Strongly Correlated Electron Materials: a Challenge for the 21 st Century Strongly Correlated Electron Materials: a Challenge for the 21 st Century Gabriel.

Strongly Correlated Electron Materials:Strongly Correlated Electron Materials: a Challenge for the 21 a Challenge for the 21stst Century Century

Gabriel KotliarGabriel Kotliar

and Center for Materials Theory

$upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, materials world network.

Colloquium Harvard University April 19 2010Colloquium Harvard University April 19 2010

11

OutlineOutline• Introduction to the problem of strongly

correlated electron systems

• Introduction to some ideas and techniques from dynamical mean field theory (DMFT)

• Application to the most correlated element Pu

• Application to copper oxides.

• Outlook

$upport : NSF -DMR , DOE-Basic Energy Sciences, MURI, NSF materials world network. 11

Collaborators. Rutgers, K. Haule, C. Weber, J. Shim T. Stanescu, M. CivelliCollaborators. Rutgers, K. Haule, C. Weber, J. Shim T. Stanescu, M. Civelli

Paris, M. Ferrero, A. Georges, L. DeLeo, P. Cornaglia, O ParcolletParis, M. Ferrero, A. Georges, L. DeLeo, P. Cornaglia, O Parcollet

Sherbrooke, A.M. Tremblay B. Kyung D. SenechalSherbrooke, A.M. Tremblay B. Kyung D. Senechal

Standard Model of Solid State PhysicsStandard Model of Solid State Physics• In many materials ( Cu, Au, …)electrons in

solids behave as waves, quasiparticles

[Sommerfeld Bloch] Metals conduct!

• The Coulomb interactions renormalize to zero at low energies [ Landau]

• Density functional theory in the LDA gives good estimates for the density and a good starting point for computing spectra [Kohn Sham ]

• First order perturbation theory in the screened Coulomb interactions [GW ] is in good agreement with experiments. [Hedin]

= [ ]k kw e

22

Different approach is needed for Different approach is needed for magnetic insulatorsmagnetic insulators

• In other materials NiO, SmCo5 …Electron behave as particles

• Solid as a collection of (open shell )atoms with localized electrons.

• Interaction among the atom (exchange) order the atomic degrees of freedom (magnetic moments….)

• Excitation spectra: multiplet theory, Hubbard bands, low wavelength collective fluctuations (spin waves)

33

Strongly Correlated MaterialsStrongly Correlated Materials•Not well described by either the fully localized picture (well Not well described by either the fully localized picture (well separated atoms) or the the band picture (weakly interacting bloch separated atoms) or the the band picture (weakly interacting bloch waves). Materials for which the standard model of the solid state waves). Materials for which the standard model of the solid state

fails. Challenging fails. Challenging non perturbative non perturbative problem. problem.

•Continuous discovery of interesting material and phenomena that Continuous discovery of interesting material and phenomena that did not fit the standard model of solid state physics. Heavy fermions did not fit the standard model of solid state physics. Heavy fermions (early 80’s) , high temperature superconductors (late 80’s), other (early 80’s) , high temperature superconductors (late 80’s), other transition metal oxides (cobaltates, manganites, vanadates….) transition metal oxides (cobaltates, manganites, vanadates….) 90’s ..high Tc in FeAs based compounds (2009).90’s ..high Tc in FeAs based compounds (2009).

44

Interesting Correlated Interesting Correlated Materials discovered by Materials discovered by serendipity and the Edisonian serendipity and the Edisonian approachapproach

Correlated materials: oxides, Correlated materials: oxides, simple recipe, rich behavior simple recipe, rich behavior

Transition metal oxidesTransition metal oxides

OxygenOxygen

transition metal iontransition metal ion

Cage : e.g 6 oxygen atoms (octahedra) Cage : e.g 6 oxygen atoms (octahedra)

or other ligands/geometryor other ligands/geometry

Build crystal with this building blockBuild crystal with this building block

or build layers separated by spacers or build layers separated by spacers

Transition metal (open shell )Transition metal (open shell )

Transition metal ionsTransition metal ions

Rare earth ionsRare earth ions

ActinidesActinides

55

LiLixxCoO2, NaCoO2, NaxxCoO2 CoO2

Battery materialsBattery materials

Thermoelectrics Thermoelectrics

VOVO2 2

Room Room temperature temperature

MIT MIT

LaLa1-x1-xSrxMnO3SrxMnO3

Colossal Colossal MagnetoresistanceMagnetoresistance

LaLa1-x1-xSrSrxxCuO4CuO4

High temperature High temperature superconductor superconductor

Probing the Electronic Structure:Photoemission Probing the Electronic Structure:Photoemission

Probability of removing

an electron and transfering energy =Ei-Ef, and momentum k

f() A() M2

e

Angle integrated spectraAngle integrated spectra

( , ) ( )dkA k A 66

a)a) Weak correlationsWeak correlations

b)b) Strong correlation: fermi liquid parameters Strong correlation: fermi liquid parameters can’t be evaluated in perturbation theory or can’t be evaluated in perturbation theory or

fermi liquid theory does not work.fermi liquid theory does not work.

A(k,A(k, A(k,A(k,

A. Georges and G. Kotliar PRB 45, A. Georges and G. Kotliar PRB 45, 6479 (1992).6479 (1992).

DMFT self consistency condition DMFT self consistency condition

1( , )

( )k

G k ii i

*

( )V Va a

a a

ww e

D =-å

impG ( )1

[ ]( ) ( )[ ][ ]n imp

nk ni k i

iww e w

=-

DS D-å

DMFT DMFT

Collective field describing the localization delocalization phenomena Collective field describing the localization delocalization phenomena ( )wD

† †

, ,

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

t c c c c U n n

1( , ) Im ( , )A k i G k i

•HubbardHubbard

•GreenGreen

Phase diagram :frustrated Hubbard model, integer Phase diagram :frustrated Hubbard model, integer filling filling M. Rozenberg G. Kotliar H. Kajuter G. Thomas PRL75, 105 (1995)

T/W

1616

Quasiparticles Quasiparticles +Hubbard +Hubbard

bandsbands

Transfer Transfer of of

spectralspectralweightweight

Mott transitionMott transition

Coherence Coherence Incoherence Incoherence CrossoverCrossover

Spectral Spectral functionsfunctions

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kH k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >® Determine energy and andDetermine energy and andself consistently from extremizing a self consistently from extremizing a functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self functional . Savrasov and Kotliar PRB 69, 245101, (2001) Full self

consistent implementation consistent implementation

99

1( , )

( ) ( )G k i

i H k i

Spectra=- Im G(k,Spectra=- Im G(k,))

DMFT meets electronic structure. LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997).

0 0

0 ff

æ ö÷ç ÷S ç ÷ç ÷ç Sè ø®

abcdU U® DMFT DMFT Bands in a Bands in a frequency frequency dependent dependent potential potential

DMFT atom in a DMFT atom in a medium described medium described ( )abwD

Dynamical Mean Field Theory DMFT Dynamical Mean Field Theory DMFT • Describes the electron both in the itinerant (wave-like) and localized

(particle-like) regimes and everything in between!. • Simpler reference systems to understand correlated solids [e.g.

harmonic oscillator]• Impurity model non gaussian reference frame (dressed atom).

Tools to think about correlated materials. Weiss field quantifies the notion of itineracy. Configuration histograms generalizes CI.

• CDMFT Reference state cluser of sites ( links, triangles, plaquettes, etc. )

1010

•Locality assumption v accurate at high T (recent comparisons with cold atoms expts. and exact Locality assumption v accurate at high T (recent comparisons with cold atoms expts. and exact numerical techniques) Kozik et. al.numerical techniques) Kozik et. al. arXiv:0907.0863 Schneider et.al. Science, 322,1520(2008)Schneider et.al. Science, 322,1520(2008) . .

•Exact in the Metzner Vollhardt limit of infinite dimensions. Exact in the Metzner Vollhardt limit of infinite dimensions.

• Impurity solvers[ Recent advances, CTQMC Gull et. al. EPL 82, 57003 (2008), P. Werner et. al. Phys. Rev. Lett. 97, 076405 (2006) Bold CTQMC , OCA K. Haule arXiv:0907.0195 Phys. Rev. B 75, 155113 (2007) …..]

• Breaks problems in two parts a) study of mean field states from b ) evaluation of their energies.

• Compare different “ mean field states” of the system for the same value of parameters. Understand “mechanism” for ordering.

• Qualitative lessons can be drawn from (a) applied to simple models. High temperature universality.

• Low temperature, multiple ordered states. Detailed comparison experiments requires realistic implementions of electronic structure, e.g.LDA+DMFT.

• Bridge between atomic information and physical properties.(Structure-Property relation ).

• Theoretical spectroscopy.

1111

Localization Delocalization in ActinidesLocalization Delocalization in Actinides

Mott Transition

. .

PuPu

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Pu anomalies Pu anomalies

Large specific Large specific heat, large heat, large Pauli-like Pauli-like

susceptilitysusceptility

Large room Large room temperature temperature resistivityresistivity

Lashley et. al. 2005Lashley et. al. 2005

The standard model of solids fails near Pu The standard model of solids fails near Pu

• Spin Density functional theory predicts that Pu , Am are magnetic, large orbital and spin moments.

• Experiments (Lashley et. al. 2005, Heffner et al. (2006)): Pu is non magnetic. No static or fluctuating moments. Susceptibility, specific heat in a field, neutron quasielastic and inelastic scattering, muon spin resonance.

•Paramagnetic LDA underestimates Volume of Paramagnetic LDA underestimates Volume of Pu Pu by by 30%30%

• Within LDA Within LDA Pu has unstable phonon modes. Pu has unstable phonon modes.

•Thermodynamic and transport properties similar to Thermodynamic and transport properties similar to other strongly correlated materials.other strongly correlated materials.

1313Pu strongly correlated paramagnetic metal Pu strongly correlated paramagnetic metal

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

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

(experiments from Wong et.al, Science, 22 August 2003)(experiments from Wong et.al, Science, 22 August 2003)1414

PhotoemissioPhotoemissionn

Gouder , Havela PRBGouder , Havela PRB

2002, 2003 2002, 2003

alpa->delta volume collapse transitionalpa->delta volume collapse transition

F0=4,F2=F0=4,F2=6.16.1

1515

J. Shim K. Haule and G J. Shim K. Haule and G Kotliar Nature 446, 513 Kotliar Nature 446, 513

(2007)(2007)..

•Physical realization of the Mott localization-delocalization Physical realization of the Mott localization-delocalization phenomena.phenomena.

•Successful testbed for the LDA+DMFT method, volume, Successful testbed for the LDA+DMFT method, volume, phonon spectra. Plutonium is non magnetic Curium is magnetic phonon spectra. Plutonium is non magnetic Curium is magnetic with large moment. with large moment.

•Delta Pu, T* ~ 800K, mixed valent 20 % fDelta Pu, T* ~ 800K, mixed valent 20 % f66 80 % f 80 % f55 strong strong coupling to the lattice. coupling to the lattice.

•Moment screening by valence fluctuations. sMoment screening by valence fluctuations. s

•Reconciles susceptibility, XAS, and specific heat expts. Reconciles susceptibility, XAS, and specific heat expts.

•Quasiparticle multiplets, material specific fingerprints in the Quasiparticle multiplets, material specific fingerprints in the photoemisson spectra of Pu compounds. Prediction : photoemisson spectra of Pu compounds. Prediction : temperature dependence .temperature dependence .

Conclusions: elemental actinides. Conclusions: elemental actinides.

1616

Smith-Kmetko phase diagram Smith-Kmetko phase diagram

Realism:Realism:

electronic electronic structure structure

(band (band theory+atomtheory+atomic physics+ic physics+

Coupling Coupling electrons to electrons to structure)structure)

++

Conclusion: elemental Conclusion: elemental actinidesactinides

1717

Electron and Hole Electron and Hole Doped Cuprates : Doped Cuprates : Similar but Yet Similar but Yet

DifferentDifferent

NCCO : Robust AF PhaseNCCO : Robust AF Phase

Comensurate MagnetismComensurate Magnetism

Lower TcLower Tc

T^2 resistivity.T^2 resistivity.

Non monotonic angle dependence of Non monotonic angle dependence of SC order parameter ……… SC order parameter ………

Review: Armitage Fournier Review: Armitage Fournier Green (2009)Green (2009)

Apical Apical oxygenoxygen

1188

Fundamental questions still unresolvedFundamental questions still unresolved• Important degrees of freedom at different energy

scales [ dx2-y2, px py, pz, dz2 ]

• Relevant effective hamiltonians, one band multiband, etc.

• Mechanism of the superconductivity [ phonons, spin fluctuations, charge fluctuations, critical fluctuations, ….]

• How to describe the underlying normal state which does not fit in the fermi liquid paradigm (reference system) experiments.

• Difference among different families, electron vs hole doped. [ t, t’ fermi surfaces, ….]

1919

LSCOLSCO

NCCONCCOSingle site DMFT (low T) look for AF and PM Single site DMFT (low T) look for AF and PM solutions for the parent compound solutions for the parent compound

NCCO magnetism is responsible for gap !NCCO magnetism is responsible for gap !

LCO is a Mott charge transfer insulator.LCO is a Mott charge transfer insulator.

2020

Building phase diagram Building phase diagram magnetization at T=0 vs magnetization at T=0 vs ..

Single siteSingle site

Two siteTwo site

2121

Optical spectroscopyOptical spectroscopy

Theory (single site DMFT) C. Weber et. al.Theory (single site DMFT) C. Weber et. al.

Onose et. al. PRB Onose et. al. PRB 69, 024504 (2004 ) 69, 024504 (2004 ) Expt. Uchida et. al. PRB. Expt. Uchida et. al. PRB. 43, 7942 (1991)43, 7942 (1991) 2222

Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev)Optical Spectral Weights in LSCO and NCCO (up to 1.5 ev)

[xx][xx]Y. Onose et al., Phys. Y. Onose et al., Phys. Rev. B, 69, 024504 Rev. B, 69, 024504

(2004).(2004).

[xx][xx] S. Uchida et al., S. Uchida et al., Phys. Rev. B 43, 7942 Phys. Rev. B 43, 7942

(1991).(1991).

[xx]S. Lupi et al., Jour of [xx]S. Lupi et al., Jour of superconductivity 17, 131 superconductivity 17, 131 (2004).(2004).

2323

Zero Temperature Moment

Include d-wave superconducting Include d-wave superconducting solutions. Four site DFMT T=0solutions. Four site DFMT T=0

2424

Conclusions LDA+DMFT NCCO vs LSCOConclusions LDA+DMFT NCCO vs LSCO

• Strength of correlations (as quantified by single site DMFT) the most fundamental difference between NCCO and LSCO compounds.

• NCCO ( < c2 )and LSCO ( > c2)straddle the Zaanen Sawatsky Allen localization delocalization boundary.

• Can be traced to the absence of apical oxygen in NCCO

(structure property relation).• Introduces subtle differences in the metallization process and

interplay of magnetism and superconductivity.

•Good agreement with many subtle experimental features in NCCOGood agreement with many subtle experimental features in NCCO

even within single site DMFT. even within single site DMFT.

•No need to use x dependent values of the interaction UNo need to use x dependent values of the interaction U

•In general, better modeling with DMFT (more sites, more orbitals etc ) better In general, better modeling with DMFT (more sites, more orbitals etc ) better

resultsresults..

2525

Bath Bath

11 22

Return to Return to models, models,

Hubbard, t-JHubbard, t-J

Link DMFT Ferrero et. al. Link DMFT Ferrero et. al. Europhys. Lett. 85, 57009 Europhys. Lett. 85, 57009 (2009) Stanescu and (2009) Stanescu and Phillips P RB,69, 245104 Phillips P RB,69, 245104 (2004). (2004).

Plaquette DMFT:Plaquette DMFT:

Lichtenstein and Lichtenstein and Kastnelson PRB (2000)Kastnelson PRB (2000)

T. Maier, et. al. 2001, T. Maier, et. al. 2001, Europhys. Lett. Europhys. Lett. 56, 563.56, 563.

Sordi et.al. . Sordi et.al. . arXiv:1002.2960

Civelli et. al. Phys. Rev. Lett. Civelli et. al. Phys. Rev. Lett. 100, 046402 (2008) 100, 046402 (2008)

Haule and Kotliar Phys. Rev. Haule and Kotliar Phys. Rev. B 76, 104509 (2007)B 76, 104509 (2007)

Kinetic Energy Kinetic Energy ExchanExchange ge

EnergyEnergy

Real Real SpaceSpace

Momentum Momentum SpaceSpace

2626

Link DMFT (high temperature results)Link DMFT (high temperature results) Real Space Picture Real Space Picture

Momentum Space Picture

T=| ↑, ↑ >T=| ↑, ↑ >

1+= 1/√2(|0, ↑> + | ↑, 0>)1+= 1/√2(|0, ↑> + | ↑, 0>)

E=|0, 0>E=|0, 0>

S=1/√2( | ↑, ↓> -| ↑↓ S=1/√2( | ↑, ↓> -| ↑↓ >)>)

OverdopedOverdopedUnderdopedUnderdoped

2727

Fermi Fermi LiquidLiquid

Holes in a sea of Holes in a sea of singletssinglets

Nodal Antinodal Dichotomy : Spectral Function A(k,Nodal Antinodal Dichotomy : Spectral Function A(k,ω→ω→0) vs k0) vs k

K.M. Shen et.al. PRL 2004K.M. Shen et.al. PRL 2004

LSCO LSCO

2X2 CDMFT2X2 CDMFT

Nodal Region Nodal Region

Antinodal Region Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005) U=16 U=16 t=1, t’=-.3t=1, t’=-.3 , Hubbard model , Hubbard model 2828

Cluster DMFT charicature of something intererstingCluster DMFT charicature of something interersting

• Plaquette and link DMFT share many similarities. • CDMFT finite T technique (finite range cumulants

or self energies) . Underdoped region at low temperatures Fermi arcs turn into pockets. T. Stanescu and GK Phys. Rev. B 74, 125110 (2006)

• Optimal doping. Maximum in the scattering rate. Surprising particle hole symmetry. Hidden QCP (K Haule and G. Kotliar PRB 76, 104509 (2007)or critical endpoint (Sordi et.al. 2010)) preempted by SC Tc.

Phase diagram from plaquette DMFT coherence scale extracted from Phase diagram from plaquette DMFT coherence scale extracted from (0,pi) cluster self energy. (0,pi) cluster self energy.

2929

Tunneling : DOS Ratio ATunneling : DOS Ratio ASS/A/AN,.N,. Theory K. Theory K.

Haule and GK. PRB (2007)Haule and GK. PRB (2007)

Exp:Bi2212 with STMExp:Bi2212 with STM

McElroy et.al. McElroy et.al.

PRL 94, 197005 (2005)PRL 94, 197005 (2005)

3030

Superconductivity evolves continuously with doping Superconductivity evolves continuously with doping

Anomalous superconductivity in the underdoped region. Coherence Anomalous superconductivity in the underdoped region. Coherence peaks decrease when gap increases. Non BCS. peaks decrease when gap increases. Non BCS.

Pushp et. al. arXiv:0906.0817

Superexchange Mechanism . Superexchange Mechanism . K. Haule and GK K. Haule and GK Phys. Rev. B 76, 104509 (2007).Phys. Rev. B 76, 104509 (2007).

D.J. Scalapino and S.R. D.J. Scalapino and S.R.

White, Phys. Rev. B White, Phys. Rev. B 58, 8222 (1998).58, 8222 (1998).

Reminiscent of Reminiscent of PW Anderson RVG Science 235, 1196 (1987) and PW Anderson RVG Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 3131

Conclusion / Outlook Conclusion / Outlook • Correlated Electron Systems: fundamental questions, promising applications. Huge phase space• DMFT (simple ? ) framework to think about electrons in solids and compute their properties. • Many succesful applications to many materials for which cannot be treated with other techniques.• In use for many materials by many groups. Qualitative and quantitative system specific results gives us confidence in the method. Example of

actinide series, cuprates. • Needed: progress in implementation!• DMFT treats local correlation well but ignores non linear interactions among long wavelength modes and topological defects and other

fluctuation effects. Limited k dependence [Connection with other approaches to strong correlation]• Starting point for more sophisticated treatments [ including long wavelength modes and their interactions as in. Stat mech]

• Needed : Fluctuations around DMFT

3232

Realistic DMFT as a tool for material explorationRealistic DMFT as a tool for material exploration

• New arenas Interfaces, junctions heterostructures, artificial materials containing correlated electrons

• Grand challenge: using theory and computation to accelerate discoveries in strongly correlated electron systems.

Separates essential ingredients [e.g. phonons, orbitals, structure Separates essential ingredients [e.g. phonons, orbitals, structure etc. ] responsible for an effect. etc. ] responsible for an effect.

Compare different “states” of the system for the same value of Compare different “states” of the system for the same value of parameters. parameters. Understand Mechanism for ordering. Understand Mechanism for ordering.

Bridge between atomic information and physical and Bridge between atomic information and physical and spectroscopical properties. [Structure-Property relation spectroscopical properties. [Structure-Property relation Design] Design]

3232

Thank you for your attention!

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

GK PRB (2006)GK PRB (2006)

What is the Superconductivity Mechanism ?What is the Superconductivity Mechanism ?

Optics and RESTRICTED SUM RULESOptics and RESTRICTED SUM RULES

0( ) ( ) [ ] ( ) [ ] ( )n s n sd Ekin T Ekin T

[Ekin]n is only defined for T> Tc, while [Ekin]s [Ekin]n is only defined for T> Tc, while [Ekin]s exists only for T<Tcexists only for T<Tc

Experiment: use of this equation implies Experiment: use of this equation implies extrapolation. extrapolation.

Theory : use of this equation implies of mean Theory : use of this equation implies of mean field picture to continue the normal state field picture to continue the normal state

below Tc.below Tc.

. 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).H.J.A. Molegraaf et al., Science 295, 2239 (2002).

A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB

72, 092504(2005) . 72, 092504(2005) .

Even within the same scheme at low T. In some region of Even within the same scheme at low T. In some region of parameters of the Hamiltonian, there are at low temperature parameters of the Hamiltonian, there are at low temperature many many solutions to the DMFT equations with different many many solutions to the DMFT equations with different broken symmetries and ever increasing unite cells. broken symmetries and ever increasing unite cells.

No unique approach to expand around DMFT. No unique approach to expand around DMFT.

Landscape of DMFT Solutions Problem Landscape of DMFT Solutions Problem

DifficultiesDifficulties

•2x2 cluster DMFT equations are considerably harder to 2x2 cluster DMFT equations are considerably harder to solve and to interpret than single site DMFT.solve and to interpret than single site DMFT.

•Uniqueness: no unique formulation of cluster DMFT Uniqueness: no unique formulation of cluster DMFT

•Realistic implementations ARE demanding.Realistic implementations ARE demanding.

Technical IssuesTechnical Issues

Correlated SuperconductivityCorrelated Superconductivity• New concepts and techniques to treat highly

incoherent normal state and the superconductivity that emerges from it.

• Coherence incoherence crossover, lines of zeros, momentum space differentiation, ……

• Proximity to the Mott transition, accounts for many observations in correlated superconductors.

• Still further developments are needed to improve cluster dynamical mean field theories (k space resolution)

• Separates local mean field effects from fluctuation effects, superconducting fluctuations[ Nernst region etc.] Fluctuations around mean field.

T. Stanescu and GK Phys. Rev. B T. Stanescu and GK Phys. Rev. B 7474, 125110 (2006), 125110 (2006)Pseudogap state pockets + lines of zeros that screend them. Pseudogap state pockets + lines of zeros that screend them.

Some similiarities with phenomenological approach developed around the same time. Some similiarities with phenomenological approach developed around the same time. Yang Rice and Zhang PRB 73 174501 (2006). R. M. Konik, T. M. Rice, A. M. Tsvelik, Yang Rice and Zhang PRB 73 174501 (2006). R. M. Konik, T. M. Rice, A. M. Tsvelik,

Phys. Rev. Lett96, 086407 (2006).Phys. Rev. Lett96, 086407 (2006).

Superexchange Mechanism . Superexchange Mechanism . K. Haule and GK K. Haule and GK Phys. Rev. B 76, 104509 (2007).Phys. Rev. B 76, 104509 (2007).

D.J. Scalapino and S.R. D.J. Scalapino and S.R.

White, Phys. Rev. B White, Phys. Rev. B 58, 8222 (1998).58, 8222 (1998).

Reminiscent of Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and PW Anderson RVB Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 3131

Superexchange Mechanism . Superexchange Mechanism . K. Haule and GK K. Haule and GK Phys. Rev. B 76, 104509 (2007).Phys. Rev. B 76, 104509 (2007).

D.J. Scalapino and S.R. D.J. Scalapino and S.R.

White, Phys. Rev. B White, Phys. Rev. B 58, 8222 (1998).58, 8222 (1998).

Reminiscent of Reminiscent of PW Anderson RVG Science 235, 1196 (1987) and PW Anderson RVG Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988) 3131

Superexchange Mechanism?Superexchange Mechanism? . . K. Haule and GK K. Haule and GK Phys. Rev. Phys. Rev. B 76, 104509 (2007).B 76, 104509 (2007).Ex= Jij(< Si. Sj >s- < Si . Sj>n)/t

D.J. Scalapino and S.R. White, Phys. Rev. B 58, D.J. Scalapino and S.R. White, Phys. Rev. B 58, 8222 (1998).8222 (1998).

How is the energy distributed How is the energy distributed in q and w ?in q and w ?

Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave boson picture slave boson picture G. Kotliar and J. Liu P.RB 38,5412 (1988)

Expts; Dai et.al. Expts; Dai et.al.

Uchida et. al. PRB. Uchida et. al. PRB. 43, 7942 (1991)43, 7942 (1991)Onose et. al. PRB Onose et. al. PRB 69, 69, 024504 (2004)024504 (2004)

““Matthias’s Rules” for High TcMatthias’s Rules” for High Tc

• Metals. Must have d electrons (not just s s-p, nor f). Stay away from oxides.

• High symmetry is good, cubic is best. Nb3Sn

• Certain electron concentrations are favored (look for peak in density of states at Fermi level)

• Stay away from theorists

• “ Do not follow my rules “

2929

ThanksThanks !!! !!!

Number of Articles using DMFTNumber of Articles using DMFT• Citation Report Topic=(Dynamical Mean Field Theory)• Refined by: Subject Areas=( PHYSICS, CONDENSED MATTER )

,.,. Theory K. Haule and GK. PRB (2007) Theory K. Haule and GK. PRB (2007)McElroy et. al.McElroy et. al.

PRL 94, 197005 (2005)PRL 94, 197005 (2005)

Exp:Bi2212 with STMExp:Bi2212 with STM

Separating overdoped and Separating overdoped and underdoped region there is an underdoped region there is an

optimal doping where the optimal doping where the scattering rate is maximal.scattering rate is maximal.

Plaquette Plaquette

T=| ↑, ↑ >T=| ↑, ↑ >

S= 1/√2(|0, ↑i + | ↑, 0i)S= 1/√2(|0, ↑i + | ↑, 0i)

E=|0, 0>E=|0, 0>

S=1/√2( | ↑, ↓> -| ↑↓ >)S=1/√2( | ↑, ↓> -| ↑↓ >)

Optimal doping: Coherence Optimal doping: Coherence scale seems to vanishscale seems to vanish

TcTc

underdopedunderdoped

overdopedoverdoped

optimallyoptimally

scattering scattering

at Tcat Tc

Comments Comments

What about broken symmetries and their quantum critical points ?What about broken symmetries and their quantum critical points ?

Charge stripes, spin stripes, staggered currents, uniform currents, bond density waves, Charge stripes, spin stripes, staggered currents, uniform currents, bond density waves, spin nematic order , spin glass, bond glass, have all been observed.spin nematic order , spin glass, bond glass, have all been observed.

Connections with S. Sachdev Connections with S. Sachdev arXiv:0910.0846

What about the electron phonon interaction, disordered introduced by dopants, oxygen What about the electron phonon interaction, disordered introduced by dopants, oxygen orbitals, realistic band structure which are out there ? orbitals, realistic band structure which are out there ?

Isolate the qualitative aspects that follow from the proximity to the Mott transition, Isolate the qualitative aspects that follow from the proximity to the Mott transition, from those that require specific broken symmetries.from those that require specific broken symmetries.

For example: formation of arcs. Competing short range order. For example: formation of arcs. Competing short range order.

Can all be studied with CDMFT, given a well defined hamiltonian. Can all be studied with CDMFT, given a well defined hamiltonian. But first things first, local physics, high temperature regimes less But first things first, local physics, high temperature regimes less

sensitive to details. sensitive to details.

Can be studied with LDA+DMFT, GW+DMFT to understand chemical trends Can be studied with LDA+DMFT, GW+DMFT to understand chemical trends within different families. within different families.

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)

LaOFeAsLaOFeAs

11111111

99

NaNaxx CoO CoO22

YBa2Cu3O7 YBa2Cu3O7

VO2VO2

55

Underdoped vs Overdoped T=0Underdoped vs Overdoped T=0

Civelli PRB (2009)Civelli PRB (2009)Stanescu Kotliar (2005)Stanescu Kotliar (2005)

F. F. Balakirev et. al. arXiv.org:0710.4612F. F. Balakirev et. al. arXiv.org:0710.4612

(2007).(2007).

Finding interesting correlated materialsFinding interesting correlated materials

Serendipity

An aptitude for making desirable discoveries by accidentAn aptitude for making desirable discoveries by accident

The Edisonian approach to innovation is The Edisonian approach to innovation is characterized by trial and error discovery rather characterized by trial and error discovery rather

than a systematic theoretical approach.than a systematic theoretical approach.

In developing the carbon microphone that In developing the carbon microphone that became the basis of telephones of the next became the basis of telephones of the next

hundred years Edison and his co-workers tried hundred years Edison and his co-workers tried hundreds of substances. hundreds of substances.

+ E+ Edisonian approachdisonian approach

66

Optimal superconductivityOptimal superconductivity

_c2_c2

Kinetic energy we can’t measure Kinetic energy we can’t measure but…….but…….

Return to models (Hubbard and t-J)Return to models (Hubbard and t-J)

• Unclear if they are superconducting in the thermodynamical limit.

• Have Mott transition that can be studied in plaquette DMFT. Important lessons that will impact further realistic studies.

• Concentrate on robust states. AF, SC, PM, PI.

• There are of course ordered states [current order, nematic order, stripe order, staggered flux order, etc. etc etc. etc. ]

Key QuestionKey QuestionWithin the Stated Assumptions (i.e. Model

Hamiltonians and Plaquette DMFT)what is the

mechanims for superconductivity.How it is connected to the underlying “normal” state

Valence-bond DMFT: bond-in-a bathValence-bond DMFT: bond-in-a bath

One patch covers the One patch covers the nodalnodal part of part of the BZ. The other covers the the BZ. The other covers the

antinodalantinodal part part

In the even/odd basis:In the even/odd basis:

- The - The even (bonding) orbital 1+2 even (bonding) orbital 1+2 is associated with the is associated with the central (nodal) central (nodal) patchpatch

- The - The odd (antibonding) orbital 1-2 odd (antibonding) orbital 1-2 is associated with is associated with the the border (antinodal) border (antinodal) patchpatch

Two-site Anderson Two-site Anderson impurity modelimpurity model

Bath Bath

11 22

2-patch DCA construction2-patch DCA construction

Valence-bond DMFT: bond-in-a bathValence-bond DMFT: bond-in-a bath

One patch covers the One patch covers the nodalnodal part of part of the BZ. The other covers the the BZ. The other covers the

antinodalantinodal part part

Bath Bath In the even/odd basis:In the even/odd basis:

- The - The even (bonding) orbital 1+2 even (bonding) orbital 1+2 is associated with the is associated with the central (nodal) central (nodal) patchpatch

- The - The odd (antibonding) orbital 1-2 odd (antibonding) orbital 1-2 is associated with is associated with the the border (antinodal) border (antinodal) patchpatch

Two-site Anderson Two-site Anderson impurity modelimpurity model

11 22

2-patch DCA construction2-patch DCA construction

Dimer and PlaquetteDimer and Plaquette. Haule and Kotliar, Civelli et. al., Sordi et. al. . Haule and Kotliar, Civelli et. al., Sordi et. al.

Ferrero et. al.Ferrero et. al.

Stanescu and Stanescu and Phillips Phillips

Early studies of plaquette and link Early studies of plaquette and link DMFTDMFT

Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, Phys. Rev. Lett. Lett. 91,91,017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Lett. 56, 563.56, 563.

Cluster DMFT Cluster DMFT

Alternative construction via cumulantsAlternative construction via cumulants

Previous methods fail near Pu Previous methods fail near Pu

• Spin Density functional theory: Pu , Am, magnetic, large orbital and spin moments.

• Experiments (Lashley et. al. 2005, Heffner et al. (2006)): Pu is non magnetic. No static or fluctuating moments. Susceptibility, specific heat in a field, neutron quasielastic and inelastic scattering, muon spin resonance…

•Paramagnetic LDA underestimates Volume of Paramagnetic LDA underestimates Volume of Pu Pu. .

•Thermodynamic and transport properties similar to Thermodynamic and transport properties similar to strongly correlated materials.strongly correlated materials.

•Plutonium: correlated paramagnetic metal. Plutonium: correlated paramagnetic metal.

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 Low energy sum rule can have T and doping dependence . For and doping dependence . For

nearest neighbor it gives the kinetic nearest neighbor it gives the kinetic energy. Use it to extract changes in energy. Use it to extract changes in

KE in superconducing state KE in superconducing state

, ,H hamiltonian J electric current P polarization

Below energy Below energy

2

2

kk

k

nk

Plutonium Plutonium

Plutonium: an anomalous metal Plutonium: an anomalous metal

Photoemssion (expt. Arko, theory Photoemssion (expt. Arko, theory Shim. al. science) Shim. al. science) Gouder , Havela PRBGouder , Havela PRB

2002, 2003 2002, 2003

Pengcheng et.al., Pengcheng et.al.,

Science Science 284284, (1999), (1999)

YBaYBa22CuCu33OO6.66.6 (Tc=62.7K) (Tc=62.7K)

41meV resonance41meV resonance

local susceptibilitylocal susceptibility

•Resonance at 0.16t~48meVResonance at 0.16t~48meV

•Most pronounced at optimal dopingMost pronounced at optimal doping

•Second peak shifts with doping (at Second peak shifts with doping (at 0.38~120meV opt.d.) and changes below 0.38~120meV opt.d.) and changes below Tc – contribution to condensation energyTc – contribution to condensation energy

Dynamical Mean Field Theory Dynamical Mean Field Theory

• Describes the electron both in the itinerant (wave-like) and localized (particle-like) regimes and everything in between!.

• Simpler reference systems to understand a problem. [e.g. harmonic oscillator]

• Follow different mean field states (phases) Compare free energies properties.

• Non Gaussian reference frame for correlated materials. • Reference frame can be cluster of sites CDMFT

1111

REVIEWS. Lichtenstein and Kastnelson PRB REVIEWS. Lichtenstein and Kastnelson PRB (2000) Stanescu, T. D., and P. Phillips, 2003, (2000) Stanescu, T. D., and P. Phillips, 2003,

Phys. Rev. Lett. Phys. Rev. Lett. 91,91,017002. DCA in 2x2 017002. DCA in 2x2 Jarrell, M., T. Maier, et. al. 2001, Europhys. Jarrell, M., T. Maier, et. al. 2001, Europhys.

Lett. Lett. 56, 563 56, 563 CUMULANTS. REFERENCE.CUMULANTS. REFERENCE.

Strongly Correlated MaterialsStrongly Correlated MaterialsNot well described by either the fully localized picture (well Not well described by either the fully localized picture (well separated atoms) or the the band picture (weakly interacting bloch separated atoms) or the the band picture (weakly interacting bloch waves)..Materials for which the standard model of the solid state waves)..Materials for which the standard model of the solid state

fails. Challenging fails. Challenging non perturbative non perturbative problem. problem.

Correlated Correlated variables variables

Continuous discovery of interesting material and phenomena that did Continuous discovery of interesting material and phenomena that did not fit the standard model of solid state physics. Heavy fermions not fit the standard model of solid state physics. Heavy fermions (early 80’s) , high temperature superconductors (late 80’s), other (early 80’s) , high temperature superconductors (late 80’s), other transition metal oxides (cobaltates, manganites, vanadates….) 90’s transition metal oxides (cobaltates, manganites, vanadates….) 90’s …………

high Tc in FeAs based compounds (2009).high Tc in FeAs based compounds (2009).

Materials discovered by serendipity and theMaterials discovered by serendipity and the

Edisonian approachEdisonian approach

( )R

( ) ( ') ( ) ( ')r r r r

44

Optics: electron doped cupratesOptics: electron doped cuprates

Building the phase diagram and testingBuilding the phase diagram and testing1 site – 2site DMFT 1 site – 2site DMFT

HartreeHartree

FockFock

Hartree Hartree FockFock

2121