Gordon Conference 2007

Superconductivity near the Mott transition: what can we learn from plaquette DMFT?

K HauleRutgers University

Les Diablerets 2007

References and Collaborators

Strongly Correlated Superconductivity: a plaquette Dynamical mean field theory study,

K. H. and G. Kotliar, cond-mat/0709.0019 (37 pages and 42 figures)

Nodal/Antinodal Dichotomy and the Energy-Gaps of a doped Mott Insulator, M. Civelli, M. Capone, A. Georges, K. H., O. Parcollet, T. D. Stanescu, G. Kotliar,

cond-mat/0704.1486.

Quantum Monte Carlo Impurity Solver for Cluster DMFT and Electronic Structure Calculations in Adjustable Base,

K. H., Phys. Rev. B 75, 155113 (2007).

Optical conductivity and kinetic energy of the superconducting state: a cluster dynamical mean field study, K. H., and G. Kotliar, Europhys Lett. 77, 27007 (2007).

Doping dependence of the redistribution of optical spectral weight in Bi2Sr2CaCu2O8+delta F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf, E. van Heumen, V. Lukovac, F. Mars

iglio, D. van der Marel, K. H., G. Kotliar, H. Berger, S. Courjault, P. H. Kes, and M. Li, Phys. Rev. B 74, 064510 (2006).

Avoided Quantum Criticality near Optimally Doped High Temperature Superconductors , K.H. and G. Kotliar, cond-mat/0605149

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Approach

Understand the physics resulting from the proximity to a

Mott insulator in the context of the simplest models. Construct mean-field type of theory and follow different “states” as a function of parameters – superconducting & normal state.

[Second step compare free energies which will depend more on the detailed modeling and long range terms in Hamiltonian…..]

Leave out disorder, electronic structure, phonons …[CDMFT+LDA second step, under way]

Approach the problem from high temperatures where physics is more local. Address issues of finite frequency– and finite temperature crossovers.

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Impurity solvers:•ED•NCA•Continuous time QMC

Cluster DMFT approach

Exact Baym Kadanoff functional of two variables Exact Baym Kadanoff functional of two variables [[,G]. ,G].

Restriction to the degrees of freedom that live on a plaquette and Restriction to the degrees of freedom that live on a plaquette and its supercell extension.. its supercell extension..

R=(0,0) R=(1,0)

R=(1,1)

[Gplaquette]

periodization

Maps the many body problem ontoMaps the many body problem onto a self consistenta self consistent impurity modelimpurity model

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Momentum versus real space

In plaquette CDMFT cluster quantities are diagonal matrices in cluster momentum base

In analogy with multiorbital Hubbard model exist well defined orbitals

But the inter-orbital Coulomb repulsion is nontrivial and tight-binding Hamiltonian in this base is off-diagonal

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(i) with CTQMC

Hubbard model, T=0.005t

on-sitelargest

nearest neighborsmaller

next nearest neighborimportant in underdoped regime

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Norm

al sta

te

T>

Tc

(0,0) orbital reasonable coherent Fermi liquid

t-J model, T=0.01t

Momentum space Momentum space differentiationdifferentiation

(,0) very incoherent around optimal doping (~0.16 for t-J and ~0.1 for Hubbard U=12t)

() most incoherent and diverging at another doping(1~0.1 for t-J and 1~0 for Hubbard U=12t)

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Momentum space Momentum space differentiationdifferentiation

t-J model, T=0.005t

Norm

al sta

te

T>

Tc

SC

sta

te

T<

<Tc

with large anomalous self-energy

…gets replaced by coherent SC state

Normal state T>Tc:Very large scattering rate atoptimal doping

(,0) orbital

T

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Fermi surface

=0.09

Arcs FS in underdoped regimepockets+lines of zeros of G == arcs

Cumulant is short in ranged:

Single site DMFT PD

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Nodal quasiparticles

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Nodal quasiparticles

the slope=vnod

almost constant

Vnod almost constantup to 20%

v dome like shape

Superconducting gap tracks Tc!M. Civelli, cond-mat 0704.1486

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Antinodal gap – two gaps

M. Civelli, using ED,cond-mat 0704.1486

Superconducting gap has

a dome like shape (like v)

Normal state “pseudogap”monotonically decreasing

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Superfluid density at low T

Low T expansion using imaginary axis QMC data. Current vertex corrections are neglected

In RVB the coefficient b~2 at low [Wen&Lee, Ioffe&Millis]

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Superfluid density close to Tc

Computed by NCA, current vertex corrections neglected

underdoped

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Anomalous self-energy and order parameter

Anomalous self-energy:•Monotonically decreasing with i•Non-monotonic function of doping

(largest at optimal doping)•Of the order of t at optimal doping

at T=0,=0Order parameter has a dome like shape and is small (of the order of 2Tc)

Hubbard model, CTQMC

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Anomalous self-energy on real axis

Computed by the NCA for the t-J model

•Many scales can be identifiedJ,t,W

•It does not change sign at certain frequency D->attractive for any

•Although it is peaked around J, it remains large even for >W

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SC Tunneling DOS

Large asymmetry at low doping

Gap decreases with doping DOS becomes more symmetric

Normal state has a pseudogapwith the same asymmetry

Computed by the NCA for the t-J model

Approximate PH symmetry at optimal doping

SC =0.08SC =0.20NM =0.08NM =0.20

also B. Kyung et.al, PRB 73, 165114 2006

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Optical conductivity

Basov et.al.,PRB 72,54529 (2005)

Low doping: two componentsDrude peak + MIR peak at 2J

For x>0.12 the two components merge

In SC state, the partial gap opens – causes redistribution of spectral weight up to 1eV

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Optical spectral weight - Hubbard versus t-J model

t-J model

J

Drude

no-U

Experiments

intraband interband transitions

~1eV

Excitations into upper Hubbard band

Kinetic energy in Hubbard model:•Moving of holes•Excitations between Hubbard bands

Hubbard model

DrudeU

t2/U

Kinetic energy in t-J model•Only moving of holes

f-sumrule

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Optical spectral weight & Optical mass

F. Carbone,et.al, PRB 74,64510 (2006)

Bi2212

Weight increases because the arcs increaseand Zn increases (more nodal quasiparticles)

mass does not diverge approaches ~1/J

Basov et.al.,PRB 72,60511R (2005)

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Temperature/doping dependence of the optical

spectral weight

Single site DMFT gives correct order of magnitude (Toshi&Capone)At low doping, single site DMFT has a small coherence scale -> big change

Cluser DMF for t-J:Carriers become more coherentIn the overdoped regime ->bigger change in kinetic energy for large

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Bi2212

~1eVWeight bigger in SC,

K decreases (non-BCS)

Weight smaller in SC, K increases (BCS-like)

Optical weight, plasma frequency

F. Carbone,et.al, PRB 74,64510 (2006)

A.F. Santander-Syro et.al, Phys. Rev. B 70, 134504 (2004)

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Phys Rev. B 72, 092504 (2005)

cluster-DMFT, Eu. Lett. 77, 27007 (2007).

Kinetic energy change

Kinetic energy decreases

Kinetic energy increases

Kinetic energy increases

Exchange energy decreases and gives

largest contribution to condensation energy same as RVB (see P.W. Anderson Physica C, 341, 9 (2000)

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Origin of the condensation energy

•Resonance at 0.16t~5Tc (most pronounced at optimal doping)•Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy

Scalapino&White, PRB 58, (1998)

Main origin of the condensation energy

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Conclusions

•Plaquette DMFT provides a simple mean field picture of theunderdoped, optimally doped and overdoped regime

•One can consider mean field phases and track them even in theregion where they are not stable (normal state below Tc)

•Many similarities with high-Tc’s can be found in the plaquette DMFT:•Strong momentum space differentiation with appearance of arcs in UR•Superconducting gap tracks Tc while the PG increases with underdoping•Nodal fermi velocity is almost constant•Superfluid density linear temperature coefficient approaches constant at low doping•Superfuild density close to Tc is linear in temperature•Tunneling DOS is very asymmetric in UR and becomes more symmetric at ODR•Optical conductivity shows a two component behavior at low doping•Optical mass ~1/J at low doping and optical weigh increases linearly with •In the underdoped system -> kinetic energy saving mechanism

overdoped system -> kinetic energy loss mechanism exchange energy is always optimized in SC state

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Issues

The mean field phase diagram and finite temperature crossover between underdoped and overdoped regime

Study only plaquette (2x2) cluster DMFT in the strong coupling limit (at large U=12t)

Can not conclude if SC phase is stable in the exact solution of the model. If the mean

field SC phase is not stable, other interacting term in H could stabilize the mean-field phase (long range U, J)

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Doping dependence of the spectral weight

F. Carbone,et.al, PRB 74,64510 (2006)

Comparison between CDMFT&Bi2212

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RVB phase diagram of the t-J m.

Problems with the RVB slave bosons: Mean field is too uniform on the Fermi

surface, in contradiction with ARPES. Fails to describe the incoherent finite

temperature regime and pseudogap regime. Temperature dependence of the penetration

depth.

Theory: [T]=x-Ta x2 , Exp: [T]= x-T a. Can not describe two distinctive gaps:

normal state pseudogap and superconducting gap

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Similarity with experiments

Louis Taillefer, Nature 447, 565 (2007).

A. Kanigel et.al., Nature Physics 2, 447 (2006)

Arcs FS in underdoped regimepockets+lines of zeros of G == arcs

de Haas van Alphen small Fermi surface

Shrinking arcs

On qualitative level consistent with

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Fermi surface

=0.09

Arcs FS in underdoped regimepockets+lines of zeros of G == arcs

Arcs shrink with T!

Cumulant is short in ranged:

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Insights into superconducting state (BCS/non-BCS)?

J. E. Hirsch, Science, 295, 5563 (2002)

BCS: upon pairing potential energy of electrons decreases, kinetic energy increases(cooper pairs propagate slower)Condensation energy is the difference

non-BCS: kinetic energy decreases upon pairing(holes propagate easier in superconductor)

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Pengcheng et.al., Science 284, (1999)

YBa2Cu3O6.6 (Tc=62.7K)

Origin of the condensation energy

local susceptibility

•Resonance at 0.16t~5Tc (most pronounced at optimal doping)•Second peak ~0.38t~120meV (at opt.d) substantially contributes to condensation energy

Scalapino&White, PRB 58, (1998)

Main origin of the condensation energy

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Similarity with experiments

Louis Taillefer, Nature 447, 565 (2007).

Arcs FS in underdoped regimepockets+lines of zeros of G == arcs

de Haas van Alphen small Fermi surface

On qualitative level consistent with