Cluster Dynamical Mean Field Approach to Strongly Correlated Materials

1
Cluster Dynamical Mean Field Approach to Strongly
Correlated Materials

• K Haule
• Rutgers University

2
References and Collaborators

• Strongly Correlated Superconductivity a
  plaquette Dynamical mean field theory study, K.
  H. and G. Kotliar, Phys. Rev. B 76, 104509
  (2007).
• 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, Phys. Rev. Lett. 100, 046402 (2008).
• Modelling the Localized to Itinerant Electronic
  Transition in the Heavy Fermion System CeIrIn5,
  J.H. Shim, K. Haule and G. Kotliar, Science 318,
  1615 (2007),
• 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 Bi2Sr2CaCu2O8delta,
  F. Carbone, A. B. Kuzmenko, H. J. A. Molegraaf,
  E. van Heumen, V. Lukovac, F. Marsiglio, 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, Phys. Rev. B 76, 092503 (2007).

Thanks to Ali Yazdani for unpublished data!
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Standard theory of solids
Band Theory electrons as waves Rigid band
picture En(k) versus k Landau Fermi Liquid
Theory applicable Very powerful quantitative
tools LDA,LSDA,GW

• Predictions
• total energies,
• stability of crystal phases
• optical transitions

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Strong correlation Standard theory fails

• Fermi Liquid Theory does NOT work . Need new
  concepts to replace rigid bands picture!
• Breakdown of the wave picture. Need to
  incorporate a real space perspective (Mott).
• Non perturbative problem.

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Non perturbative methods
On site correlations usually the strongest
-gt Mott phenomena at integer fillings
Successful theory which can describe Mott
transition Dynamical Mean Field Theory
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Mott phenomena at half filling
Single site DMFT
Georges, Kotliar, Krauth, Rozenber, Rev. Mod.
Phys. 1996
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Dynamical Mean Field Theory
For a given lattice site, DMFT envisions the
neighboring sites on the lattice as a Weiss
field of conduction electrons, exchanging
electrons with that site.
Maps lattice model to an effective quantum
impurity model
More rigorously DMFT sumps up all local diagrams
(to all orders in perturbation theory)
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DMFT in single site approximation
Successfully describes spectra and response
functions of numerous correlated materials
Mott transition in V2O3 LaTiO3 actinides
(Pu,) Lanthanides (Ce,) and far to many to
mention all
, KH et al. 2007
, KH et al. 2003
Recent review
(G. Kotliar S. Savrasov K.H., V. Oudovenko O.
Parcollet and C. Marianetti, RMP 2006).
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Band structure and optics of heavy fermion CeIrIn5
Non-f spectra at 10K
300K
J.H. Shim, KH, and G. Kotliar, Science 318, 1618
(2007).
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Later verified by Yang Pines
Remarkable agreement with Y. Yang D. Pines
cond-mat/0711.0789!
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High Tc Need non-local self-energy
d-wave pairing 2x2 cluster-DMFT necessary to
capture the order parameter
Fermi surface evolution with doping can not be
understood within single site DMFT.
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Cluster DMFT approaches

• Momentum space approach-Dynamical cluster
  approximation
• (Hettler, Maier, Jarrel)
• Real space approach Cellular DMFT
  (Kotliar,Savrasov,Palson)

In the Baym Kadanoff functional, the interacting
part F is restrictied to the degrees of freedom
(G) that live on the cluster.
Maps the many body problem onto a self consistent
impurity model
FGplaquette

• Impurity solvers
• Continuous time QMC
• Hirsh-Fye QMC
• NCA
• ED

periodization
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An exact impurity solver, continuous time QMC -
expansion in terms of hybridization
K.H. Phys. Rev. B 75, 155113 (2007) P Werner,
PRL (2007) N. Rubtsov PRB 72, 35122 (2005).
General impurity problem
Diagrammatic expansion in terms of hybridization
D Metropolis sampling over the diagrams

• Exact method samples all diagrams!
• No severe sign problem

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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..

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

• Leave out disorder, electronic structure, phonons
  
• CDMFTLDA second step, under way

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S(iw) with CTQMC
next nearest neighbor important in underdoped
regime
on-site largest
nearest neighbor smaller
Hubbard model, T0.005t
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Momentum space differentiation
gets replaced by coherent SC state
with large anomalous self-energy
t-J model, T0.005t
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SC Tunneling DOS
K. H. and G. Kotliar, Phys. Rev. B 76, 104509
(2007).
NM d0.20
SC d0.08
SC d0.20
NM d0.08
Large asymmetry at low doping
Gap decreases with doping DOS becomes more
symmetric
Asymmetry is due to normal state DOS -gt Mottness
Computed by the NCA for the t-J model
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Ratio AS/AN
Ratio more universal, more symmetric
With decreasing doping gap increases, coherence
peaks less sharp-gtNon BCS
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Ratio AS/AN
CDMFT calculation
ratio almost symmetric
Pronounced dip-hump feature can not be fitted
with BCS
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Non-BCS
DOS in normal state decreases
Gap increases
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Normal state DOS and SC gap
CDMFT
Not using realistic band structure (t)
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Gap changes, mode does not
J.C. Davis, Nature 442, 546 (2006)
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Where does the dip-hump structure come from?
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Eliashberg theory
Real part constant
phonon frequency
W
D0
Gap up to D0W
No scattering up to D0W
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Dip-hump structure
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Phenomenology
Similar frequency dependence of gap recently
introduced by W.Sacks and B. Doucot PRB
74,174517 (2006) to fit experiments.
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Fermi surface
d0.09
Cumulant is short in ranged
Arcs FS in underdoped regime pocketslines of
zeros of G arcs
Single site DMFT PD
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Nodal quasiparticles
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Nodal quasiparticles
Vnod almost constant up to 20
the slopevnod almost constant
vD dome like shape
Superconducting gap tracks Tc!
M. Civelli, cond-mat 0704.1486
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Two energy scales in Raman Spectrum in the SC
State of Underdoped Cuprates
doping
Energy scale of peak in antinodal (nodal) region
increases (decreases) with decreasing doping in
underdoped cuprates.
Le Tacon et al, Nat. Phys. 2, 537 (2006)
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Evolution of Nodal and Antinodal energy scales
with doping
Le Tacon et al, Nat. Phys. 2, 537 (2006)
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Antinodal gap two gaps
M. Civelli, using ED, PRL. 100, 046402 (2008).
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Optical conductivity

• Low doping two components
• Drude peak MIR peak at 2J

• For xgt0.12 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
f-sumrule
Excitations into upper Hubbard band
Drude
t-J model
J
no-U
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Optical spectral weight Optical mass
mass does not diverge approaches 1/J
Bi2212
F. Carbone,et.al, PRB 74,64510 (2006)
Weight increases because the arcs increase and Zn
increases (more nodal quasiparticles)
Basov et.al., PRB 72,60511R (2005)
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Temperature/doping dependence of the optical
spectral weight
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Optical weight, plasma frequency
Weight bigger in SC, K decreases (non-BCS)
Weight smaller in SC, K increases (BCS-like)
A.F. Santander-Syro et.al, Phys. Rev. B 70,
134504 (2004)
F. Carbone,et.al, PRB 74,64510 (2006)
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Kinetic energy change
Kinetic energy increases
cluster-DMFT, Eu. Lett. 77, 27007 (2007).
Kinetic energy decreases
Phys Rev. B 72, 092504 (2005)
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 0.16t5Tc (most pronounced at
  optimal doping)
• Second peak 0.38t120meV (at opt.d)
  substantially contributes to condensation energy

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Conclusions

• Plaquette DMFT provides a simple mean field
  picture of the
• underdoped, optimally doped and overdoped regime
• One can consider mean field phases and track them
  even in the
• region where they are not stable (normal state
  below Tc)
• Many similarities with high-Tcs 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
• Tunneling DOS As/An has a dip hump dip structure
  -gt comes from the structure
• in the anomalous self-energy
• Optical conductivity shows a two component
  behavior at low doping
• Optical mass 1/J at low doping and optical weigh
  increases linearly with d
• In the underdoped system -gt kinetic energy saving
  mechanism
• overdoped system -gt kinetic energy loss
  mechanism
• exchange energy is always optimized in SC
  state