The Mott Transition and the Challenge of Strongly Correlated

Electron Systems. G. Kotliar

Physics Department and Center for Materials Theory

Rutgers

PIPT Showcase Conference UBC Vancouver May 12th 2005

The Standard Model of Solid State Physics.

• Itinerant limit. Band Theory. Wave picture of the electron in momentum space. . Pauli susceptibility.

• Localized model. Real space picture of electrons bound to atoms. Curie susceptibility at high temperatures, spin-orbital ordering at low temperatures.

Correlated Electron Materials

• Are not well described by either the itinerant or the localized framework .

• Compounds with partially filled f and d shells. Need new starting point for their description. Non perturbative problem. New reference frame for computing their physical properties.

• Have consistently produce spectacular “big” effects thru the years. High temperature superconductivity, colossal magneto-resistance, huge volume collapses……………..

Large Metallic Resistivities

21 1 1( )

(100 )MottF Fe k k l

cmh

Transfer of optical spectral weight non local in frequency Schlesinger et. al. (1994), Vander Marel

(2005) Takagi (2003 ) Neff depends on T

Breakdown of the standard model of solids.

• Qualitative Issues : i.e. Large metallic resistivities. Breakdown of the rigid band picture. Anomalous transfer of spectral weight in photoemission and optics……………

• The quantitative tools of the standard model fail (i.e. density functional + GW )

Two paths for calculation of electronic structure of

strongly correlated materials

Correlation Functions Total Energies etc.

Model Hamiltonian

Crystal structure +Atomic positions

DMFT ideas can be used in both cases.

1( , ) Im[ ( , )] Im[ ]

( , )k

A k G kk

MODEL HAMILTONIAN AND OBSERVABLES

Limiting case itinerant electrons( ) ( )kk

A

( ) ( , )k

A A k

( ) ( ) ( )B AA

Limiting case localized electrons

Hubbard bands

Local Spectral Function

A BU

† †

, ,

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

t c c c c U n n

Parameters: U/t , T, carrier concentration, frustration :

Limit of large lattice coordination1

~ d ij nearest neighborsijtd

† 1~i jc c

d

†

,

1 1~ ~ (1)ij i j

j

t c c d Od d

~O(1)i i

Un n

Metzner Vollhardt, 89

1( , )

( )k

G k ii i

Muller-Hartmann 89

Dynamical Mean-Field Theory

A. Georges, G. Kotliar Phys. Rev. B 45, 6497(1992)Review: G. Kotliar and D. Vollhardt Physics Today 57,(2004)

†

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 t

G i

ww

w

D =D - -

D

å

† †

, ,

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

t c c c c U n n

1( )] ( )

( )[ ]

1( )[ ]

( )]

[

[[ ]

n n nn

nk n n k

i i iG i

G ii i t

w m w ww

ww m w

+ - S =D -D

D =+ - S -å

Mott transition in V2O3 under pressure

or chemical substitution on V-site. How does the electron go from localized to itinerant.

The Mott transition and Universality

Same behavior at high tempeartures, completely

different at low T

T/W

Phase diagram of a Hubbard model with partial frustration at integer filling.  M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995). .

COHERENCE INCOHERENCE CROSSOVER

Transfer of optical spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

Transfer of optical spectral weight

M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P

Metcalf Phys. Rev. Lett. 75, 105 (1995)

Anomalous transfer of optical spectral weight, NiSeS. [Miyasaka and Takagi

2000]

Anomalous Resistivity and Mott transition Ni Se2-x Sx

Crossover from Fermi liquid to bad metal to semiconductor to paramagnetic insulator. M. Rozenberg G. Kotliar H. Kajueter G Tahomas D. Rapkikne J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995)

Single-site DMFT and expts

Conclusions: lessons from the application of DMFT to toy model.

• Three peak structure, quasiparticles and Hubbard bands.

• The Mott transition is driven by transfer of spectral weight from low to high energy as we approach the localized phase.

• The method can describe coherent and incoherent phenomena and their crossover. Access to non perturbative regime . Real and momentum space.

• Theory and experiments begin to agree on a broad picture.

Realistic Descriptions of Materials and a First Principles Approach to

Strongly Correlated Electron Systems.

• Incorporate realistic band structure and orbital degeneracy.

• Incorporate the coupling of the lattice degrees of freedom to the electronic degrees of freedom.

• Predict properties of matter without empirical information.

LDA+DMFT V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). • Realistic band structure and orbital degeneracy.

Describes the excitation spectra of many strongly correlated solids. .

Spectral Density Functionals. Chitra and Kotliar PRB 2001 Savrasov et. al. Nature (2001) Savrasov and Kotliar PRB (2005)

•Determine the self energy , the density and the structure of the solid by extremizing a functional of these quantities. Coupling of electronic degrees of freedom to structural degrees of freedom.

Mott Transition in the Actinide Series

Pu phases: A. Lawson Los Alamos Science 26, (2000)

LDA underestimates the volume of fcc Pu by 30%.

Within LDA fcc Pu has a negative shear modulus.

LSDA predicts Pu to be magnetic with a 5 b moment. Experimentally it is not.

Treating f electrons as core overestimates the volume by 30 %

Total­Energy­as­a­function­of­volume­for­Total­Energy­as­a­function­of­volume­for­PUPU

(Savrasov, Kotliar, Abrahams, Nature ( 2001)Non magnetic correlated state of fcc Pu.

Double well structure and Pu Qualitative explanation of negative thermal expansion[ G. Kotliar J.Low

Temp. Physvol.126, 1009 27. (2002)]See also A . Lawson et.al.Phil. Mag. B 82, 1837 ]

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.

Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003

Inelastic X Ray. Phonon energy 10 mev, photon energy 10 Kev.

E = Ei - EfQ =ki - kf

DMFT­­Phonons­in­fcc­DMFT­­Phonons­in­fcc­-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)

J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003

First Principles DMFT Studies of Pu

• Pu strongly correlated element, at the brink of a Mott instability, which could not be described within the standard model.

• Quantitative computations : total energy, photoemission spectra and phonon dispersions of delta Pu.

• Qualitative Insights and quantitative studies. Double well. Alpha and Delta Pu.Other Pu anomalies.

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

Density­functional­based­electronic­structure­calculations: Non­magnetic­LDA/GGA­predicts­volume­50%­off.­ Magnetic­GGA­corrects­most­of­error­in­volume­but­gives­m~6B

(Soderlind et.al., PRB 2000). Experimentally,­Am­has non­magnetic­f6­ground­state­with­J=0­(7F0)

Experimental­Equation­of­State­(after Heathman et.al, PRL 2000)

Mott Transition?“Soft”

“Hard”

Mott transition in open (right) and closed (left) shell systems.

Realization in Am ??

S S

U U

TLog[2J+1]

Uc

~1/(Uc-U)

J=0

???

Tc

0

1 2

( , ) ( )

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

kx ky kx ky

Cluster Extensions of Single Site DMFTCaptures new physics beyond single site DMFT , i.e. d wave superconductivity, and other novel aspects of the Mott transition in two dimensional systems.

Conclusions Future Directions• DMFT: Method under development, but it

already gives new insights into materials…….• Exciting development: cluster extensions. Allows

us to see to check the accuracy of the single site DMFT corrections, and obtain new physics at lower temperatures and closer to the Mott transition where the single site DMFT breaks down.

• Development of DMFT allow us to focus on deviations of experiments from mean field theory.

• DMFT and RG developments

Some References

• Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68 , 13, (1996).

• Reviews: G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti. Submitted to RMP (2005).

• Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004)