Second CMSN Coordination Meeting
Predicting 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
CaFe2As2 under pressure
A. Kreyssig et.al, arXiv: 0807.3032
CMSN network for correlated materials
Collaborative Project. Shared Posdocs/Students
RUTGERS
IOWA
DOE
BES
ARIZONA
UC DAVIS
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
[G,W ]  TrLnG  Tr[G01  G 1 ]G  TrLnW  Tr[VC1  W 1 ]W  Ehartree  [G,W ]
2
2
Introduce projector Gloc
Wloc
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 self energy for Si
Beyond GW
0.04
0.03
0.008
0.02
0.006
0.01
0.004
Sigma s GW
Sigma p GW
0
-0.01
0.002
D Sigma s
D Sigma p
0
-0.02
-0.002
-0.03
-0.004
-0.04
0
1
2
-0.006
0
Coordination Sphere
1
2
Coordination Sphere
GW and DMFT
DMFT
00 ( )  00
( )
GW
0 R ( )  0 R ( )
R neq 0
.
• 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
LDA+DMFT as an approximation to the general GW+DMFT
scheme
S
®
æ0
ç
ç
ç
ç
è0
ö
0
÷
÷
+
÷
÷
S dmft ff - Edcø
æ[Wloc ]spd ,sps [Wloc ]spd , f
Wloc (iw) = ççç
çè [Wloc ] f ,spd
[Wloc ] ff
æVxc[k ]spd ,sps
ç
ç
ç
ç Vxc[k ] f ,spd
è
é0
ù
0
ö
÷
ê
ú
÷
®
÷
÷
ø
ê0 Uabcd ú
ë
û
Vxc[k ]spd , f
Vxc[k ] ff
U is parametrized in
terms of Slater
integrals F0 F2 F4 ….
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.
Recent Work on determining U, and Edc using SC GW.
12
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)
Singlet formation. S (singlet),T
(triplet) N=2 singlet, triplet
E (empty) N=0
1+ states with 1 electron in + orb
• Momentum Space Picture: High T
Underdoped region: arcs shrink as T is reduced. Overdoped
region FS sharpens as T is reduced.
17
Reminiscent of PW Anderson RVB Science 235, 1196 (1987) and slave
Superexchange
Mechanism?
K. Haule
and
GK Phys.
boson
picture G. Kotliar
and J. Liu P.RB
38,5412
(1988)
.
(2007).
Ex=Rev.
Jij(< BSi.76,
Sj >104509
<
Si
.
Sj>n)/t
s
How is the energy distributed
in q and w ?
D.J. Scalapino and S.R. White, Phys. Rev. B 58,
8222 (1998).
Expts; Dai et.al.
Early DMFT predictions
Unconventional SC
Phonon Tc<1K
Importance of correlations
Mass 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
A. Kutepov K. Haule S. Savrasov and G. Kotliar PRB 82, 045105 (2010)
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 moment
very sensitive to J-Huds:
Yin et. al. arXiv: 1007.2867
Correlation phase diagram and
ordered moment of Hunds metals. Yin
et al.
Zhiping Yin et. al.
17
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.
M
X
BaFe2As2
3
D. Singh and M. H Du arXiv:0803.0429 )
Cvektovic and Tezanovic arXiv:0804.4678)
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
F0=4,F2=6.1
F0=4.5,F2=7.15
F0=4.5,F2=8.11
Curium has large magnetic moment and orders
antiferromagnetically Pu does is non magnetic.
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 coupling
Hubbard U is not the “relevant” parameter.
LaO1-0.1F0.1FeAs
The Hund’s coupling brings correlations!
Specific heat within LDA+DMFT
for LaO1-0.1F0.1FeAs at U=4eV
Prediction
LDA value
For J=0 there is negligible mass enhancement at U~W!
The coupling between the Fe magnetic moment and the mean-field medium
(As-p,neighbors Fe-d) becomes ferromagnetic for large Hund’s coupling!
J~0.35 gives correct order ofMagnitude r.
KHaule, G. Kotliar,
Problem for us: Experimental evidences for
weak correlations. NO SATELITES
NOT SEEN
• 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.
wc=3000cm-1 ~. .3 ev
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
Nature Physics 5, 647 (2009)
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 metal
Completely 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
Some DMFT Reviews
• 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).
Thanks for your
Attention!!
Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation
theory in the screened interactions (Hedin 1965)
G
1
1
0KS
 G
+[
-
VKS
]
-
Dynamical Mean Field Theory. Cavity Construction.
A. Georges and G. Kotliar PRB 45, 6479 (1992).
å
J ij Si S j - h å Si
i, j

 (t
i , j  ,
D (w) =
ij
H MF = - heff So
i
†
H Anderson
† Imp   (V c0 A +c.c). 
 d ij )(c c j  c j ci ), U  ni ni
†
i
 A A  



†
,
c0† c0  Uc0†c0c0† c0
,
i
å
a
*
Va Va
w - ea
A()
b
b
b
¶
c
(
t
)[
+ m- D (t - t ')]cos (t ') + U ò no­ no¯
òò
¶
t
0 0
0
†
os
10
Impurity Solver
D ( w) ®
A( w),
® Gimp (iwn ),
atomic levels
S (iwn )[D ]
Machine for
summing all local
diagrams in PT in
U to all orders.
mi = th[b å J ij m j + h ]
1
iwn - S (iwn )[D ] B - D (iwn ) +
Gimp (iwn )[D ]
j
Quantifying the degree of
localization/delocalization
Glatt (iwn , k )[D ] =
1
[iwn + m+ t (k ) - S imp (iwn )[D ]]
Gimp (iwn )[D ] =
å
k
G latt (iwn , k )[D ]
8
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++
1
G ( k , i ) 
i  t (k )  (i )
Spectra=- Im G(k,)
U
®U
S
abcd
t (k )
®
®
æ0
ö
0
÷
çç
÷
ççè0 S ff - Edc÷
÷
ø
æH [k ]spd ,sps
çç
ççè H [k ] f ,spd
| 0 > ,|­ > ,|¯> ,|­ ¯>
®
Determine energy and and  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)
dft
lda + dmf
loc ,
G [r ] ¾ ¾
®G
t[G
r ,U ]
ö
H [k ]spd , f ÷
÷
÷
H [k ] ff ÷
ø
| LSJM J g... >
12
Main steps in DMFT
• 1) Solve for atomic shell in a medium, Gloc Ploc loc
and Wloc [Impurity Solver]
• 2) Embed loc 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) ………
Impurity
Solver
An exact impurity solver,
continuous time QMC - expansion in terms of hybridization
P. Werner et. al. PRL (2006) K.H.aule Phys. Rev. B 75, 155113 (2007)
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
Same expansion using diagrammatics – real axis solver
every atomic state represented with a unique pseudoparticle
atomic eigenbase - full (atomic) base
, where
Luttinger Ward functional
NCA
OCA
general AIM:
SUNCA
(
)
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 ?