Exact double-counting in combining the Dynamical Mean Field Theory and the Density Functional Theory

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by  Kristjan Haule

Exact double-counting in combining the Dynamical Mean Field Theory and the
Density Functional Theory
Kristjan Haule
Department of Physics and Astronomy, Rutgers University, Piscataway, USA
(Dated: January 15, 2015)
We propose a continuum representation of the Dynamical Mean Field Theory, in which we were
able to derive an exact overlap between the Dynamical Mean Field Theory and band structure
methods,suchastheDensityFunctionalTheory. Theimplementationofthisexactdouble-counting
shows improved agreement between theory and experiment in several correlated solids, such as the
transitionmetaloxidesandlanthanides. Previouslyintroducednominaldouble-countingisinmuch
betteragreementwiththeexactdouble-countingthanmostwidelyusedfullylocalizedlimitformula.
5
1 PACSnumbers: 71.27.+a,71.30.+h
0
2
n Understanding the electronic structure of materials tional of the electron Green’s function G, and takes the
a with strong electronic correlations remains one of the formΓ[G]=−Tr((G−01−G−1)G)+Trlog(−G)+ΦVc[G].
J great challenges of modern condensed matter physics. The first part is the material dependent part, in which

4 The first step towards calculating the electronic struc- G−1(rr(cid:48);ω) = [ω + µ + ∇2 − V (r)]δ(r − r(cid:48)), and
0 ext
1
ture of solids has been achieved by obtaining the single- the second two terms are universal functionals of the

particle band dispersion E(k) within the density func- Green’s function G(rτ,r(cid:48)τ(cid:48)) and the Coulomb interac-
]
l tional theory (DFT) in the local density approximation tion V (r−r(cid:48)). In the exact theory, Φ [G] contains all
e c Vc
- (LDA) [1], which takes into account correlation effects skeletonFeynmandiagram,constructedbyGandVc[13].
r
only in a limited extent. In the language of Luttinger Ward functional, different
t
s
To account for the many-body correlation effects be- approximations can then be looked at as different ap-
.
at yondLDA,moresophisticatedmethodshavebeendevel- proximations to the interacting part ΦVc[G].
m oped. Amongthem,oneofthemostsuccessfulschemesis The Density Functional Theory can be derived by ap-
d- tthheedpyronbalmemicaolfmdeesacnr-ifibeinldgtchoerorerlyat(iDonMeFffTec)t[s2]i.nIatpreepriloacdeics pErox[ρim(ra)t],ingwthheereexEact faunndctiEonal ΦarVec[Gth]ebyHEarHtr[eρe(r)a]n+d
xc H xc
n
lattice by a strongly interacting impurity coupled to a the exchange-correlation functionals, respectively. The
o
self-consistent bath [3]. This method was first developed stationarity condition gives the DFT equations, i.e.,
c
[ to solve the Hubbard model, but it was soon realized [4] G−1 − G−1 = (V [ρ] + V [ρ])δ(r − r(cid:48))δ(τ − τ(cid:48)), be-
0 H xc
that it can also be combined with the LDA method, to cause δE [ρ]/δG = δ(r − r(cid:48))δ(τ − τ(cid:48)) δE [ρ]/δρ =
1 xc xc
v give more material specific predictions of correlation ef- δ(r−r(cid:48))δ(τ−τ(cid:48))Vxc[ρ]. Notethatinthislanguage,exact
8 fectsinsolids. TheLDA+DMFTmethodachievedgreat DFT appears as an approximation to the exact Green’s
3 success in the past two decades, as it was successfully function, where the exact self-energy is approximated by
4 applied to numerous correlated solids [5]. The combina- a static and local potential. Note also that the static
3
tion of the two methods, nevertheless lead to a problem approximation is a consequence of truncating the vari-
0
of somewhat ambiguous way of subtracting the part of ableofinterest, namelyreplacingfullG(r,τ,r(cid:48),τ(cid:48))byits
.
1 correlations which are accounted for by both methods. diagonalcomponentsρ(r)=δ(r−r(cid:48))δ(τ−τ(cid:48))G(rτ,r(cid:48)τ(cid:48)).
0
5 The so-called double-counting (DC) term was usu- IntheLuttinger-Wardfunctionallanguage,theDMFT
1 ally approximated by the formula first developed in the appears as an approximation where the Green’s function
: context of LDA+U, and was evaluated by taking the in the Φ functional is replaced by its local counterpart
v
i atomic limit for the Hubbard interaction term [6, 7]. G → Glocal , and the Coulomb repulsion Vc by screened
X Many other similar schemes were proposed recently [8– interaction V → U, namely ΦDMFT = Φ [G ]. [5]
c U local
r 11], but rigorous derivation of this double-counted inter- Note that the DMFT functional has exactly the same
a
action within DMFT and LDA is missing to date. Here form as the exact functional Φ [G], because all the
Vc
we propose a new method of calculating the overlap be- skeleton Feynman diagrams constructed by G and
local
tween DMFT and a band-structure method (either DFT U are summed up by DMFT, while in DFT the func-
or GW), and we explicitly evaluate this DC functional tional E [ρ] is unknown, and further approximation is
xc
within LDA+DMFT applied to well studied transition necessary. Thetruncationofthevariableofinterestfrom
metal oxides such as SrVO3, LaVO3, and most studied G to Glocal leads in DMFT to self-energy, which is also
lanthanide metal, the elemental Cerium. local in space, but it keeps its dynamic nature. Other
To compare different approximations in the same lan- approximationssuchasHartree-FockorGWcanbesim-
guage, it is useful to cast them into the form of the ilarly derived by replacing Φ [G] by some limited set
Vc
Luttinger Ward functional [5, 12, 13], which is a func- of Feynman diagrams, i.e., truncation in space of Feyn-
2
man diagrams, rather than truncation of the variable of overlap between the two methods. The Hartree term
interest. is accounted for exactly in the LDA method, and has
There is some kind of disconnect between the DMFT the form EVHc[ρ] = 21(cid:82) drdr(cid:48)ρ(r)ρ(r(cid:48))Vc(r − r(cid:48)), while
functional ΦDMFT[G ], and the LDA functional in DMFT it takes the following form EH,DMFT =
E [ρ(r)], moUstly becaloucsael the auxiliary systems for the 1(cid:82) drdr(cid:48)(Pˆρ(r))(Pˆρ(r(cid:48)))Vλ(r − r(cid:48)), which can also be
xc 2 c
twomethodsareverydifferent. Theauxiliarysystemfor written as EH,DMFT = EH [Pˆρ], where Pˆρ = δ(r −
Vλ
LDA approximation is the uniform electron gas problem c
r(cid:48))δ(τ−τ(cid:48))G (rτ,r(cid:48)τ(cid:48))=δ(r−r(cid:48))δ(τ−τ(cid:48))PˆG(rτ,r(cid:48)τ(cid:48)),
definedforcontinuum,intheabsenceofcomplexityofthe local
andEH[ρ]istheexactHartreefunctionaldefinedabove.
solid. On the other hand, DMFT is usually associated Vc
TheHartreecontributiontotheDCwithinLDA+DMFT
with the lattice model like Hubbard model, where map-
(or any other band structure method which includes ex-
pingtothelocalproblemreducestotheAndersonimpu-
act Hartree term) is thus EH [Pˆρ] [15]. This DC term
ritymodel,whichdoesnothaveawell-definedcontinuum Vλ
c
thus corresponds to truncating Green’s function G and
representation. The double-counting problem occurs be-
Coulomb interaction V by their local counterparts, i.e.,
cause it is not clear what is the overlap between the two c
G→PˆG and V →Vλ.
methods, i.e., what physical processes are accounted for c c
For approximations, which truncate in the space
in one and what in the other method.
of Feynman diagrams (such as Hartree-Fock or GW
It is useful to represent the DMFT method in the
method), one can obtain the DMFT double-counting by
continuum representation. Such representation is not
applying both the truncation in space of Feynman dia-
unique, but physical intuition can guide the map-
grams as well as the DMFT truncation in variables of
ping. Here we propose to look at the DMFT prob-
interest. For the case of GW method, one can check di-
lem as the approximation, which solves exactly the
agram by diagram that the corresponding DMFT Feyn-
problem defined by some auxiliary Green’s function
man diagram is obtained by replacing G by PˆG and V
G =PˆGandCoulombrepulsionreplacedbyYukawa c
local byVλ ineachdiagram,justlikeitwasdoneaboveforthe
short-range interaction Vλ = e−λ|r−r(cid:48)|. We have in c
c |r−r(cid:48)| Hartree term. More precisely, the GW functional can be
mind some projector Pˆ, which is very local, and trun- written as ΦGW[G]=EH − 1Trlog(1−V G∗G), where
Vc Vc 2 c
cates the Green’s function to a region mostly concen- G∗G=P istheconvolutionoftwoGreen’sfunctions(po-
trated inside the muffin-tin sphere. It can for ex- larization function). The GW+DMFT double-counting
ample be defined by a set of quasi-atomic orbitals is thus EH,DMFT − 1Trlog(1−Vλ(PˆG)∗(PˆG)), which
Glocal(r,r(cid:48)) = (cid:80)L,L(cid:48)(cid:104)r|ΦL(cid:105)(cid:104)ΦL|G|φL(cid:48)(cid:105)(cid:104)φL(cid:48)|r(cid:48)(cid:105) where can be shortly writte2n as ΦGVWλ [PˆGc ].
(cid:104)r|Φ (cid:105) = u (r)Y (r) are spheric harmonics times lo- c
L l L In the case of DFT+DMFT, the expansion in terms
calized radial wave function. Note that this trunca-
of Feynman diagrams is not possible, however, to iden-
tion of the Green’s function G(r,r(cid:48)) to its local coun-
tify the overlap between the two methods, this is not
terpart parallels the truncation of the Green’s function
essential. Clearly, the double-counting in DFT+DMFT
to its diagonal component in theories that choose den- is obtained by the same procedure of replacing G by PˆG
sity as the essential variable, i.e., ρ(r)=G(rτ,r(cid:48)τ(cid:48))δ(r− and V by Vλ in the DFT functional. Since the DFT
r(cid:48))δ(τ −τ(cid:48)). The screening λ in Yukawa interaction Vλ c c
c also truncates the Green’s function to its diagonal com-
has to be large enough such that the interaction be- ponents only (ρ=δ(τ −τ(cid:48))δ(r−r(cid:48))G) the DC is a func-
tween electrons on neighboring sites is negligible. The tionalofthelocalchargeonlyρ =Pˆρ. DCthustakes
local
DMFT can then give an exact Luttinger-Ward func-
the form
tional Φ [PˆG], i.e., containing all local Feynman dia-
Vλ
gramsconcstructedbyPˆGandVλ, definedinthecontin- ΦDDFCT+DMFT =EVHλ[Pˆρ]+EVXλC[Pˆρ].
c c c
uum [14]. The stationarity condition for the Luttinger- In LDA method, the exchange-correlation functional is
WardfunctionalgivestheDMFTequationsG−1−G−01 = obtainedfromtheenergyoftheuniformelectrongas. To
Pˆ (δΦVcλ[Glocal]/δGlocal). obtaintheLDA+DMFTdouble-counting,onethusneeds
Theprecisedeterminationofthescreeningλisbeyond to solve the problem of the electron gas with the density
thescopeofthispaper. However,wenoticethatoncethe that contains only ”local” charge Pˆρ, which interacts by
CoulombinteractionU inDMFTisknown,thescreening the screened Yukawa interaction Vλ. [24]
c
length λ is uniquely determined by U through the ma- Includingtheexactdouble-counting,theLDA+DMFT
trix elements of the Yukawa interaction in DMFT basis. Φ functional is thus
Notice that Hund’s coupling J is not a free parameter
ΦLDA+DMFT[G]=EH[ρ]+EXC[ρ]+Φ [PˆG]−
in this parametrization, but is uniquely determined by λ Vc Vc Vcλ
through Yukawa form of the Coulomb interaction. [24] −EH [Pˆρ]−EXC[Pˆρ], (1)
Vλ Vλ
c c
After the mapping of the DMFT method to the
continuous (r,r(cid:48)) Hilbert space, where DFT exchange- where ΦVλ[PˆG] is the DMFT functional which contains
c
correlation is defined, it is easy to see what is the all Feynman diagrams constructed from PˆG and Vλ.
c
3
This is the central equation of this paper, as it defines Ce-α n V /U
f dc
theLDA+DMFTapproximationincludingtheexactDC. exact 0.997 0.424
The saddle point equations give the LDA+DMFT set of
nominal 1.002 0.500
equations
FLL 1.035 0.533
TABLE I: LDA+DMFT valence and DC potential for α-Ce
δΦ [G ]
G−1−G−1 =Pˆ Vcλ local + (2) atT =200K.ThelocalCoulombrepulsioninCeisU =6eV.
0 δG
local
(cid:32)δEHXC[ρ] δEHXC[ρ ](cid:33)
Vcλ −Pˆ Vcλ local δ(r−r(cid:48))δ(τ −τ(cid:48))
δρ δρ
local where
λ(α +α λ)
log(1+a )= 0 1 (4)
where we used EHXC[ρ] ≡ EH[ρ]+EXC[ρ] and PˆG ≡ 1 1+α λ2+α3λ4+α λ6
2 4
Glocal. log(1+a )= λ2(β0+β1λ) (5)
The only difference between functional Eq. 1, and the 2 1+β2λ2+β3λ4
usual LDA+DMFT implementation, is the presence of λ3(γ +γ λ)
EVHcλXC. Thisisthesemi-localexchangeandLDAcorrela- log(1+a3)= 1+0γ2λ12 (6)
tionfunctionaloftheelectrongasinteractingbyYukawa log(1+a )=λ4(δ +δ λ2) (7)
interaction. The semi-local exchange-density εx [ρ] (de- 4 0 1
Vλ
fined by Ex[ρ]=(cid:82) drρ(r)εx[ρ(r)]), can be compcuted an- The best fit gives the following coefficients:
alytically, and takes the following form
α =[1.2238912,7.3648662,9.6044695,
i
−0.7501634,0.0207808]×10−1
C
εxVcλ[ρ]=−rsf(x) βi =[5.839362,11.969474,10.156124,1.594125]×10−2
γ =[8.27519,5.57133,17.25079]×10−3
i
where δi =[5.29134419,0.0449628225]×10−4 (8)
Finally,thecorrelationpotentialisVc = Vλc=0 + εcλ=0 ,
1 4arctan(2x) (12x2+1)log(1+4x2) λ A(rs,λ) C(rs,λ)
f(x)=1− − + , where A(r ,λ) = 1+(cid:80)4 a rn and C(r ,λ) = 3[1+
6x2 3x 24x4 s n=1 n s s
(cid:80)4 a rn]2/(cid:80)4 n a rn. We take the unscreened
n=1 n s n=1 n s
correlation energy density εc (and unscreened po-
C = 3(cid:0) 9 (cid:1)1/3, r = (cid:16) 3 (cid:17)1/3, and x = (cid:0)9π(cid:1)1/3 1 . tential) from the standard paλr=am0 etrization of quantum
2 4π2 s 4πρ 4 λrs Monte Carlo results, hence the G W calculation is only
The exchange potential Vx = δ Ex[ρ] is then Vx = 0 0
δρ Vλ usedforrenormalizationofcorrelationsbyscreeningwith
c
4εx + 1 Cxdf. Yukawa form.
3 Vcλ 3rs dx
In the following we present results for some of the
The correlation part requires solution of the homoge-
most often studied correlated solids, namely, elemen-
neous electron gas problem interacting with Yukawa re-
tal Cerium, SrVO and LaVO . We used three differ-
pulsion, which was solved by QMC [16–18]. Here we 3 3
ent forms of DC functional: i) ”exact”, which we intro-
want to have an analytic expression for correlation en-
duced above, ii) ”FLL” stands for fully localized limit
ergy at arbitrary λ and r . It is well established that
s
form introduced in Ref. 6, which has the simple form
G W gives quite accurate correlation energy of the
0 0
V =U(n−1/2)−J/2(n−1), and n stands for the cor-
electron gas [19, 20], especially when computed from dc
relatedoccupancy, c)andthe”nominal”DC,introduced
the Luttinger-Ward functional Γ[G]. We thus repeated
in Ref. 9, 10. The ”nominal” V takes the same form as
G W calculation for the electron gas, but here we use dc
0 0
”FLL” formula, but n in the formula is replaced by the
Yukawa interaction. We evaluate the total energy using
closestintegervalue(n0 =[n]),andhencen0corresponds
Luttinger-Ward functional of GW to achieve high accu-
to so-called nominal valence. We use LDA+DMFT im-
racy. We then fit the correlation energy in the range
plementation of Ref. 9.
of physically most relevant r ∈ [0,10] and screenings
s
The physical properties of correlated materials are
λ∈[0,3](λismeasuredinBohrradiusinverse)withthe
very sensitive to the value of the local occupancy n ,
following functional form: f
and n is sensitive to the value of DC. In table I we
f
show results for elemental Cerium in the α phase. All
εc = εcλ=0 (3) three DC functionals give very similar correlated occu-
Vcλ 1+(cid:80)4n=1anrsn pancies nf, and all are very close to nominal valence
4
SrVO3 n n n Vt2g/U Veg/U LaVO3(t2g-only) n Va1g/U Veg(cid:48)/U
t2g+eg t2g eg dc dc t2g dc dc
exact 2.223 1.507 0.716 1.384 1.406 exact 2.014 1.195 1.193
nominal 2.251 1.541 0.710 1.443 1.444 nominal 2.074 1.450 1.450
FLL 2.529 1.699 0.830 1.943 1.943 FLL 2.099 1.544 1.544
TABLEII:LDA+DMFTresultsforSrVO atT =200Kand TABLE III: LDA+DMFT results for LaVO at T = 200K
3 3
U =10eV. Both t2g and eg orbitals are treated by DMFT. and U =10eV. Only t2g orbitals are treated by DMFT.
5 LaVO3(t2g+eg) n n n Va1g/U Veg(cid:48)/U Veg
experiment t2g+eg t2g eg dc dc dc
V] 4 FnLoLminal exact 2.444 2.048 0.397 1.596 1.599 1.665
1/e3 exact nominal 2.344 2.032 0.312 1.458 1.458 1.458
OS [2 FLL 2.706 2.167 0.540 2.114 2.114 2.114
D
1
TABLE IV: LDA+DMFT results for LaVO at T = 200K
3
08 6 4 2 0 2 4 and U = 10eV. Both t2g and eg orbitals are treated by
Energy [eV]
DMFT.
FIG. 1: (Color online) LDA+DMFT total density of states
for SrVO using three different DC potentials. Experimental
3
photoemissionisreproducedfromRef.22. (parameterslisted case is presented in Table II and spectra in Fig. 1. One
in table II).
can notice that the exact and the nominal DC give very
similar n , while the FLL formula gives 14% larger n .
d d
This is because the value of the DC potential is substan-
n0 =1. The actual value of the DC potential V differs
dc tiallylarger(≈40%)whenusingFLLascomparedtoex-
for less than 0.1U, which leads to almost indistinguish-
actcase. Itisneverthelesscomfortingtoseethat40%er-
able spectra on the real axis, and from the previously
rorindouble-countingstilldoesnotleadstomajorfailure
published results [9], hence we do not reproduce them
of LDA+DMFT. We plot the spectra in Fig. 1, to show
here. We found a general trend in all materials stud-
how this change in V leads to shift of oxygen-p spec-
dc
ied that the exact DC is somewhat smaller then given
tra relative to vanadium-d states. For the exact DC, the
by FLL formula. For Ce, the Hartree contribution to
oxygen peak positions match very well with the experi-
DC potential is V = n U ≈ 0.997U, the semi-local
H f mentallymeasuredspectra. Thenominalvalenceisquite
exchange contribution is V ≈−0.485U and LDA corre-
x closetotheexactspectra,whileFLLformulaleadstoan
lation is V ≈ −0.088U, hence the total DC potential is
c upward shift of oxygen for roughly 0.6eV, which is still
V +V +V ≈0.424U,whichisslightlysmallerthanFLL
H x c relativelysmallcomparedtothedifferenceinthedouble-
formula U(n −1/2)−J/2(n −1)≈0.533U or nominal
f f counting potentials, which is VFLL−Vexact ≈5.37eV.
formulaU(n0−1/2)−J/2(n0−1)=0.5U. Itisinterest- dc dc
f f
ing to note that the semi-local exchange used in LDA is
quitedifferentfromtheexactexchangevalue. Thelatter (a) experiment
is only |V | = Un/14 ≈ 0.071U, a substantially smaller eV]10 t2g-only FLL
F 1/ nominal
value then the semi-local exchange |Vx| ≈ 0.485U. This OS [ 5 exact
shows why DC within LDA+DMFT is so different from D
the Hartree-Fock value of the DMFT self-energy, i.e., 0
8 6 4 2 0 2 4
(b)
Σ(ω =∞). V]
e10 t2g+eg
traNnesxittiownemperteaslenotxitdeestwsiftohrnSormVOin3a,llywhsiinchgleisealecmtreotanlliinc OS [1/ 5
the t2g shell. Near the Fermi level E , there are mostly D
F
t2g states. Themajorityofeg statesareaboveEF,how- 0 8 6 4 2 0 2 4
Energy [eV]
ever, due to strong hybridization with oxygen some part
of eg orbitals also gets filled. There are two ways the
FIG. 2: (Color online) LDA+DMFT total density of states
DMFT method can be used here. In the first case, one
for LaVO using the three different DC formulas. (a) only
3
can treat only the t2g shell within DMFT. The vast ma- t2g orbitals are treated by DMFT (b) both t2g and eg or-
jorityofDMFTcalculationsforSrVO3weredoneinthis bitals are treated dynamically. Experimental photoemission
way. Inthiscase, allthreeDCpotentialsagaingivevery is reproduced from Ref. 23.
similarresultsandthespectraisalmostindistinguishable
frompreviouslypublishedresultsinRef.10. Onecanalso Next we present results for the Mott insulating ox-
treat dynamically with DMFT the entire d shell. This ide LaVO , which is solved in two ways, i) treating only
3
5
the t2g orbitals dynamically with DMFT, presented in [2] A. Georges, and G. Kotliar, Phys. Rev. B 45, 6479
Table III and Fig. 2a, and ii) treating both t2g and eg (1992).
with DMFT. In the first case, the valences are similar in [3] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg,
Rev. Mod. Phys.68, 13 (1996).
allthreedouble-countingformulas. Thet2g occupancyis
[4] Anisimov, Poteryaev, Korotin, Anokhin, Kotliar J.
veryclosetonominalvalue2. Theexactdouble-counting
Phys.: Condens. Matter 9, 7359 (1997); Lichtenstein,
is again smaller than given by FLL or nominal formula,
Katsnelson Phys. Rev. B 57, 6884 (1998).
whichleadstoaslightlylargersplittingbetweenoxygen- [5] For a review see: G. Kotliar, S. Y. Savrasov, K. Haule,
pandV-dstates,i.e.,slightupwardshiftofoxygenstates V. S. Oudovenko, O. Parcollet, and C. A. Marianetti,
in Fig. 2a. In case ii) displayed in Fig. 2b and tabulated Rev. Mod. Phys. 78, 865 (2006).
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14211 (1994).
byDMFT,theFLLformuladramaticallyfails,asitover-
[7] V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein,
estimates the valence, i.e., nFLL−nexact ≈ 0.26. While
d d J. Phys.: Condens. Matter 9, 767 (1997).
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[8] M. Karolak, G. Ulm, T. Wehling, V. Mazurenko, A.
underestimated by FLL formula. The nominal valence, Poteryaev, A. Lichtenstein, Journal of Electron Spec-
however, gives very similar results as the exact DC. This troscopy and Related Phenomena 181 11 (2010).
improvement of nominal DC as compared to FLL was [9] K. Haule, C.-H. Yee, and K. Kim, Phys. Rev. B 81,
pointed our in Refs. 9, 10, and was found to hold not 195107 (2010).
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075136 (2014).
The t2g occupancy n in the nominal and exact DC is
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very close to nominal value of 2, equal to the scheme i)
B 89, 245133 (2014).
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This work was supported Simons foundation under
85, 17007 (2009).
project ”Many Electron Problem”, and by nsf-dmr
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