Pis'ma v ZhETF, vol.86, iss. 10, pp.769-774
© 2007 November 25
Cluster Dual Fermion Approach to Nonlocal Correlations
H. Hafermann+, S.Brener+, A.N.Rubtsov, M. I. Katsnelson*, A. I. Lichtenstein+1ï Department of Physics, Moscow State University, 119992 Moscow, Russia + L Institute of Theoretical Physics, University of Hamburg, 20355 Hamburg, Germany * Institute for Molecules and Materials, Radboud University, 6525 ED Nijmegen, The Netherlands
Submitted 13 September 2007 Resubmitted 18 October 2007
We formulate a general cluster Dual Fermion Approach to nonlocal correlations in crystals. The scheme allows the treatment of long-range correlations beyond the cluster DMFT and nonlocal effects in realistic calculations of multiorbital systems. We show that the simplest approximation exactly corresponds to the free-cluster DMFT. We apply this approach to the one-dimensional Hubbard model. Already the first dual-fermion correction to the free cluster leads to a drastic improvement of the calculated Green function.
PACS: 71.10.Fd, 71.15.—m, 71.27.+a
One of the successful routes to the description of strongly correlated systems is the Dynamical Mean Field Theory (DMFT) [1, 2]. It is commonly accepted now that this approach typically catches the most essential correlation effects, e.g., the physics of the Mott-Hubbard transition [1, 2]. The method was implemented successfully into realistic electronic structure calculations [3, 4], which now is a standard tool in the microscopic theory of strongly correlated systems [5]. In the DMFT, the many-body problem for the crystal is split into a single-particle lattice problem and the many-body problem for an atom in a self-consistently determined Gaussian fermionic bath. The self-energy in the DMFT approach is local in space but frequency dependent. However, there are many phenomena for which non-local correlations are important and often the relevant correlations are even long-ranged. The examples are Luttinger-Liquid formation in low-dimensional systems [6, 7], non-Fermi-Liquid behavior due to van-Hove singularities in two dimensions [8], the physics near quantum critical points [9] or d-wave pairing in high-Tc superconductors [10]. Obviously, the DMFT is not sufficient for the description of such systems. To treat these nonlocal correlations it is desirable to combine local intersite many-body phenomena, like the formation of RVB singlets [7], and long-range correlations. The former can be taken into account within various cluster approaches. They include the so-called Dynamical Cluster approximation (DCA) [11], real space periodic [12] and free cluster approaches [13], as well as the Cellular-DMFT [14](CDMFT) and the variational cluster approach [15].
e-mail: alichtenôphysnet.uiii-hamburg.de
Recently, steps have been taken to go beyond DMFT and to treat long-range correlations. One of them is the Dynamical Vertex approximation [16] and similar approaches [17, 18], where a diagrammatic expansion around the DMFT solution is made. A principally new scheme with a fully renormalized expansion called Dual Fermion Approach has been proposed[19]. It is based on the introduction of new variables in the path integral representation. This approach yields very satisfactory results already for the lowest-order corrections, while the schemes proposed in Refs. [16-18] operate with infinite diagrammatic series and require the solution of complicated integral equations. A scheme similar to the Dual Fermion approach has been discussed earlier in terms of Hubbard operators [20], but without attempts to use it in a practical calculation.
In this letter we formulate a general cluster (or multiorbital) Dual Fermion scheme for non-local correlations. Similar to known cluster methods we consider a system with local interaction and assume that most of the correlations are located within the cluster. We point out however that the remaining long-range part of the correlations is physically important, and take it into account within a diagrammatic expansion of a special kind. By transforming the original interacting problem to so-called dual fermion variables we are able to include the local contribution to the self-energy into a bare propagator of the dual fermions and achieve much faster convergence of the perturbation expansion. An outcome of the scheme is the Green function of the original variables restored from a certain exact relation. Our method allows the treatment of clusters or multiorbital atoms within the Dual Fermion framework and can describe long-range
9 IlHCbMa b ?K3T<J> tom 86 Bbin.9-10 2007 769
correlations in real systems. We test the scheme for the half-filled one-dimensional Hubbard chain and demonstrate its superiority over short-range cluster methods.
Our goal is to find an (approximate) solution to a general multiband problem described by the imaginary time action
5[с*,с]=- ((гш + м)1 - hkiT)mm, сшк1Ттi +
cukcrmm'
+ ^Гнш[с*,а]. (l)
i
Here hbv is the one-electron part of the Hamiltonian, ш = (2n + l)ir/(3,n = 0, ±1,... are the Matsubara frequencies, ¡3 and fi are the inverse temperature and chemical potential, respectively, a =f, 4- labels the spin projection, то,to' are orbital indices and c*,c are Grass-mannian variables. The index i labels the lattice sites and the k-vectors are quasimomenta. It is important to note that can be any type of interaction inside the multiorbital Hint atom. The only requirement and our main assumption is that it is local:
Hint[c*i,ci] = У drC/1234C*1C*2Ci4Cj3 , (2) « 0
where U is the general symmetrized Coulomb vertex and e.g. 1 = {wiTOicrj} comprehends frequency-, orbitaland spin degrees of freedom and summation over these states is implied.
The formalism is equally applied within the cluster (super-site) formalism. In this case, i and то label clusters and atoms within the cluster, repectively, while к runs over the reduced supercell Brillouin zone. In order to capture the local physics, we introduce a cluster impurity problem just in the spirit of CDMFT[12, 14, 13] in the form
^imp[c , с] = — 'У ^ Cujcrm ((^ — ^wcr)mmi Сшсгт.' +
wcr
+ Hint[c*,c], (3)
where Д is an as yet unspecified hybridization matrix describing the interaction of the impurity cluster with an electronic bath. We suppose that all properties of the impurity problem, i.e. the single-particle Green function gaw and the irreducible vertices 7W, 7^, etc. are known. Our goal is to express the Green function Gwk and vertices 7 of the original lattice problem via these quantities.
Since Д is local, one may formally rewrite the original lattice problem in the following form:
S[c*,c] — / ^ 5jmp[c* i(T, Cuicr] i
~ Cwk(rm (A¡ü<T — ^ktj)mm* Coiko-ro' •
wk crmm'
(4)
We introduce spinors сшк(Т = (... ,сшксTm, ■■■), с*к<т = = (... , с*к(ТГО,...). Omitting indices, the Gaussian identity that facilitates the transformation to the dual variables in matrix-vector notation is
J exp (-f*Af - Г Be - с*ж) V[f, Г] = = det(A) exp (c*BA-1Bcj , (5)
which is valid for arbitrary complex matrices A and В. In order to decouple the non-local term in Eq. (4), we choose
A = 9йа - h^y1 g~l , В = , (6)
where gwa is the Green function matrix of the local impurity problem in orbital space (то, то'). Using this identity, the lattice action can be rewritten in the form
S[c*,c,f*,f] = ^Ssite,» +
i
+ Y1 [Скст - h^y1 g~l w] , (7)
u'k'T
where
^ ^ S$ite,¿ = ^ ^ 5¡mp[c¿ , C¿] + i i
+ ClV 9wlcui<r + cwia 9wl^uicr- * (8)
Here the summation in the last term over states labeled by к has been replaced by the equivalent summation over all sites. The Gaussian identity can further be used to establish an exact relation between the lattice Green function and the dual Green function. To this end, the partition function of the lattice is written in the two equivalent forms
Z= yexp(-S[c*,c])2>[c,c*] =
= Zfj J exp (—S[c*,c,f*,f])D[f, f*]D[c, c*], (9)
where
Zf =]J det [g Ш<Т (A LiJCT - hue) 9w<j] ■ (10)
u'k'T
Письма в ЖЭТФ том 86 вып. 9-10 2007
By taking the functional derivative of the partition function, Eq. (9), one can obtain the following exact relationship between the dual and lattice Green functions:
Gwko- = (AW<T - hkv) 1 + + (9u,<r (A UJ<T hua)) Gt k<r ((A UJ<T -h^a)9wa) , (11)
where the lattice Green function is defined via the imaginary time path integral as
G12 —
~ J dc^expi-Sic'^Vic,^] (12)
and similarly for the local Green function g and dual Green function Gd with Z and S replaced by the corresponding expressions.
We now wish to derive an action depending on the dual variables only. This can be achieved by integrating out the original variables c,c*. The crucial point is that this can be done for each site separately:
J exp ( —Sgite [c*, Cj, f* , fj) X>[cj, c*] =
- 7. g-(S»» Cl. t-i.+ViIl?,*])
(13)
This equation can be seen as the defining equation for the dual potential V[f*,f]. Since Ssjte contains the impurity action, expanding the remaining part of the exponential and integrating out the original variables corresponds to averaging over the impurity degrees of freedom. Equating the resulting expressions by order, one finds that the dual potential in the lowest order approximation is given
by
F[f*,f] = J 5>m4«?l«?2fi4fi3
(14)
where
(4) _ -1-1 [ imp inip,0 ] _1 _
'1234 — 9ll'922' [ai'2'3'4' A1'2'3'4'J 9z'z9v
imp.O
Xl234 - 514523 - 513524
(15)
is the fully antisymmetric irreducible vertex. The local two-particle Green function of the impurity model is defined as
imp _
A1234 —
— f
^imp J
c1c2c3c4 exp (-Sîmp[c*, c]) V[c, c*]
(16)
The dual action now depends on dual variables only and can be written as
Sd[f*, f] = - £ Çk(T(G^(J + E m;, :fi] •
(17)
u'k'T
The bare dual Green function is given by
= —5W<T [5010- + (Aw<7 - ftkcr) 1j 9wo
and the dual self energy reads
=(gajL)-1-(G:
Ju;k er
d X ct?k(T/
(18)
(19)
Let us introduce non-local part £ as the difference between the self-energy £ of the lattice problem and it's DMFT value, i.e. E //
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