Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and...

125
Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal, P. Sémon, B. Kyung, G. Kotliar Trieste, 6 August, 2012

Transcript of Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and...

Page 1: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Mott transition, Hubbard model andsuperconductivity: an introduction

A.-M. Tremblay

G. Sordi, K. Haule, D. Sénéchal,

P. Sémon, B. Kyung, G. Kotliar

Trieste, 6 August, 2012

Page 2: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

How to make a metal

Courtesy, S. Julian

ω

r

Page 3: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Superconductivity

Page 4: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Attraction mechanism in the metallic state

p

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Attraction mechanism in the metallic state

p’

Page 6: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Attraction mechanismin the metallic state

-p

Page 7: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Attraction mechanismin the metallic state

-p’

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Attraction mechanism in the metallic state

p -p

p’

-p’

Page 9: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

#1 Cooper pair, #2 Phase coherence

EP = ∑p,p′ Up−p′ψp↑,−p↓ψp′↑,−p′↓∗

EP=∑p,p′

Up−p′ ⟨ψp↑,−p↓ ⟩ψp′↑,−p′↓∗

+ ψp↑,−p↓ ψp′↑,−p′↓∗

|BCSθ⟩ =…+eiNθ|N⟩ + eiN+2θ|N + 2⟩ +… .

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Half-filled band is metallic?

Page 11: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Half-filled band: Not always a metal

NiO, Boer and Verway

Peierls, 1937 Mott, 1949

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« Conventional » Mott transition

T

p

Page 13: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

t

t’t’’

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

Hubbard model

1931-1980

tEffective model, Heisenberg: J = 4t2 /U

Page 14: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Superconductivityand attraction?

Page 15: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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)

Page 16: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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 » inducedby 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

Page 19: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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

Page 21: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Theoretical Methods

Example of some that have been usedto search for d-wave

superconductivity in Hubbard model

Page 22: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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, FLEXproceedings 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

Page 25: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Outline

• More on the model

• Method DMFT– Validity

– Impurity solvers

• FiniteT phase diagram– Normal state

• First order transition

• Widom line and pseudogap

• T=0 phase diagram– The « glue »

• SuperconductivityT finite

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The correct model?

Cuprates as doped Mott insulators

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Spectral weight transfer

Meinderset al.PRB 48, 3916 (1993)

Page 28: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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 lowenergy states aboveω = 0 scales as 2x +Not as 1+x as in Fermi liquid

Meinderset al.PRB 48, 3916 (1993)

Page 29: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Charge transfer insulator

Eskes, Meinders, Sawatzky, PRL 67, 1991

Page 30: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Outline

• More on the model

• Method DMFT– Validity

– Impurity solvers

• FiniteT phase diagram– Normal state

• First order transition

• Widom line and pseudogap

• T=0 phase diagram– The « glue »

• SuperconductivityT 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

2dHubbard: 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)

Page 34: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Self-consistency

impiωn

−1= Gσ

0−impiωn

−1− Σσiωn

Nc ∫ ddk

2πd1

G

kσ0 iωn

−1−Σσiωn

= Gσ

impiωn

Page 35: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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

Page 36: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

DMFT as a stationnarypoint

Page 37: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Whenis cluster DMFT OK?Example: The Mott transition

Page 38: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Local moment and Mott transition

T

U

n = 1, unfrustratedd = 3 cubic lattice

J = 4t2 /UAFM

Slater Heisenberg

Mott

Page 39: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Local moment and Mott transition

T

U

n = 1, d = 2square lattice

M I

Understanding finite temperature phase froma mean-field theory down to T = 0

Page 40: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Size dependence

Gull, Parcollet, MillisarXiv:1207.2490v1 Kancharla et al. PRB 77, 184516 (2008)

Page 41: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Size dependence near FS

2 4 8 16

Sakai et al. arXiv:1112.3227

Page 42: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Understandingfinite temperature phase froma mean-field theorydown to T = 0

• Fermi liquid– Start fromFermi sea

– Self-energy analytical

– One to one correspondence of elementary excitations

– Landau parameters

– Long-wavelengthcollective 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

Page 43: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

+ 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

Page 44: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Some many-body theory for the Hubbard model

Page 45: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

U = 0

ciσ =1N∑k

eik⋅r i ckσ

H = ∑k,σkckσ

ckσ

|Ψ⟩ = ∏k,σ

ckσ |0⟩

Page 46: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

tij = 0

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

U

U2N

|Ψ⟩ = ∏i

ci↑ ∏

j

cj↓ |0⟩

Page 47: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Green’s function: free electrons, atomic limit

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

Gkσiωn =1

iωn−k−μ

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

⟨n⟩ = 1 Gσiωn =1/2

iωn+U2

+1/2

iωn−U2

Page 48: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Self-energyand all that

Gσiωn =1/2

iωn+U2

+1/2

iωn−U2

H = −∑<ij>σ

t i,j ciσ cjσ + cjσ

ciσ + U∑ini↑ni↓H = −∑

<ij>σt i,j ciσ

cjσ + cjσ ciσ + U∑

ini↑ni↓

Self-energy in the atomic limit for n = 1

Gσiωn =1

iωn+U2−Σiωn

Σiωn =U2+

U2

iωn

Gkσiωn =1

iωn−k−μ−Σkσiωn

Gkσ−1iωn = Gkσ

0−1iωn − Σkσiωn

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Impurity solvers

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C-DMFT

K

Page 51: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

CDMFT + ED

Caffarel and Krauth, PRL (1994)

++

-

-

++

-

-

Sarma Kancharla

No Weiss field on the cluster!

Marcello Civelli

Page 52: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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

Page 53: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Monte Carlo method

Gull, Millis, Lichtenstein, Rubtsov, Troyer, Werner, Rev.Mod.Phys. 83, 349 (2011)

Page 54: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Monte Carlo: Markov chain

• Ergodicity

• Detailed balance

Page 55: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Reminder on perturbation theory

exp−βHa + Hb = exp−βHaUβ

∂Uβ

∂β= −HbβUβ

Uβ = 1 − ∫0

β

dτHbτ + ∫0

β

dτ ∫0

τ

dτ ′HbτHbτ′ +…

Page 56: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Partition functionas sumover configurations

Z =TrexpHa + Hb

Page 57: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Updates

Page 58: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Solvingcluster in a bath problem

• Continuous-time Quantum Monte Carlo calculations to sumall 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).

Page 59: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Expansion in powers of the hybridization

Pm =

⟨m|e−βHlocdjkτk djk

† τk′ …dj1

τ1dj1′

† τ1′ |m⟩

∑n

⟨n|e−βHlocdjkτk djk

′† τk

′ …dj1τ1dj1

′† τ1

′ |n⟩

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Signproblem

P. Sémon, A.-M.S. Tremblay, (unpub.)

Page 61: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Outline

• More on the model

• Method DMFT– Validity

– Impurity solvers

• FiniteT phase diagram– Pseudogap normal state

• First order transition

• Widom line and pseudogap

• T=0 phase diagram– The « glue »

• SuperconductivityT finite

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The normal state pseudogap

Page 63: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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 fromn = 1

Page 64: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Three broadclasses 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)

Page 65: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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)

Δ = kBT = ℎvFΔk

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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

Page 67: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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

Page 69: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Hole-dopedcase: Competingphases?

Leboeuf, Doiron-Leyraud et al. PRB 83, 054506 (2011)

Page 70: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Pseudogap fromMott 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)

Page 71: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Doping-inducedMott 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 fromDMFT, first order transition + critical

point governs phase diagram

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C-DMFT

K

Continuous-time Quantum Monte Carlo calculations to sumall 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 drivenMott transition, t’ = 0

Method t’ Orbital selective

U Criticalpoint

Ref.

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 drivenMott 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

Page 76: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

First order transition at finite doping

Page 77: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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)

SmallerD and S

Page 78: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Densityof states

Page 79: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Densityof states

Khosaka et al. Science 315, 1380(2007);

Page 80: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Densityof states

Page 81: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Spin susceptibility

Underdoped Hg1223Julien et al. PRL 76, 4238 (1996)

Page 82: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Pseudogap T* alongthe Widom line

Page 83: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

The Widom line

Giovanni SordiGiovanni SordiGiovanni SordiKristjan HauleKristjan HauleKristjan Haule

Patrick SémonPatrick Sémon

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

Page 84: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

What is the Widom line?

Page 85: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Pseudogap T* alongthe Widom line

Page 86: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Summary: normal state

• Mott physics extends waybeyond half-filling

• Pseudogap is a phase

• Pseudogap T* is a Widom line

• High compressibility (stripes?)

Page 87: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Outline

• More on the model

• Method DMFT– Validity

– Impurity solvers

• FiniteT phase diagram– Normal state

• First order transition

• Widom line and pseudogap

• SuperconductivityT=0 phase diagram– The « glue »

• SuperconductivityT finite

Page 88: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

A bit of physics: superconductivityand repulsion

Page 89: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Cartoon «BCS» weak-coupling picture

Δ p = −

12V ∑ p ′

Up − p ′ Δ p ′

Ep ′

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)

Page 90: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

A cartoon strong coupling picture

H MF = ∑k⃗ ,σ

k⃗ck⃗ ,σ† c k⃗ ,σ − 4Jmm̂− Jdd̂ + d̂ † + F 0

J ∑⟨i,j ⟩

Si ⋅ Sj = J ∑⟨i,j ⟩

12

c i†σ⃗c i ⋅

12

c j†σ⃗c j

d = ⟨d̂⟩ = 1/N ∑k⃗

cosk x − cosk y ⟨c k⃗ ,↑c− k⃗ ,↓ ⟩

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 diagramn = 1

Phase diagram

Exact diagonalization as impuritysolver (T=0).

Page 92: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Theoretical phase diagramBEDT

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 diagram

Exact diagonalization as impuritysolver (T=0).

Page 94: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

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

Page 98: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Consistent withfollowing experiments

H. Mukuda, Y. Yamaguchi, S. Shimizu, … A. Iyo JPSJ 77, 124706 (2008)

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Magnetic phase diagramof YBCO

Haug, … Keimer, New J. Phys. 12, 105006 (2010)

Page 100: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Materials dependent properties

C. Weber, C.-H. Yee, K. Haule, and G. Kotliar, ArXiv e-prints (2011), 1108.3028.

Page 101: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

T = 0 phase diagram

The glue

Page 102: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Im Σan and electron-phonon in Pb

Maier, Poilblanc, Scalapino, PRL (2008)

Page 103: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

The glue

Wakimoto … BirgeneauPRL (2004)

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

Page 104: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

The glue and neutrons

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

Page 105: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Outline

• More on the model

• Method DMFT– Validity

– Impurity solvers

• FiniteT phase diagram– Normal state

• First order transition

• Widom line and pseudogap

• T=0 phase diagram– The « glue »

• SuperconductivityT 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

Page 108: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Cuprates (doping driventransition)

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Cuprates (doping driventransition)

Pseudogap vs pair

F. Rullier-Albenque, H. Alloul, and G.Rikken,

Phys. Rev. B 84, 014522 (2011).

Page 110: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Larger clusters

• Is there a minimal size cluster where Tc

vanishes before half-filling?

• Learn something fromsmall clusters as well

• Local pairs in underdoped

Page 111: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Meaningof 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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Fluctuatingregion

Dubroka et al. 106, 047006 (2011)

Infrared response

Page 113: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Tpair

Kondo, Takeshi, et al. Kaminski Nature Physics2011, 7, 21-25

ARPESBi2212

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T2

Magnetoresistance, LSCOFluctuating vortices

Patrick M. Rourke, et al. Hussey Nature Physics 7, 455–458 (2011)

Page 115: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Giant proximityeffect

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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Gaussianamplitude fluctuations in Eu-LSCO

Chang, Doiron-Leyraud et al.

Page 118: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Phase fluctuations and disorder?

A. T. Bollinger et al. & I. Božović, 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).

Page 120: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Superconductivityin underdoped vs BCS

Page 121: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

First-order transition leaves its mark

Page 122: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Summary

• Below the dome finiteT critical point (not QCP) controls normal state

• First-order transition destroyed but traces in the dynamics

• Pseudogap differentfrom pairing.

• ActualTc in underdoped– Competing order

– Long wavelengthfluctuations (see O.P.)

– Disorder

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André-Marie Tremblay

Sponsors:

Page 124: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

Mammouth

Page 125: Mott transition, Hubbard model and superconductivity: an ... · Mott transition, Hubbard model and superconductivity: an introduction A.-M. Tremblay G. Sordi, K. Haule, D. Sénéchal,

C’est fini…Merci

Thank you