. B 89, 165113/1-6 (2014)

Scaling with J and with t’
J = 4t2 /U

Alexis Reymba
U=9

A. Reymbaut, Marion, A. Fratino, G. Sordi, P. Sémon, AMT, unpublishedARPES

Kancharla et al. PRB 77 (2008)

Summary

Giovanni Sordi

Patrick Sémon

Strongly correlated
Superconductivity
Sordi et al. PRL 108, 216401 (2012)
Fratino et al.
Sci. Rep. 6, 22715 (2016)

Lorenzo Fratino

Superconductiviy in Doped Mott insulatorn = 1, d = 2 square latticeTc controlled by J

Fratino et al.
Sci. Rep. 6, 22715

Some experiments that suggest Tc < Tpair < T*

T. Kondo et al. PRL 111 (2013)
Kondo, Takeshi, et al. Kaminski Nature Physics 2011, 7, 21-25
A. Pushp, Parker, … A. Yazdani, Science 364, 1689 (2009)
Lee …Tajima (Osaka) https://arxiv.org/pdf/1612.08830
Patrick M. Rourke, et al. Hussey Nature Physics 7, 455–458 (2011)

An organizing principle
Fratino et al.
Sci. Rep. 6, 22715E. Gull and A. J. Millis
Phys. Rev. B 88, 075127

Theory, see also
Jarrel PRL(2004), Gull
Millis PRB(2014)
Experiments:
Bontemps,Santander-Syro
Van der Marel …

Superconductiviy in Doped Mott insulatorn = 1, d = 2 square latticeAn organizing principle

Fratino et al. Sci. Rep. 6, 22715
See also
Jarrell
Gull Millis PRB 90, 041110(R)Experiments:
Deutscher et al, PRB 2005;
Molegraaf et al, Science 2002;
Carbone et al, PRB 2006
Giannetti et al, Nat Comm 2014

Part III

Intermediate correlations :TPSC e-doped

Dominic Bergeron

Bumsoo Kyung

V. Hankevych

A.-M. Daré

Pseudogap in the normal state e-doped

A. Subromanian

A. VenneInfluence of Mott transition
away from half-fillingn = 1, d = 2 square lattice

Method 2 : Small to intermediate U/t
Two-particle self-consistentVilk, AMT J. Phys. I France, 7, 1309 (1997);
REVIEW:AMT Theoretical methods for strongly correlated electrons
Ed. F. Mancini et al. Springer 2011
Mahan, Many-Particle Physics 3rd edition

Yury Vilk

TPSC: general ideas• Approach– Non perturbative: 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->Vilk, AMT J. Phys. I France, 7, 1309 (1997);
Theoretical methods for strongly correlated electrons also (Mahan, 3rd)

Benchmarks

U=+4
(,0)

TPSC

Monte Carlo

Many-Body(0,0)

Flex(0,)( 4 , 2)( 4, 4)
( 2 , 2)(0,  )

(0,0)
-5.00

0.00

5.00

-5.00

0.00

5.00

/t

Calc. + QMC: Moukouri et al. P.R. B 61, 7887 (2000).

-5.00

0.00

5.00

Fermi surface plots
Hankevych, Kyung, A.-M.S.T., PRL, sept. 2004

U=5.75t’=-0.175t, t’’=0.05t
t=350 meV, T=200 K
U=6.25
Hubbard repulsion U has to…
be not too large

15%
U=6.2510%

U=5.75increase forsmaller doping
B.Kyung et al.,PRB 68, 174502 (2003)

U/t smaller, at least near optimal doping

h-dopedD. Sénéchal

e-doped
Sénéchal, AMT, PRL (2004)
Weber, Haule, Kotliar, PRB (2010)

e-doped pseudogap
E. M. Motoyama et al.. Nature 445, 186–189 (2007).

Vilk criterion

Thermal de Broglie wavelengthΔ  k BT∇ k  Δk  k BT th vFTΔk  kvBTF
2
 th

k BTv F

Hot spots from AFM quasi-static scattering
Mermin-Wagnerd=2
 ~ thVilk, AMT (1997)
Kyung, et al.
PRL, 2004

Armitage et al. PRL 2001

Conclusion

 ~ th

Mammouth

Review: A.-M.S.T. arXiv: 1310.1481

A.-M.S. Tremblay“Strongly correlated superconductivity”
Chapt. 10 : Emergent Phenomena in Correlated Matter Modeling andSimulation, Vol. 3, E. Pavarini, E. Koch, and U. Schollwöck (eds.)
Verlag des Forschungszentrum Jülich, 2013

D. Sénéchal

Bumsoo Kyung

The glue
Kyung, Sénéchal, Tremblay, Phys. Rev. B 80, 205109 (2009)
Sénéchal, Day, Bouliane, AMST, Phys. Rev. B 87, 075123 (2013)
A. Reymbaut et al. PRB 94 155146 (2016)Im an and electron-phonon in Pb
Maier, Poilblanc, Scalapino, PRL (2008)

The glue
Kyung, Sénéchal, Tremblay, Phys. Rev. B
80, 205109 (2009)

Wakimoto … Birgeneau
PRL (2004)

The glue and neutrons

Wakimoto … Birgeneau PRL (2007);
PRL (2004)