Theoretical and Experimental Magnetism Meeting Theoretical and Experimental Magnetism Meeting Gabriel Kotliar Rutgers University Support :National Science

Theoretical and Experimental Magnetism MeetingTheoretical and Experimental Magnetism Meeting

Gabriel KotliarGabriel Kotliar

Rutgers University

Support :National Science Foundation. Department of Energy (BES).

3-4 August Cosener house , Abingdon, Oxfordhisre UK3-4 August Cosener house , Abingdon, Oxfordhisre UK

Dynamical Mean Field Approach to Dynamical Mean Field Approach to strongly Correlated Electronsstrongly Correlated Electrons

OutlineOutline• Motivation. Introduction to DMFT ideas.

• Application to the late actinides.

• Application to Cuprate Supeconductors.

Collaborators M. Civelli K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers)

S. Savrasov (UCDavis ) A.M. Tremblay B. Kyung V. Kancharla

(Sherbrook) M. Capone (Rome) O Parcollet(Saclay).

The Mott transition across in actinides The Mott transition across in actinides

Cuprate Superconductors: doping the Mott Cuprate Superconductors: doping the Mott insulator. insulator.

DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).DMFT Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). H Happy appy marriage of atomic and band physics.marriage of atomic and band physics.

Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti (to appear in RMP).

1( , )

( )k

G k ii i

Extremize a functional of the local spectra. Local self energy.

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993)..

Mott transition in one band model. Review Georges et.al. RMP 96

DMFT + electronic structure methodDMFT + electronic structure method

Technical Implementation is Involved. Different Impurity Solvers. [ED-NCA- Expansions in t and U , etc ] Different forms of Self consistency conditions for the bath in the clusters case. Different levels of complexity in the description of the electronic structure, simple models to all electron calculations. Review: G. Kotliar, S. Savrasov K. Haule, V. OudovenkoO Parcollet, C. Marianetti . Review of Modern Physics 2006.

Basic idea of DMFT: reduce the quantum many body problem to a one site or a cluster of sites problem, in a medium of non interacting electrons obeying a self-consistency condition. (A. Georges et al., RMP 68, 13 (1996)). DMFT in the language of functionals: DMFT sums up all local diagrams in BK functional

Basic idea of DMFT+electronic structure method (LDA or GW): For less correlated bands (s,p): use LDA or GWFor correlated bands (f or d): with DMFT add all local diagrams. Gives total energy and spectra

Mean Field ApproachMean Field Approach Follow different “states” as a function of

parameters.

• Second step compare free energies.

• Work in progress. Solving the DMFT equations are non trivial.

Configurational cordinate, doping, T, U, structureConfigurational cordinate, doping, T, U, structure

T

Curie-Weiss

Tc

Photoemission and Localization Photoemission and Localization Trends in ActinidesTrends in Actinides

alpa->delta volume collapse transition

Curium has large magnetic moment and orders antifPu does is non magnetic.

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

The “DMFT-The “DMFT-valence” in the valence” in the late actinideslate actinides

Minimum in melting curve and divergence of the Minimum in melting curve and divergence of the compressibility at the Mott endpointcompressibility at the Mott endpoint

( )dT V

dp S

Vsol

Vliq

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Resistivity of Am under pressure. J. C. Griveau Rebizant Lander and Kotliar PRL 94, 097002 (2005).

Photomission Spectra of Am under pressure. Sunca. Onset of Photomission Spectra of Am under pressure. Sunca. Onset of

mixed valence. Savrasov Haule Kotliar (2005)mixed valence. Savrasov Haule Kotliar (2005)

Theoretical Approach [P.WAnderson,1987]Theoretical Approach [P.WAnderson,1987]

• Connection of the cuprate anomalies to the proximity to a doped Mott insulator without magnetic long range order.[Spin Liquid]

• Study low energy one band models, Hubbard and t-J.

Needed. a good mean field theory Needed. a good mean field theory of the problem. of the problem.

RVB physics requires a plaquette RVB physics requires a plaquette as a reference frame. as a reference frame.

.. • A functional of the cluster Greens function. Allows the

investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. al T. Maier et. al. (2000) . ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8t )

• Use exact diag ( Krauth Caffarel 1995 ) and vertex corrected NCA as a solvers to study larger U’s and CDMFT as the mean field scheme.

CDMFT study of cuprates

RVB phase diagram of the Cuprate RVB phase diagram of the Cuprate Superconductors. Superexchange.Superconductors. Superexchange.

Flux-S+iD spin liquid. [Affleck and Marston , G Kotliar]

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria

N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)

Superconductivity in the Hubbard model Superconductivity in the Hubbard model role of the role of the Mott transitionMott transition and and influence of the super-exchangeinfluence of the super-exchange. .

( work with M. Capone et.al V. Kancharla.et.al ( work with M. Capone et.al V. Kancharla.et.al CDMFT+ED, 4+ 8 sites t’=0) . CDMFT+ED, 4+ 8 sites t’=0) .

cond-mat/0508205cond-mat/0508205 Anomalous superconductivity in doped Anomalous superconductivity in doped Mott insulator:Mott insulator:Order Parameter and Superconducting Gap . Order Parameter and Superconducting Gap . They scale together for small U, but not for large U. S. They scale together for small U, but not for large U. S. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G.

Kotliar andA.Tremblay. Cond mat Kotliar andA.Tremblay. Cond mat 05082050508205 M. Capone M. Capone (2006). (2006).

Superconducting DOS Superconducting DOS

=.08

= .16

Superconductivity is destroyed by transfer of spectral weight.. Similar to slave bosons d wave RVB. M. Capone et. al

Doping Driven Mott transiton at low temperature, in 2d Doping Driven Mott transiton at low temperature, in 2d ((U=16 t=1, t’=-.3U=16 t=1, t’=-.3 ) Hubbard model ) Hubbard model

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k0) vs kK.M. Shen et.al. 2004

2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)Senechal et.al Senechal et.al PRL94 (2005)PRL94 (2005)

Nodal Antinodal Dichotomy and pseudogap. T. Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302Stanescu and GK cond-matt 0508302

Optics and RESTRICTED SUM RULESOptics and RESTRICTED SUM RULES

0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk


0( ) ( ) ( ) ( )n s n sd T T T T

<T>n is defined for T> Tc, while <T>s exists only for T<Tc . <T>n is a strong is a strong function of temperature in the function of temperature in the normal state. Carbone et. al normal state. Carbone et. al (2006) .(2006) .

Hubbard model

DrudeU

t2/U t

t-J model

J-t

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 in t-J K Haule and GK

Kinetic energy decreases

Kinetic energy increases

Kinetic energy increases

Exchange energy decreases and gives

largest contribution to condensation energy

cond-mat/0503073

Scalapino White PRB 58, 8222 (1988)Scalapino White PRB 58, 8222 (1988)

i jk

d. >= 3 b( ) )cos ."( , i jS S q R Rk

Haule and Kotliar (2006) Haule and Kotliar (2006) Coarsed grained or “local Coarsed grained or “local “ susceptibility around “ susceptibility around

(())

ConclusionConclusion

• DMFT versatile tool for advancing our understanding, and predicting properties of strongly correlated materials.

• Theoretical spectroscopy in the making.

Substantial work is needed to refine the tool.

• Great opportunity for experimental-theoretical interactions.

• Refine the questions and our understanding by focusing on differences between the DMFT results and the experiments.

Mean-Field : Classical vs QuantumMean-Field : Classical vs Quantum

Classical case Quantum case

Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

†

0 0 0

( )[ ( ')] ( ')o o o oc c U n nb b b

s st m t t tt ¯

¶+ - D - +

¶òò ò

( )wD

†( )( ) ( )

MFo n o n SG c i c is sw w D=- á ñ

1( )

1( )

( )[ ][ ]

nk

n kn

G ii

G i

ww e

w

=D - -

D

å

,ij i j i

i j i

J S S h S- -å å

MF eff oH h S=-

effh

0 0 ( )MF effH hm S=á ñ

eff ij jj

h J m h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Anomalous Self EnergyAnomalous Self Energy. (from Capone et.al.) Notice the . (from Capone et.al.) Notice the remarkable increase with decreasing doping! True remarkable increase with decreasing doping! True

superconducting pairing!! U=8tsuperconducting pairing!! U=8t

Significant Difference with Migdal-Eliashberg.

<l.s> in the late actinides [DMFT results: <l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]K. Haule and J. Shim ]

-U

J. Tobin et.al. PRB 72,085109 J. Tobin et.al. PRB 72,085109 (2005)(2005)

XAS and EELS

Double well structure and Double well structure and Pu Pu Qualitative explanation of negative thermal expansion[Lawson, A. C., Roberts J. A., Martinez, B., and Richardson, J. W., Jr. Phil. Mag. B, 82, 1837,(2002). G. Kotliar J.Low Temp. Physvol.126, 1009 27. (2002)]

Natural consequence of the conclusions on the model Hamiltonian level. We had two solutions at the same U, one metallic and one insulating. Relaxing the

volume expands the insulator and contract the metal.

F(T,V)=Fphonons+F(T,V)=Fphonons+FinvarFinvar

““Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag. Invar model “ for Pu-Ga. Lawson et. al.Phil. Mag.

(2006) Data fits if the excited state has zero stiffness.(2006) Data fits if the excited state has zero stiffness.

References and CollaboratorsReferences and Collaborators

• References:• M. Capone et. al. in preparation• M. Capone and G. Kotliar cond-mat cond-mat/0603227 • Kristjan Haule, Gabriel Kotliar cond-mat/0605149• M. Capone and G.K cond-mat/0603227• Kristjan Haule, Gabriel Kotliar cond-mat/0601478

• Tudor D. Stanescu and Gabriel Kotliar cond-mat/0508302• S. S. Kancharla, M. Civelli, M. Capone, B. Kyung, D.

Senechal, G. Kotliar, A.-M.S. Tremblay cond-mat/0508205• M. Civelli M. Capone S. S. Kancharla O. Parcollet and G.

Kotliar Phys. Rev. Lett. 95, 106402 (2005)

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT within plaquette Cellular DMFT

• Rich Structure of the normal state and the interplay of the ordered phases.

• Work needed to reach the same level of understanding of the single site DMFT solution.

• A) Either that we will understand some qualitative aspects found in the experiment. In which case the next step LDA+CDMFT or GW+CDMFT could be then be used make realistic modelling of the various spectroscopies.

• B) Or we do not, in which case other degrees of freedom, or inhomogeneities or long wavelength non Gaussian modes are essential as many authors have surmised.

• Too early to tell, talk presented some evidence for A.

.

Correlations Magnetism and Structure across the Correlations Magnetism and Structure across the actinide series : a Dynamical Mean Field Theory actinide series : a Dynamical Mean Field Theory PerspectivePerspective

Plutonium Futures Asilomar July 9-13 (2006).

G.Kotliar Physics Department and Center for Materials Theory Rutgers University.

.

Collaborators K. Haule (Rutgers ) Ji-Hoon Shim (Rutgers) S. Savrasov (UCDavis ) A.M. Tremblay B. Kyung (Sherbrook) M. Capone (Rome) O Parcollet(Saclay).

Expts. : M. Fluss J. C Griveaux G Lander A. Lawson A. Migliori J.Singleton J.Smith J Thompson J. Tobin

Support: DOE- BES DOE-NNSA .

M. Capone and GK cond-mat 0511334 . Competition fo M. Capone and GK cond-mat 0511334 . Competition fo superconductivity and antiferromagnetism. superconductivity and antiferromagnetism.

Temperature dependence of the spectral Temperature dependence of the spectral weight of CDMFT in normal state. weight of CDMFT in normal state.

Carbone, see also ortholani for CDMFT. Carbone, see also ortholani for CDMFT.

Finite temperature view of the phase Finite temperature view of the phase diagram t-J model.diagram t-J model.

K. Haule and GK (2006) K. Haule and GK (2006)

OutlineOutline

• Introduction. Mott physics and high temperature superconductivity. Early Ideas: slave boson mean field theory. Successes and Difficulties.

• Dynamical Mean Field Theory approach and its cluster extensions.

• Results for optical conductivity.

• Anomalous superconductivity and normal state.

• Future directions.

UPS of alpha-UUPS of alpha-U

- He I (hv=21.21eV), He II (hv=40.81eV)- f-electron features is enhanced in He II spectra.Opeil et al. PRB(2006)

GGA

-LDA+DMFT reproduces peaks near -1eV, 0.3eV, and EF-The peak near -3eV corresponds to U 6d states.

n_f=2.94

0

1 2

( , ) ( )

( )(cos cos ) ( )(cos .cos ) .......latt k

kx ky kx ky

Cluster Extensions of Single Site DMFT

Many Techniques for solving the impurity model: QMC, (Fye-

Hirsch), NCA, ED(Krauth –Caffarel),

IPT, …………For a review see Kotliar et. Al to appear in RMP

(2006)

n_5/2=2.41n_7/2=0.53

How is the Mott insulatorHow is the Mott insulatorapproached from the approached from the

superconducting state ?superconducting state ?

Work in collaboration with M. Capone M Civelli O Parcollet

• In BCS theory the order parameter is tied to the superconducting gap. This is seen at U=4t, but not at large U.

• How is superconductivity destroyed as one

approaches half filling ?

Superconducting State t’=0Superconducting State t’=0

• Does it superconduct ?• Yes. Unless there is a competing phase.• Is there a superconducting dome ?• Yes. Provided U /W is above the Mott

transition .• Does the superconductivity scale with J ?• Yes. Provided U /W is above the Mott

transition .• Is superconductivity BCS like?• Yes for small U/W. No for large U, it is RVB like!

• The superconductivity scales

with J, as in the RVB approach.

Qualitative difference between large and small U. The superconductivity goes to zero at half filling ONLY above the Mott transition.

•Can we connect the Can we connect the superconducting state with the superconducting state with the “underlying “normal” state “ ? “underlying “normal” state “ ?

What does the underlying “normal” What does the underlying “normal” state look like ?state look like ?

Follow the “Follow the “normal state”normal state” with doping. with doping. Civelli et.al. PRL 95, Civelli et.al. PRL 95, 106402 (2005)106402 (2005)

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k U=16 t, 0) vs k U=16 t,

t’=-.3t’=-.3

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

K.M. Shen et.al. 2004

2X2 CDMFT

Nodal Antinodal Dichotomy and pseudogap. T. Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302Stanescu and GK cond-matt 0508302


0( ) ,eff effd P J

iV

, ,eff eff effH J P

2

0( ) ,

ned P J

iV m

Low energy sum rule can have T and doping dependence . For nearest neighbor it gives the kinetic energy. Use it to extract changes in KE in superconducing state

, ,H hamiltonian J electric current P polarization

Below energy

2

2

kk

k

nk

Larger frustration: t’=.9t U=16tLarger frustration: t’=.9t U=16tn=.69 .92 .96n=.69 .92 .96

M. Civelli M. CaponeO. Parcollet and GK M. Civelli M. CaponeO. Parcollet and GK

PRL (20050PRL (20050

Add equation for the difference between Add equation for the difference between the methods. the methods.

• Can compute kinetic energy from both the integral of sigma and the expectation value of the kinetic energy.

• Treats normal and superconducting state on the same footing.

•

. Spectral weight integrated up to 1 eV of the three BSCCO . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-films. a) under-

doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼overdoped, Tc=63 K; the fulloverdoped, Tc=63 K; the full

symbols are above Tc (integration from 0+), the open symbols symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the below Tc, (integrationfrom 0, including th weight of the

superfuid).superfuid).

H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) . Recent review:

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT within plaquette Cellular DMFT

• Rich Structure of the normal state and the interplay of the ordered phases.

• Work needed to reach the same level of understanding of the single site DMFT solution.

• A) Either that we will understand some qualitative aspects found in the experiment. In which case LDA+CDMFT or GW+CDMFT could be then be used to account semiquantitatively for the large body of experimental data by studying more realistic models of the material.

• B) Or we do not, in which case other degrees of freedom, or inhomgeneities or long wavelength non Gaussian modes are essential as many authors have surmised.

• Too early to tell, talk presented some evidence for A.

.

IssuesIssues• What aspects of the unusual properties of the cuprates

follow from the fact that they are doped Mott insulators using a DMFT which treats exactly and in an umbiased way all the degrees of freedom within a plaquette ?

• Solution of the model at a given energy scale, Physics at a given energy • Recent Conceptual Advance: DMFT (in its single site a

cluster versions) allow us to address these problems. • A) Follow various metastable states as a function of doping.• B) Focus on the physics on a given scale at at time. What is

the right reference frame for high Tc.

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc. t-J limit.

• Slave boson approach. <b> coherence order parameter. singlet formation order parameters.Baskaran Zhou Anderson , (1987)Ruckenstein Hirshfeld and Appell (1987) .Uniform Solutions. S-wave superconductors. Uniform RVB states.

Other RVB states with d wave symmetry. Flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) . Spectrum of excitation have point zerosUpon doping they become a d –wave superconductor. (Kotliar and Liu 1988). .

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

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Dynamical Mean Field Theory. Cavity Construction.Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).A. Georges and G. Kotliar PRB 45, 6479 (1992).

Reviews: A. Georges W. Krauth G.Kotliar and M. Rozenberg RMP (1996)G. Kotliar and D. Vollhardt Physics Today (2004).

Classical case Quantum case

A. Georges, G. Kotliar (1992)

Mean-Field : Classical vs QuantumMean-Field : Classical vs Quantum

†

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ¯

¶+ D-

¶- +òò ò

( )wD

†( )( ( ) )) (

MFo n oo n n Sc i c iG i s ss ww w D=- á ñ

( )

(()

)

11

([ ]

)[ ]n

n

kn

G i

G it ki m

w

wwD

D

=- - +

å

,ij i j i

i j i

J S S h S- -å å

eMF offhH S=-

effh

00 ( )MF effH hm S=á ñ

ijff jj

e mh J h= +å

† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

Easy!!!

0 [ ]S th heffbá ñ=

Hard!!!QMC: J. Hirsch R. Fye (1986)NCA : T. Pruschke and N. Grewe (1989)PT : Yoshida and Yamada (1970)NRG: Wilson (1980)

• Pruschke et. al Adv. Phys. (1995) • Georges et. al RMP (1996)

IPT: Georges Kotliar (1992). .QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell

(1993),ED:Caffarel Krauth and Rozenberg (1994)Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999),……………………………………...

DMFT Qualitative Phase diagram of a DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillingfrustrated Hubbard model at integer filling

T/W

Georges et.al. Georges et.al. RMP (1996) RMP (1996)

Kotliar Kotliar Vollhardt Vollhardt

Physics Today Physics Today (2004)(2004)

Single site DMFT and kappa organics. Qualitative phase Single site DMFT and kappa organics. Qualitative phase

diagram Coherence incoherence crosoverdiagram Coherence incoherence crosover. .

Finite T Mott tranisiton in CDMFT Finite T Mott tranisiton in CDMFT O. Parcollet O. Parcollet

G. Biroli and GK PRL, 92, 226402. (2004))G. Biroli and GK PRL, 92, 226402. (2004))

CDMFT results Kyung et.al. (2006)CDMFT results Kyung et.al. (2006)

.. • Functional of the cluster Greens function. Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Can study different states on the same footing allowing for the full frequency dependence of all the degrees of freedom contained in the plaquette.

• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS -(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • b--------> b(k), ----- (), k• Better description of the incoherent state, more general

functional form of the self energy to finite T and higher frequency.

CDMFT : methodological comments

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

Further extensions by periodizing cumulants rather than self energies. Stanescu and Further extensions by periodizing cumulants rather than self energies. Stanescu and GK (2005)GK (2005)

Early SB DMFT. Early SB DMFT. • There are two regimes, one overdoped one underdoped.• Tc has a dome-like shape.• High Tc superconductivity is driven by superexchange.• Normal state at low doping has a pseudogap a low doping

with a d wave symmetry.

• Normal State at low temperatures.

Dependence on periodization scheme. Dependence on periodization scheme.

Energetics and phase separation. Right Energetics and phase separation. Right U=16t Left U=8t U=16t Left U=8t

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

t’=0

Phase diagram

Temperature Depencence of Integrated spectral weight

E Energy difference between the normal E Energy difference between the normal and superconducing state of the t-J model. and superconducing state of the t-J model.

K. Haule (2006)K. Haule (2006)

ConclusionConclusion• DMFT studies of electrons and lattice

displacements.• Valence changes and transfers of spectral weight. [

Consistent picture of Pu-Am-Cm].• Alpha and delta Pu, screened (5f)^5 configuration.

Differ in the degree of screening. Different views [ Pu non magnetic (5f)^6, Pu

magnetic ] • Magnetism and defects. • Important role of phonon entropy in phase

transformations .

LS vs jj coupling in Am and CmLS vs jj coupling in Am and Cm

Temperature dependence of the spectral Temperature dependence of the spectral weight of CDMFT in normal state. weight of CDMFT in normal state.

Carbone, see also Toschi et.al for CDMFT. Carbone, see also Toschi et.al for CDMFT.

UPS of alpha-UUPS of alpha-U

- He I (hv=21.21eV), He II (hv=40.81eV)- f-electron features is enhanced in He II spectra.Opeil et al. PRB(2006)

GGA

-LDA+DMFT reproduces peaks near -1eV, 0.3eV, and EF-The peak near -3eV corresponds to U 6d states.

n_f=2.94

<l.s> in the late actinides [DMFT results: <l.s> in the late actinides [DMFT results: K. Haule and J. Shim ]K. Haule and J. Shim ]

-U

Why is Epsilon Pu (which is smaller than delta Why is Epsilon Pu (which is smaller than delta Pu) stabilized at higher temperatures ??Compute Pu) stabilized at higher temperatures ??Compute phonons in bcc structure.phonons in bcc structure.

Phonon entropy drives the Phonon entropy drives the epsilon delta phase transitionepsilon delta phase transition

• Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

• At the phase transition the volume shrinks but the phonon entropy increases.

• Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.







ApproachApproach

• Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models.

[ Leave out disorder, electronic structure,phonons …]

• Follow different “states” as a function of parameters.

[Second step compare free energies which will depend more on the detailed modelling…..]

• Work in progress. The framework and the resulting equations are very non trivial to solve.

Approach the Mott point from the right Am under Approach the Mott point from the right Am under pressurepressure

Density functional based electronic structure calculations: Non magnetic LDA/GGA predicts volume 50% off. Magnetic GGA corrects most of error in volume but gives m~6B

(Soderlind et.al., PRB 2000). Experimentally, Am has non magnetic f6 ground state with J=0 (7F0)

Experimental Equation of State (after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Am equation of state. LDA+DMFT.New acceleration Am equation of state. LDA+DMFT.New acceleration technique for solving DMFT equations S. Savrasov K. technique for solving DMFT equations S. Savrasov K.

Haule G. Kotliar cond-mat. 0507552 (2005)Haule G. Kotliar cond-mat. 0507552 (2005)

Photoemission spectra using Hubbard I solver [Lichtenstein Photoemission spectra using Hubbard I solver [Lichtenstein and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat and Katsnelson, PRB 57, 6884,(1998 ), Svane cond-mat

0508311] and Sunca . [Savrasov Haule and Kotliar cond-mat 0508311] and Sunca . [Savrasov Haule and Kotliar cond-mat 0507552] Hubbard bands width is determined by multiplet 0507552] Hubbard bands width is determined by multiplet

splittings.splittings.

DMFT Phonons in fcc DMFT Phonons in fcc -Pu-Pu

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa)

Theory 34.56 33.03 26.81 3.88

Experiment 36.28 33.59 26.73 4.78

( Dai, Savrasov, Kotliar,Ledbetter, Migliori, Abrahams, Science, 9 May 2003)

(experiments from Wong et.al, Science, 22 August 2003)

Phonon entropy drives the Phonon entropy drives the epsilon delta phase transitionepsilon delta phase transition

• Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

• At the phase transition the volume shrinks but the phonon entropy increases.

• Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.







-U

Why is Epsilon Pu (which is smaller than delta Why is Epsilon Pu (which is smaller than delta Pu) stabilized at higher temperatures ??Compute Pu) stabilized at higher temperatures ??Compute phonons in bcc structure.phonons in bcc structure.

t’=0

Phase diagram

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

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

.. • AFunctional of the cluster Greens function. Allows the investigation of the normal state

underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. T. Maier et. al. (2000) . ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8t )

• Use exact diag ( Krauth Caffarel 1995 ) and vertex corrected NCA as a solvers to study larger U’s and CDMFT as the mean field scheme.

• Recently (K. Haule and GK ) the region near the superconducting –normal state transition temperature near optimal doping was studied using NCA + DCA-CDMFT .

• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS -(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • b--------> b(k), ----- (), k• Extends the functional form of self energy to finite T and higher frequency.• Larger clusters can be studied with VCPT CPT [Senechal and Tremblay, Arrigoni, Hanke ]

CDMFT study of cuprates

Exact Baym Kadanoff functional ofwo variables. Exact Baym Kadanoff functional ofwo variables. ,G]. Restric to the ,G]. Restric to the degrees of freedom that live on a plaquette and its supercell extension.. degrees of freedom that live on a plaquette and its supercell extension.. Maps the many body problem onto Maps the many body problem onto a self consistenta self consistent impurity model impurity model

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP in Press. Tremblay Kyung Senechal cond-matt 0511334

Problems with the approach.Problems with the approach.

• Stability of the MFT. Ex. Neel order. Slave boson MFT with Neel order predicts AF AND SC. [Inui et.al. 1988] Giamarchi and L’huillier (1987).

• Mean field is too uniform on the Fermi surface, in contradiction with ARPES.[Penetration depth, Wen and Lee ][Raman spectra, sacutto’s talk, Photoemission ]

• Description of the incoherent finite temperature regime.

Development of DMFT in its plaquette version may solve some of these problems.!!

Documents

Theoretical and Experimental Magnetism Meeting Theoretical and Experimental Magnetism Meeting Gabriel Kotliar Rutgers University Support :National Science