Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko

Collaborators: G.Kotliar, S. Savrasov, V. Oudovenko

Kristjan Haule, Physics Department and

Center for Materials TheoryRutgers University Rutgers University

Electronic Structure of Strongly CorreElectronic Structure of Strongly Correlated Electron Materials: A Dynamicalated Electron Materials: A Dynamica

l Mean Field Perspective.l Mean Field Perspective.

• Application of DMFT to real materials (Spectral density functional approach). Examples: – alpha to gamma transition in Ce, optics near

the temperature driven Mott transition. – Mott transition in Americium under pressure

• Extensions of DMFT to clusters. Examples:– Superconducting state in t-J the model– Optical conductivity of the t-J model

OverviewOverview

V2O3Ni2-xSex organics

Universality of the Mott transitionUniversality of the Mott transition

First order MIT

Critical point

Crossover: bad insulator to bad metal

1B HB model 1B HB model (DMFT):(DMFT):

Coherence incoherence crossover in the Coherence incoherence crossover in the

1B HB model (DMFT)1B HB model (DMFT)

Phase diagram of the HM with partial frustration at half-fillingPhase diagram of the HM with partial frustration at half-filling

M. Rozenberg et.al., Phys. Rev. Lett. M. Rozenberg et.al., Phys. Rev. Lett. 7575, 105 (1995)., 105 (1995).

mappingmapping

fermionic bathfermionic bath

Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.Basic idea of Spectral density functional approach: instead of using functionals of the density, use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission]

Dynamical Mean Field TheoryDynamical Mean Field Theory

Density functional theory

Dynamical mean field theory:

observable of interest is the electron densityobservable of interest is the electron density

observable of interest is the local Green's function observable of interest is the local Green's function

(on the lattice(on the lattice uniquely defined uniquely defined))

fermionic bathfermionic bath

mappingmapping

exact BK

functional

DMFT

approximation

DFT & DMFT from the unifying point of viewDFT & DMFT from the unifying point of view

LDA+U corresponds to LDA+DMFT when impurity is solved in the Hartree Fock approximation

observable of interestobservable of interest is the "local“is the "local“ Green's functionsGreen's functions (spectral (spectral function)function)

Currently feasible approximations: LDA+DMFT and GW+DMFT:

G. Kotliar et.al., cond-mat/0511085

Spectral density functional theory

basic idea: sum-up all local diagrams for electrons in correlated orbitals

f1

L=3,S=1/2 J=5/2

f6

L=3,S=3 J=0

Periodic table

Cerium

Ce overview

volumes exp. LDA LDA+U 28Å3 24.7Å3

34.4Å3 35.2Å3

•Transition is 1.order•ends with CP

isostructural phase transition ends in a critical point at (T=600K, P=2GPa)

(fcc) phase

[ magnetic moment

(Curie-Wiess law),

large volume,

stable high-T, low-p]

(fcc) phase

[ loss of magnetic

moment (Pauli-para),

smaller volume,

stable low-T, high-p]

with large

volume collapse

v/v 15

LDA and LDA+U

f DOStotal DOSvolumes exp. LDA LDA+U

28Å3 24.7Å3

34.4Å3 35.2Å3

ferromagnetic

LDA+DMFT alpha DOS

TK(exp)=1000-2000K

LDA+DMFT gamma DOS

TK(exp)=60-80K

Photoemission&experiment

Fenomenological Landau approach:Kondo volume colapse (J.W. Allen, R.M. Martin, 1982)Kondo volume colapse (J.W. Allen, R.M. Martin, 1982)

•A. Mc Mahan K Held and R. Scalettar (2002)

•K. Haule V. Udovenko and GK. (2003)

Optical conductivity

*

+

+ K. Haule, et.al., Phys. Rev. Lett. 94, 036401 (2005)

* J.W. van der Eb, A.B. Ku’zmenko, and D. van der Marel, Phys. Rev. Lett. 86, 3407 (2001)

Americium

Americium

"soft" phase

f localized

"hard" phase

f bonding

Mott Transition?

f6 -> L=3, S=3, J=0

A.Lindbaum, S. Heathman, K. Litfin, and Y. Méresse, Phys. Rev. B 63, 214101 (2001)

J.-C. Griveau, J. Rebizant, G. H. Lander, and G.KotliarPhys. Rev. Lett. 94, 097002 (2005)

Am within LDA+DMFT

S. Y. Savrasov, K. Haule, and G. KotliarPhys. Rev. Lett. 96, 036404 (2006)

F(0)=4.5 eV F(2)=8.0 eVF(4)=5.4 eV F(6)=4.0 eV

Large multiple effects:

Am within LDA+DMFT

nf=6

Comparisson with experiment

from J=0 to J=7/2

•“Soft” phase very different from Cenot in local moment regime since J=0 (no entropy)

•"Hard" phase similar to Ce,

Kondo physics due to hybridization, however, nf still far from Kondo regime

nf=6.2

Different from Sm!

Exp: J. R. Naegele, L. Manes, J. C. Spirlet, and W. MüllerPhys. Rev. Lett. 52, 1834-1837 (1984)

Theory: S. Y. Savrasov, K. Haule, and G. KotliarPhys. Rev. Lett. 96, 036404 (2006)

V=V0 Am IV=0.76V0 Am IIIV=0.63V0 Am IV

Beyond single site DMFT

What is missing in DMFT?

•Momentum dependence of the self-energy m*/m=1/Z

•Various orders: d-waveSC,…

•Variation of Z, m*, on the Fermi surface

•Non trivial insulator (frustrated magnets)

•Non-local interactions (spin-spin, long range Columb,correlated hopping..)

Present in DMFT:•Quantum time fluctuations

Present in cluster DMFT:•Quantum time fluctuations•Spatially short range quantum fluctuations

The simplest model of high Tc’s

t-J, PW Anderson

Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)

BK Functional, Exact

cluster in k space cluster in real space

t’=0

Phase diagram

What can we learn from “small” Cluster-DMFT?

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

J. E. Hirsch, Science, 295, 5563 (2226)

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)

D van der Marel, Nature 425, 271-274 (2003)

cond-mat/0601478

Optical conductivity

overdoped

optimally doped

Bi2212

~1eVWeight bigger in SC,

K decreases (non-BCS)

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

Optical weight, plasma frequency

D. van der Marel et.al., in preparation

Hubbard model

DrudeU

t2/U

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

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

Hubbard versus t-J model

Phys Rev. B 72, 092504 (2005)

cluster-DMFT, cond-mat/0601478

Kinetic energy change

Kinetic energy decreases

Kinetic energy increases

Kinetic energy increases

Exchange energy decreases and gives

largest contribution to condensation energy

cond-mat/0503073

electrons gain energy due to exchange energyholes gain kinetic energy (move faster)

underdoped

electrons gain energy due to exchange energyhole loose kinetic energy (move slower)

overdoped

BCS likesame as RVB (see P.W. Anderson Physica C, 341, 9 (2000),

or slave boson mean field (P. Lee, Physica C, 317, 194 (1999)

Kinetic energy upon condensation

J

J J

J

Optics mass and plasma frequency

Extended Drude model

•Within DMFT, optics mass is m*/m=1/Z and diverges at the Mott transition •Plasma frequency vanishes as 1/Z (Drude shrinks as Kondo peak shrinks)

•In cluster-DMFT optics mass constant at low doping doping

•Plasma frequency vanishes because the active (coherent) part of

the Fermi surface shrinks

line: cluster DMFT (cond-mat 0601478),symbols: Bi2212, Van der Marel (in preparation)

Optimal doping: Powerlaws

cond-mat/0605149D. van der Marel et. al., Nature 425, 271 (2003).

Optimal doping: Coherence scale seems

to vanish

Tc

underdoped

overdoped

optimally

scattering at Tc

Local density of states of SC

•V shaped gap (d-wave)•size of gap decreases with doping•CDMFT-optimall doping PH symmetric

STM study Cluster DMFT

1 Larger doping6 Smaller doping

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

YBa2Cu3O6.6 (Tc=62.7K)

41meV resonance

local susceptibility

•Resonance at 0.16t~48meV•Most pronounced at optimal doping•Second peak shifts with doping (at 0.38~120meV opt.d.) and changes below Tc – contribution to condensation energy

Pseudoparticle insight

N=4,S=0,K=0

N=4,S=1,K=()

N=3,S=1/2,K=(,0)

N=2,S=0,K=0

A()

’’()PH symmetry,

Large

• LDA+DMFT can describe interplay of lattice and electronic structure near Mott transition. Gives physical connection between spectra, lattice structure, optics,.... – Allows to study the Mott transition in open and closed shell cases. – In both Ce and Am single site LDA+DMFT gives the zeroth order

picture– Am: Rich physics, mixed valence under pressure.

• 2D models of high-Tc require cluster of sites. Some aspects of optimally doped, overdoped and slightly underdoped regime can be described with cluster DMFT on plaquette:– Evolution from kinetic energy saving to BCS kinetic energy cost

mechanism– Optical mass approaches a constant at the Mott transition and

plasma frequency vanishes– At optimal doping: Physical observables like optical conductivity

and spin susceptibility show powerlaw behavior at intermediate frequencies, very large scattering rate – vanishing of coherence scale, PH symmetry is dynamically restored, 41meV resonance appears in spin response

Conclusions

local in localized LMTO base

Impurity problem (14x14):

LDA

Impurity solver

DMFT SCC *

*

LDA+DMFT implementation

Carbone et.al., in preparation

Comparison of spectral weightcluster DMFT / Bi2212

Spectral weight (kinetic energy) changes faster

with T in overdoped

system – larger coherence scale

Partial DOS

4f

5d

6s

Z=0.33

More complicated f systems

•Hunds coupling is important when more than one electron in the correlated (f) orbital•Spin orbit coupling is very small in Ce, while it become important in heavier elements

The complicated atom embedded into fermionic The complicated atom embedded into fermionic

bath (with crystal fileds) is a serious chalange so solve!bath (with crystal fileds) is a serious chalange so solve!

Coulomb interaction is diagonal in the base of total LSJ -> LS base

while the SO coupling is diagonal in the j-base -> jj base

Eigenbase of the atom depends on the strength of the Hund's couling and

strength of the spin-orbit interaction

Mott transition (B. Johansson, 1974):Mott transition (B. Johansson, 1974):

Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):Kondo volume colapse (J.W. Allen, R.M. Martin, 1982):

Classical theories

Hubbard modelHubbard model

Anderson (impurity) modelAnderson (impurity) model

changes and causes Mott tr.changes and causes Mott tr.

changes changes →→ chnange of T chnange of TKK

bath

either constant or

taken from LDA and rescaled

spd electrons pure spectatorsspd electrons pure spectators

hybridization with spd electrons is crucialhybridization with spd electrons is crucial

f electrons insulating

f electrons in local moment regime

(Lavagna, Lacroix and Cyrot, 1982)(Lavagna, Lacroix and Cyrot, 1982)

Fenomenological Landau approach:

LDA+DMFTab initio calculationab initio calculation

is self-consistently determinedis self-consistently determined

contains tcontains tffff and V and Vfdfd hopping hopping

bath for AIMbath for AIM

Kondo volume colapse model resembles DMFT picture:Kondo volume colapse model resembles DMFT picture:

Solution of the Solution of the Anderson impurity modelAnderson impurity model → → Kondo physicsKondo physics

DifferenceDifference: : with DMFT the lattice problem is solved (and therefore with DMFT the lattice problem is solved (and therefore Δ must self-consistently determined) while in KVC Δ is calculated for a fictious impurity (and needs to be rescaled to fit exp.)

In KVC scheme there is no feedback on spd bans, hence optics is not much affected.

An example

Atomic physics of selected Actinides

Basov, cond-mat/0509307

optics mass and plasma w

Two Site CDMFTTwo Site CDMFT in the 1D Hubbard in the 1D Hubbard modelmodel

M.Capone M.Civelli V. Kancharla C.Castellani and Kotliar, PRB 69,195105 (2004)

Slave particle diagrammatic impurity solvers

NCA

OCA

TCA

Luttinger Ward functional

every atomic state represented with a unique pseudoparticle

atomic eigenbase - full (atomic) base

, where

general AIM:

( )

SUNCA vs QMCtwo band Hubbard model, Bethe lattice, U=4D

three band Hubbard model,

Bethe lattice, U=5D, T=0.0625D

three band Hubbard model,

Bethe lattice, U=5D, T=0.0625D