Second CMSN Coordination MeetingPredicting Superconductivity from First

Principles.

Gabriel Kotliar Physics Department and Center for Materials

Theory Rutgers University

Dallas Texas March (2011)

Outline

• CMSN overview• The DMFT strategy• How it works for models• Why it will work on pnictides.

Scientific Challenge

• Predict Non Phonon Mediated Superconducting Tc’s starting from first principles.

• Signature Problem iron pnictides, CaFe2As2

A. Kreyssig et.al, arXiv: 0807.3032

CaFe2As2 under pressure

IOWA

CMSN network for correlated materialsCollaborative Project. Shared Posdocs/Students

RUTGERS

UC DAVIS ARIZONA

DOEBES

Outline

• CMSN overview• The DMFT strategy• How it works for models• Why it will work on pnictides.

DMFT• Designed to treat strongly correlated electron materials [ for

example Mott transition problem] but treats well many other situations……MIGDAL-ELIASHBERG THEORY was the first (albeit aproximate)

Dynamical Mean Field Theoryl

• Designed to compute one electron spectral functions, photoemission and BIS

• Designed to treat finite electronic temperature• Can in principle treat superconductivity and other orders• Combines ideas of physics (bands ) and chemistry (local CI)• It is a relatively new method. Still rapidly developing.

Functional formulation of realistic DMFT PT in W and G [Chitra and GKotliar].

1 1 1 10

1 1[ , ] [ ] [ ] [ , ]

2 2 C hartreeG W TrLnG Tr G G G TrLnW Tr V W W E G W

Introduce projector Gloc Wloc

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0 1 2

Sigma s GW

Sigma p GW

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0 1 2

D Sigma s

D Sigma p

GW self energy for Si Beyond GW

Coordination Sphere Coordination Sphere

GW+DMFT Why it should work ? GW+DMFT proposed and fully implmented in the context of a a one orbital lattice model.P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). Test various levels of self consistency in Gnonloc Pinonloc P.Sun and GK PRL (2004). S. Savrasov and GK [ PRB 2003] Biermann, F.Aryasetiawan. and A. Georges, PRL 90, 86402 (2003) Test notion of locality in LMTO basis set in various materials. N. Zeyn S. Savrasov and G. Kotliar PRL 96, 226403, (2006). Include higher order graphs, first implementation of GW+DMFT (with a perturbative impurity solver).

GW and DMFT

.

• P. Sun and G. Kotliar, PRB 66, 85120 (2002)

• S. Biermann, F.Aryasetiawan. and A. Georges, PRL 90, 86402 (2003)[ Nickel !]

• S. Savrasov and GK New Theoretical Approaches to Strongly Correlated Systems, A.M. Tsvelik Ed., Kluwer Academic Publishers 259-301, (2001) arXiv:cond-mat/0208241

)()( 0000 DMFT

)()( 00 GWRR

R neq 0

, ,

,

[ ] [ ]( )

[ ] [ ]abcd

0 0

0 Uloc spd sps loc spd f

locloc f spd loc ff

W WW i

W Ww

æ ö÷ç ÷=ç ÷ç ÷çè ø

é ùê ú®ê úë û

Recent Work on determining U, and Edc using SC GW.

12

LDA+DMFT as an approximation to the general GW+DMFT scheme

, ,

dmft ,

0 0 [ ] [ ]

0 [ ] [ ]spd sps spd f

ff f spd ff

Vxc k Vxc k

Edc Vxc k Vxc k

æ ö æ ö÷ ÷ç ç÷ ÷S +ç ç÷ ÷ç ç÷ ÷ç çS -è ø è ø®

Recent calculations using B3LYP hybrid + DMFT for Ce2O3. D. Jacob K. Haule and GK EPL 84, 57009 (2008)

Various implementations over the years, more precise basis sets, better projectors, better impurity solvers.

U is parametrized in terms of Slater integrals F0 F2 F4 ….

DMFT Phonons in fcc d-Pu

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

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

Outline

• CMSN overview• The DMFT strategy• How it works for models• Why it will work on pnictides.

Hubbard model : plaquette in a medium.

Lichtenstein and Kastnelson PRB (2000) 16

Link DMFT. Normal state Real Space Picture. Ferrero et. al. (2010) (similar to plaquette Haule and GK) (2006)

• Momentum Space Picture: High T

Singlet formation. S (singlet),T (triplet) N=2 singlet, tripletE (empty) N=01+ states with 1 electron in + orb

Underdoped region: arcs shrink as T is reduced. Overdoped region FS sharpens as T is reduced. 17

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

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

How is the energy distributed in q and w ?

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

Expts; Dai et.al. 18

Early DMFT predictions

Unconventional SCPhonon Tc<1K

Importance of correlationsMass enhancement 3-5

How strong is local Coulomb repulsion?Calculations for BaFe2As2 with all valence states included in DMFT (not just d-orbitals) self consistent GW computation of U=(Wloc^-1+Pi_loc)^-1

F0=U ~ 5eV

But U does not give rise to correlations or

sizable magnetic moment

J-Hunds ~0.7-0.8eV

Strongly enhances quasiparticle mass,

Ordered magnetic momentvery sensitive to J-Huds:

Yin et. al. arXiv: 1007.2867

A. Kutepov K. Haule S. Savrasov and G. Kotliar PRB 82, 045105 (2010)

Correlation phase diagram and ordered moment of Hunds metals. Yin et al.

17

Zhiping Yin et. al.

Neutron spectroscopy with LDA+DMFT Theory: H. Park , K. Haule and GK

Experiments taken from arXiv:1011.3771

Next Step

• Get a minimal cluster DMFT equations to derive superconductivity.

• Crosscheck total energies with slave boson methods. [ interactions with Yonxing Xin and Jorg Schmalian (Iowa) , Xi Dai (Beijin)]

• Crosscheck the vertex functions with those obtained with linear response [ Savrasov, Xiangan Wan]

View from k space band theory.

BaFe2As2

D. Singh and M. H Du arXiv:0803.0429 )

Cvektovic and Tezanovic arXiv:0804.4678)

X

M

3

Building phase diagram magnetization at T=0 vs d.

Single site

Two site

19

Photoemission

Havela et. al. Phys. Rev. B 68, 085101 (2003)

K.Haule J. Shim and GK Nature 446, 513 (2007) Photoemission in Actinides

alpa->delta volume collapse transition

Curium has large magnetic moment and orders antiferromagnetically Pu does is

non magnetic.

F0=4,F2=6.1

F0=4.5,F2=7.15

F0=4.5,F2=8.11

N Zeyn S. Savrasov and G. K PRL 96, 226403 (2006)

Cutoff Radius R

Challenges

• Optimal choice of projectors.• Basis sets [LMTO, LAPW, plane

waves+PAW’s…..]• Optimal description of the “weakly correlated

sector” [ dft , GW, hybrids ]• Cluster DMFT • Determination of the screened F0, F2, F4

FeSe1-0.08, (Tc=27K @ 1.48GPa), Mizuguchi et.al., arXiv: 0807.4315

Broad Range of Viewpoints

• D. J Singh and M.H. Du Phys. Rev. Lett. 100, 237003 (2008). Itinerant magnetism.

• LDA+Spin Fluctuations. • Haule K, Shim J H and Kotliar G Phys. Rev. Lett.

100, 226402 (2008) Correlated “Bad Semi-Metal” (U< Uc2) Multi-orbital model. Z ~0.2–0.3.

• LDA+DMFT +extensions• Q.Si and E.Abrahams Phys. Rev. Lett. 101, 076401.

(2008). Localized picture,frustration.• t-J model S=3/2 1/2

Importance of Hund’s couplingHubbard U is not the “relevant” parameter.

The Hund’s coupling brings correlations!

Specific heat within LDA+DMFTfor LaO1-0.1F0.1FeAs at U=4eV

LDA value

For J=0 there is negligible mass enhancement at U~W!J~0.35 gives correct order ofMagnitude .r

The coupling between the Fe magnetic moment and the mean-field medium (As-p,neighbors Fe-d) becomes ferromagnetic for large Hund’s coupling!

KHaule, G. Kotliar,

LaO1-0.1F0.1FeAs

Prediction

Problem for us: Experimental evidences for weak correlations. NO SATELITES

• XES: no lower Hubbard band or sharp quasiparticle peak• XAS: XAS and RIXS spectra are each qualitatively similar to

Fe metal• XPS: itinerant character of Fe 3d electrons• V. I. Anisimov, et al, PhysicaC 469, 442 (2009)• W. L. Yang, et al, PRB 80, 014508 (2009) • Soft underbelly of the approach when one approaches very

itinerant systems... Many different estimates of the effective paramaters, U, J, etc in the literature, which leads to very different results.

NOT SEEN

wc=3000cm-1 ~. .3 ev

Nature Physics 5, 647 (2009) M. M. Qazilbash,1,, J. J. Hamlin,1 R. E. Baumbach,1 Lijun Zhang,2 D. J. Singh,2 M. B. Maple,1 and D. N. Basov1

Photoemission reveals now Z ~ .3

Freq. dep. U matrix well parametrized by F0 F2 F4

F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2

Z =1/2 for x2- y2 and z2 , Z =1/3 f xz; yz zx orbitals.

Theory: Kutepove et.al. Expt:. Qazilbash,et.al

DMFT F0 = 4:9 eV, F2 = 6:4 eV and F4 = 4:3 eV., nc=6.2

DOS [kutepov et.al. 2010]

There is transfer of spectral weight to high energies, spectral weight is conserved. But the DOS is featuresless no satellites, and resembles the LDA!Big difference between oxides and pnictides.

Theory: Kutepov et.al. Expt Brouet et.al.

Magnetic moment .95 muB LDA ~ 2 muB, expt 1 muB

Photoemission Spectra

DMFT Valence Histogram Kutepov et. al. (2010)

Completely different than that of a weakly correlated metalCompletely different from that of an oxide!The width is determined by F2 and F4 and the hybridization with As which is spread over many ev’s. All atomic states have weight!But the states are spread over a scale much larger than the bandwidth

• K. Held. Adv. in Physics, 56:829, 2007.

• A.Georges, G. K., W. Krauth and M. J. Rozenberg, Reviews of . Modern Physics 68, 13 (1996).

• G. Kotliar S. Savrasov K. Haule O. Parcollet V.Oudvenko and C.

Marianetti Reviews of Modern Physics 78, 865-951, (2006).

• G. Kotliar and D. Vollhardt Physics Today, Vol 57, 53 (2004).

Some DMFT Reviews

Thanks for your Attention!!

+ [ - ]

KSV10KSG 1G

Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965)

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

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0 0 0

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s st m tt

t t ¯

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t c c c c U n n

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a a

ww e

D =-å

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, , ,

( +c.c). H c A A A c c UcV c c c

A(w)

10

( )wD ®

latt ( ,1

G [ ]( ) [( ) ]

)[ ]n imp

nn

ik ii

ktw m

ww+ + - S

DD

=

latt( ) G ([ [)] ] ,imp n nk

G i i kw wD D=å 8

[ ]ijij

jm mJth hb= +å1

( )[ ]( )

( )[ ]imp n

n n ni i iG iw

w w wD +DD

- S -B

atomic levels

Quantifying the degree oflocalization/delocalization

( ),

( ),

( )[ ]n

imp n

A

G i

i

w

w

wS D

®

Impurity Solver

Machine for summing all local diagrams in PT in U to all orders.

, ,

,

[ ] [ ]( )

[ ] [ ]spd sps spd f

f spd ff

H k H kt k

H k H k

æ ö÷ç ÷ç ÷ç ÷çè ø®

| 0 ,| , | , | | ... JLSJM g> > ¯> ¯> >® Determine energy and and S self consistently from extremizing a functional : the spectral density functional . Chitra and Kotliar (2001) . Savrasov and Kotliar (2001) Full self consistent implementation . Review: Kotliar et.al. RMP (2006)

12

1( , )

( ) ( )G k i

i t k i

Spectra=- Im G(k,w)

LDA+DMFT. V. Anisimov, A. Poteryaev, M. Korotin, A. Anokhin and G. Kotliar, J. Phys. Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsnelson (1998) LDA++

0 0

0 ff Edc

æ ö÷ç ÷S ç ÷ç ÷ç S -è ø®

,[ ] [ , ]dft lda dmf loct G Ur r+G ¾¾®G

abcdU U®

Main steps in DMFT

• 1) Solve for atomic shell in a medium, Gloc Ploc Sloc and Wloc [Impurity Solver]

• 2) Embed Sloc Ploc to obtain the solid greens functions. [Embedding]

• 3) Project the full greens function to get the local greens function of the relevant shell.

[Projection or Truncation]• 4) Recompute the medium in which the atom is

embedded. [ Weiss fields]• Postprocessing: evaluate total energies, A(k,omega)

sigma(omega) ………

ImpuritySolver

General impurity problem

Diagrammatic expansion in terms of hybridization D+Metropolis sampling over the diagrams

•Exact method: samples all diagrams!•Allows correct treatment of multiplets

P. Werner et. al. PRL (2006) K.H.aule Phys. Rev. B 75, 155113 (2007)

An exact impurity solver, continuous time QMC - expansion in terms of hybridization

NCA

OC

A

SU

NC

A

Luttinger Ward functional

every atomic state represented with a unique pseudoparticle

atomic eigenbase - full (atomic) base

, where

general AIM:

Same expansion using diagrammatics – real axis solver

( )

DMFT : the middle way

• More expensive than density functional theory ( because it targets spectral properties)

• Less expensive than direct application of QMC or CI (because it only uses these tools locally )

• Utilizes advances in electronic structure [ DMFT can be built on top of LDA, hybrid-DFT, GW ] and techniques such as QMC or CI, and its various levels of approx to solve the impurity problem.

• Greens function method, based on a judicious use of the local approximation. Solved the Mott transition problem in the context of the model Hamiltonians. Goal, combine those ideas with technology from electronic structure methods to understand and predict properties of correlated materials.

• Testing methods: “simple” models, experiments, predictive power ?