arX

iv:1

105.

6158

v1 [

cond

-mat

.str

-el]

31

May

201

1

Dual boson approach to collective excitations in correlated fermionic systems

A. N. Rubtsov∗

Department of Physics, Moscow State University, 119992 Moscow, Russia

M. I. KatsnelsonRadboud University Nijmegen, Institute for Molecules and Materials, 6525AJ Nijmegen, The Netherlands

A. I. LichtensteinI. Institute of theoretical Physics, University of Hamburg, 20355 Hamburg, Germany

(Dated: June 1, 2011)

Based on path-integral formalism we develop a general theory of a boson decomposition for non-local interactions in lattice fermion models which allows us to describe the fermionic degrees offreedom and collective charge and spin excitations on equal footing. The efficient perturbation the-ory in the interaction of fermionic and bosonic degrees of freedom which is valid for both weak andstrong coupling limits and interpolates between them is constructed. Zero-order approximation ofthis theory corresponds to the extended dynamical mean-field theory (EDMFT), a regular way tocalculate nonlocal corrections to the EDMFT being provided. The method is especially suitable forconsideration of collective magnetic and charge excitations and allows to calculate their renormal-ization with respect to “bare” RPA-like characteristics. As an illustration it is shown that effectivesuperexchange interactions in the half-filled Hubbard model can be derived within the dual-ladderapproximation.

PACS numbers: 71.27.+a,71.45.Gm,74.20.Mn

I. INTRODUCTION

A self-consistent description of electrons, magnons, phonons and other fermionic and bosonic degrees of freedomin solids is still a challenging problem for strongly correlated systems where simplest approximations are usually notsufficient. One of the classical examples is the theory of itinerant electron magnetism, where the same d-electrons formsthe correlated energy bands as well as quasi-localized magnetic moments and magnon spectrum1. In this problemthe conventional random phase approximation (RPA) gives a wrong description of finite temperature effects whereasthe proper corrections to RPA can be successfully calculated only in a very narrow vicinity of the Stoner instabilitypoint1,2. The main difficulty is the coexistence of localized (atomic-like) and itinerant features in the behavior ofd-electrons in solids3–5. Recent progress in the theory of correlated electron spectrum is related with developmentof the Dynamical Mean Field Theory (DMFT)6, which maps the problem of fermions with local interactions on acrystal lattice onto self-consistent numerically exact solution of effective impurity problem. The latter can be solvedvery accurately within the Continuous Time Quantum Monte Carlo (CT-QMC) scheme7 since there is no serious signproblem even in a multiorbital case. As a result, both atomic-like features such as multiplet formation and banddispersion can be taken into account simultaneously8,9. Combination of the DMFT approach with realistic electronicstructure calculations10,11, in particular, for transition metal systems5,12 opens a new way for description of electronicdegrees of freedom is solids. A generalization of the DMFT approach to the non-local interactions and consistentdescription of collective bosonic excitations is just in the period of intensive discussions10,13–15 and first attempts ofapplications to realistic systems within a so-called GW+DMFT approach16.We construct here a theory, that is capable to describe systems with strongly correlated electrons interacting with

collective bosonic modes. These modes can appear explicitly in an original Hamiltonian (for instance, phonons) orarise from electron-electron interactions (like plasmons or magnons). We assume that the Bose excitations cannotbe considered as an ideal Bose gas and boson-boson correlations are important. This is the case of weak itinerantferromagnets2 and, more generally, any systems near quantum phase transition points17.A simultaneous description of fermionic and bosonic correlations on the equal-footing was started within the Ex-

tended Dynamical Mean Feald (EDMFT) framework13,14. The strong-coupling generalization of the EDMFT18 givesa useful extension of the Hubbard-I like perturbation expansion of lattice models19. The EDMFT properly includesthe physics of local fermionic and bosonic correlations. We argue, however, that formation of the collective modesshould be related to essentially non-local electron correlations, which can be described, at least, by ladder series orby the Random Phase Approximation (RPA) in a weak-coupling regime20–22. Even this case, quite elementary fromthe point of view of general quantum many-body physics, is difficult for the EDMFT, because of uncontrollable localapproximation.In the present paper we construct a diagrammatic series on the top of the EDMFT, which allows to take into account

2

the non-local physics of a strongly correlated system with different collective modes in a regular way. We utilize adual-transform approach proposed for fermionic ensembles23 which is, actually, a smart replacement of variables inthe path integral representation for the Green functions. On the other hand, it is important to note that presenttheory is not a direct extension of dual-fermion approach23. We will see that, even for the Hubbard-like models withon-cite interactions only, it gives a different result, because of an account of collective modes (magnons).

II. DEFINITIONS

We consider a lattice fermionic problem with the intersite hopping and generic non-local interactions. For simplicity,we present here formulae for a single-band periodic lattice, but a generalization of all expressions to a multiorbitalcase is straightforward. The action is

S =∑

r

Sat[c†r, cr] +

∑

r,R 6=0,ω,σ

εRc†rωσcr+Rωσ +

∑

r,R 6=0,Ω,j,j′

V jj′

RΩρ∗jrΩρj′r+RΩ (1)

Hereafter c†, c are Grassmann variables, the indices r, R numbering lattice sites and translations, k stays for wave-vector, τ is imaginary time, ω = 2π(n + 1/2)T and Ω = 2πnT are respectively fermionic and bosonic Matsubarafrequencies (n is integer), and σ is a spin index. Prefactors β−1 and N−1 are impied for sums over frequenciesand wavevectors, respectively (β is the inverse temperature and N is the number of lattice sites). The quantity

ρ denotes spin-projection or density operator: ρj ≡∑

σσ′ c†σsjσσ′cσ′ − ρ0, where sj with j = 0, x, y, z are Pauli

matrices (s0 = 1), and the quantity ρ0 is introduced to satisfy the condition < ρ >= 0. For the Coulomb interactionor attraction mediated by phonons, only the density-density part V 00

RΩ is important; isotropic exchange interactioncorresponds to V xx = V yy = V zz , etc. We also keep in mind an important case of the Hubbard model, V = 0.Within the path integral representation an atomic action Sat is given by:

Sat = −∑

ωσ

(iω + µ)c†ωσcωσ + U

∫ β

0

c†↑c↑c†↓c↓dτ. (2)

We will be interested in a single-electron Green’s function Grτ = − < crτc†r=0,τ=0 > and a bosonic correlator

Xrτ = − < ρrτρ∗r=0,τ=0 >. It is an important point that an account of the bosonic quantity is not technical: we

suppose that collective modes are relevant for the physics of the system.

III. LOCAL APPROXIMATIONS

The general idea of effective-medium approximations is to reduce the lattice problem (1) to so-called impurity model,describing a single atom (or small cluster) in a Gaussian surrounding. For a system with bosonic and fermionic degreesof freedom, a single-site impurity problem is described by the action

Simp = Sat +∑

ω

∆ωc†ωcω +

∑

Ω

ΛΩρ∗ΩρΩ. (3)

The impurity problem is supposed to be much simpler for a numerical or analytical solution. We later suppose

that its Green’s functions: gτ = − < cτc†τ=0 >imp and χτ = − < ρτρ

∗τ=0 >imp are known, as well as higher-order

correlators. In practice, they can be calculated by the CT-QMC scheme7 or by exact diagonalization if the dimensionof the Hilbert space is not too large.For so-called local approximations, only the Green’s functions of the impurity model gω, χΩ are taken into account.

In a nutshell, the initial problem (1) is replaced by an auxiliary lattice with Gaussian nodes; those nodes are charac-terized by Green’s functions gω, χΩ. Such a consideration gives well-known expressions for lattice Green’s functions14,whose are denoted with calligraphic letters in the paper:

Gωk =1

g−1ω +∆ω − ǫk

(4)

XΩk =1

χ−1Ω + ΛΩ − Vk

where k is the wave vector and Fourier transforms of hopping and interaction parameters are introduced in the obviousway. One can see that the fermionic and bosonic self-energies of these expressions are k-independent, that is, local inspace.

3

The hybridization functions ∆ and Λ should be defined in such a way, that the impurity problem provides us theon-site (local) properties of the solution for lattice problem. Such a requirement is quite physically reasonable. Also,it is closely related with the extremal properties of the Feynman functional in the dual fermion approach24.Most common are so-called static and dynamic mean-field theories. In the static approach, the hybridization

functions ∆ and Λ are chosen to be frequency independent. Their values are defined from the equalities for the staticparts of Green’s functions:

∑

ω

gMFω =

∑

ωk

GMFωk , (5)

∑

Ω

χMFΩ =

∑

Ωk

XMFΩk

In the dynamical EDMFT approach ∆ω and ΛΩ are adjusted so that local parts of the lattice Green functionscoincide with corresponding quantities of the impurity problem for all frequency (or time) arguments :

gω =∑

k

Gωk, (6)

χΩ =∑

k

XΩk

It is worthwhile to note that a standard definition of the EDMFT14 introduces an auxiliary bosonic field called φ,and the second line of Eq.(6) is re-expressed as a condition for the Green’s function Xφ of that bosonic field. At theEDMFT level, the conditions for X and Xφ are equivalent. We discuss this issue in detail in Appendix and show thatsimilar (although not the same) bosonic field can be introduced in non-local approximations beyond the EDMFT.Some “mixed” local approximations can also make sense. For instance, if the hopping is absent (or negligible), it

is natural to consider a scheme with ∆ = 0 and to adjust bosonic hybridization only. Contrary, Λ = 0 corresponds tothe “purely fermionic” DMFT scheme6.

IV. DUAL TRANSFORMATION

It is easy to make a connection between the lattice and impurity actions:

S =∑

r

Simp[c†r, cr] +

∑

ω,k,σ

(εk −∆ωσ) c†ωkσcωkσ +

∑

Ω,k,j,j′

(

V jj′

Ωk − Λjj′

Ω

)

ρ∗ΩkjρΩkj′ (7)

Further, we use a general matrix form of the Hubbard-Stratonovich transformation to decouple the term

c†ωkσEωkσcωkσ with Eωkσ = ∆ωσ − ǫk into the pair of dual fermionic Grassmann-fields f †ωkσ, fωkσ and the term

ρ∗ΩkjWjj′

Ωk ρΩkj′ with W jj′

Ωk = Λjj′

Ω − V jj′

Ωk into the pair of dual bosonic complex-fields η∗Ωkj , ηΩkj (matrix notation is

used to shorten the expressions):

∫

D[

c†, c]

ec†Ec =

∫

D[

c†, c]

det(

α−1f Eα−1

f

)

∫

D[

f †, f]

e−f†αfE−1αff+f†αf c+c†αff

∫

D [ρ∗, ρ] eρ∗Wρ =

∫

D [ρ∗, ρ] det(

αbW−1αb

) ∫

D [η∗, η] e−η∗αbW−1αbη+η∗αbρ+ρ∗αbη,

(8)

where αf , αb are arbitrary, but local in space scaling matrices.Note that whereas the decoupling of Grassmanian variables is mathematically straigforward, more comments are

needed on the second line of Eq.(8). The first comment is that ρ is Hermitian, so that ρ∗−Ω−kj = ρΩkj , and the integral

D[ρ∗, ρ] is in fact over the variables from a half-space. Dual variables inherit this property, η∗−Ω−kj = ηΩkj .

The other comment is that the matrix αbW−1αb is not positively defined, and integrals over D[η∗, η] should be

re-defined to ensure the convergence. Here we describe the corresponding procedure, following, in particular, Ref.14,25.For illustration purposes, we suppose α to be just a real constant factor; the generalization to complex matrix α is

straightforward. Let us introduce a basis where W−1 is diagonal. Let us number the states in this basis with indexl, and consider some matrix element Wll. For a positive real part of Wll, the corresponding part of action can bedecoupled as follows:

eρ′lWllρ

′l+ρ′′

l Wllρ′′l = αW−1

ll α

∫

dη′∫

dη′′eη′lαρ

′l−η′′

l αρ′′l −η′

lαW−1

llαη′

l−η′′l αW−1

llαη′′

l , (9)

4

where we explicitly introduced Hermitian and anti-Hermitian parts of ρ = ρ′ + iρ′′. One can see that for η′l, η′′l being

interpreted as real and imaginary part of a complex number η, this decoupling becomes an explicit form of the thesecond line of (8), written for a particular state l. The integral limits for positive Wll can be chosen along the realaxis, from −∞ to +∞ for both η′l and η′′l .The key observation is that for Wll having a negative real part, the decoupling (9) still holds, up to the integration

path along imaginary axis, from −i∞ to +i∞ for η′l and in the opposite direction, that is +i∞ to −i∞, for η′′l .Thus, the integration over D[η∗, η] can be considered as a symbolic notation for the integration over D[η′, η′′] withη = η′ + iη′′, η = η′ − iη′′ substituted in the integrand. The integration path over each component of vectors η′, η′′ isexplicitly defined for the basis where αWα is diagonal: the integration goes along the real or imaginary axis, respectto a sign of the real part of diagonal matrix element.One can check that such a definition of

∫

D[η∗, η] still allows to use standard algebraic manipulations with path

integrals. In particular, to obtain the Green’s function of dual variables G = −i < η ·η∗ > one uses a regular procedurewith a differentiation with respect to αW−1α, as it can be explicitly shown from (9). On the other hand, the statisticalproperties of dual variables can be unusual because of the complex integration path. For instance, similarly to thefermionic dual Green’s function24, < η · η∗ >=< η′ · η′ > + < η′′ · η′′ > is not necessary positive-defined andconsequently ImG is not always-negative.Let us continue with the derivation of the dual formalism. Hubbard-Stratonovich decoupling in the partition

function with the action (1) gives (in a matrix form)

∫

D[c†c]e−∑

r Simp[c†r,cr]+c†(∆−ε)c+ρ∗(Λ−V )ρ = det(α−1

f (∆− ǫ)α−1f ) det(αb(Λ− V )−1αb)×

×∫

D[c†c]∫

D[f †f ]∫

D[η∗η]e−∑

rSimp[c

†r,cr]+f†αfc+c†αff+ρ∗αbη+η∗αbρ−f†αf (∆−ε)−1αff−η∗αb(Λ−V )−1αbη.

(10)

Integrating out the “localized” fermionic Grassmann variables c†, c, we arrive with the action in dual variables (withα = g−1

ω for the fermionic and α = χ−1Ω for the bosonic part):

S = −∑

ωk G−1ωk f

†ωkfωk −

∑

Ωk X−1Ωk η

∗ΩkηΩk +

∑

j U [ηj , fj , f†j ]

G−1ωk = g−1

ω (εk −∆ω)−1g−1

ω − g−1ω

X−1Ωk = χ−1

Ω (Vk − ΛΩ)−1χ−1

Ω − χ−1Ω

U [η, f, f †] =∑

ωΩ

(

λωΩη∗Ωf

†ω+Ωfω + λ∗

ωΩηΩf†ω fω+Ω

)

+1

4

∑

ωω′Ω γωω′Ωf†ω+Ωf

†ω′−Ωfωfω′ + ...,

(11)

where γ is a full four-point fermionic vertex of the impurity model

γωω′Ω =< cω+Ωcω′−Ωc

†ωc

†ω′ >imp −gωgω′(δΩ+ω−ω′ − δΩ)

gω+Ωgω′−Ωgωgω′

(12)

(δ is the Kronecker symbol), and λ is a “mixed” quantity

λωΩ =− < cω+Ωc

†ωρΩ >imp − < ρ >imp gωδΩ

gωgω+ΩχΩ. (13)

The second term in the nominator typically equals zero as < ρ >imp vanishes.

Values of γ and λ are related, since ρΩ =∑

ω′ sσσ′c†ω′σcω′−Ω,σ′ ,

λΩω = χ−1Ω

(

1−∑

ω′

γω,ω′,Ωgω′gω′−Ωsσσ′

)

(14)

There is an important note that for the Gaussian impurity problem γ vanishes, but λ takes a finite value of χ−1Ω .

There are exact relations between fermionic and bosonic Green’s functions of original and dual variables:

Gωk = (∆ω − εk)−1g−1

ω Gωkg−1ω (∆ω − ǫk)

−1 + (∆ω − ǫk)−1;

XΩk = (ΛΩ − Vk)−1χ−1

Ω XΩkχ−1Ω (ΛΩ − Vk)

−1 + (ΛΩ − Vk)−1;

(15)

5

The relation for the fermionic Green’s functions is derived in Ref.24. The idea is to substitute the right-hand sideof Eq.(10) in the formula Gωk = − 1

Z∂Z∂εωk

. The last term (∆ω − ǫk)−1 appears from the differentiation of the

determinant. The bosonic relation is derived quite similarly, but the signs appearing during the derivation shouldbe discussed. First, the expression for bosonic Green’s function has the different sign, XΩk = 1

Z∂Z

∂VΩk. This gives

−1 factors at X and X. Second, since the bosonic Hubbard-Stratonovich transformation contains an inverse of thedeterminant, the same different sign appears for the last term (ΛΩ−Vk)

−1. Therefore, signs of all terms are changed,compared to the fermionic relation between the real and dual fermionic Green’s functions. Finally, the entire resultingrelationship between the real and dual bosonic Green’s functions obeys exactly the same form, similar to the resultsof a supersymmetric Hubbard-Stratonovich transformation18 .It is useful to introduce the self-energy (for fermions) and polarization-operator (for bosons) corrections to the

effective-medium approximation, defined as

Σ′ = G−1 − G−1; Π′ = X−1 −X−1. (16)

We remind that G,X are given by the expressions (4).A self-energy and polarization operator of dual ensemble are given by the following expressions:

Σ = G−1 − G−1; Π = X−1 − X−1. (17)

Being re-expressed in these quantities, the exact relations Eq. (15) obey a particularly simple form:

(Σ′ωk)

−1 = Σ−1ωk + gω;

(Π′Ωk)

−1 = Π−1Ωk + χΩ;

(18)

One can immediately observe from these formulas that the effective-medium expressions (4) correspond to the Gaussian

approximation for the dual variables, so that Σ, Π (and consequently Σ′,Π′) vanish.Following the effective medium paradigm we introduce a generalized self-consistent condition for ∆ω and ΛΩ similar

to the Eq.(6), but for ”exact” fermionic and bosonic Green’s functions:

gω =∑

k

Gωk, (19)

χΩ =∑

k

XΩk

V. BASIC DIAGRAMS

The simplest non-local physics can be already described in EDMFT. In our formalism this corresponds to a zero-order approximation (“no diagrams”) Σ = 0 and Π = 0, and the condition (6). We intend to preserve the advantagesof EDMFT, but include also the formations and interactions with collective modes. For a “purely fermionic” Hubbard-like theory it means an account of ladder series in dual space26.For the present theory with the dual action of Eq. (11), there are two basic Green’s function lines: fermionic

(full-line with arrow) and bosonic (wavy-line) as well as two interaction vertices: a triangle (corresponding to the dualboson-fermion interaction - λ) and a square (describing the dual-fermion interaction - γ), respectively (Fig. 1). Thesimplest diagram for generating dual functional includes the electron-boson coupling as shown in Fig. 2. One can alsodraw the renormalized triangle electron-boson vertex λ due to interactions with fermionic ladder (Fig. 3 ) series (Fig.4 ). In this case one of the triangle vertices in Fig. (2) can be renormalized. We will later show that the lowest-orderdiagram Fig. (2) actually catches most of the physics we are interested in, so we shall concentrate on this case.The diagrams shown in Figs. (2,4) are the minimal set to describe the formation of collective modes in a strongly

correlated system, with an explicit account of electron-boson coupling.

VI. SINGLE-LOOP APPROXIMATION

In this paper, we will concentrate on non-local effects in bosonic modes, that appear in approximations beyond theEDMFT. Those effects do appear even when the nonlinearity of the impurity problem is absent or can be neglected,

6

G~

X~

W

l

g

w

w w+W

w

w

W

w-W

w+W

´ ´

FIG. 1: (Color online) Basic builfing blocks for dual-boson diagram diagrams

FIG. 2: (Color online) Simplest diagram for dual boson generating functional....

...

...

...

FIG. 3: (Color online) Ladder-type of dyagram for generating functional.

= +

FIG. 4: (Color online) Diagrammatic equation for the renormalized triangle vertex.

7

FIG. 5: (Color online) Bosonic dual self-energy in the lowest-order approximation.

FIG. 6: (Color online) Fermionic dual self-energy in the lowest-order approximation.

because λ does not vanish. For that case, it is natural to consider a bosonic self-energy being just a two-lines bubble(Fig. 5):

ΠΩK =∑

ωk

λΩωGωkGΩ+ω,K+kλ∗Ωω . (20)

Since γ is neglected, λωΩ = χ−1Ω . Simple manipulations with the second lines of Eqs. (4), (16) and (18) give

XΩK =1

(

χΩ +∑

ωk GωkGΩ+ω,K+k

)−1

+ ΛΩ − VK

. (21)

Similar non-local contribution to fermionic self-energy can be introduced to describe a renormalization of the fermionicGreen’s function G. Corresponding diagram contains one fermionic and one bosonic line (Fig. 6). We will however

concentrate on the analysis of bosonic quantities and for simplicity use the EDMFT result G instead of renormalizedG propagator. It makes a further simplification possible. Indeed, in the local approximation G and Gdual differ inlocal part only23,24:

Gωk = Gωk − gω. (22)

Further, since we suppose γ = 0 and since < ρ >= 0, the bosonic Green’s function for the impurity problem becomesa product of two fermionic ones, χΩ =

∑

ω gωgΩ+ω. Substituting the expression Eq. (22) in (21), we obtain

XΩK =1

(∑

ωk GωkGΩ+ω,K+k)−1

+ ΛΩ − VK

. (23)

For Λ = 0, this expression is clearly related to the Random Phase Approximation (RPA). To get the “textbook”RPA, one should completely neglect interaction terms of the impurity problem, so that gω = (iω − ∆ω)

−1 andGω,k = (iω − εk)

−1. The scheme with exactly calculated gω can be called RPA+DMFT.A theory using the single-bubble approximation (20) with exact λ (that is, without putting λ to χ−1) is essen-

tially the GW+EDMFT approach14. There is slight difference about the self-consistency condition, as we explain inAppendix. It is also important that for the diagram calculation of the GW+EDMFT, the local part of the bosonicGreen’s function is excluded ad hoc. As one can observe from (43), for an EDMFT choice of Bose variables, it actu-ally does not vanish exactly. So, our consideration proposes certain modification of the GW+EDMFT. Anyhow, suchmodification should not affect the results of the GW+EDMFT strongly. Actual advantage of our consideration is apossibility to go beyond the domain of applicability of the GW+EDMFT, that is the situation when single-bubbleapproximation becomes incorrect. Such a case is considered in the next section.

VII. LADDER SUMMATION IN A STRONG-COUPLING LIMIT

The single-bubble approximation becomes invalid, when vertex γ is large. There is a general situation of this kind:four-point vertex of an impurity problem with magnetic moment formed

8

that gives a large contribution at low frequencies. This simply means that for isolated atoms the magnetic momentsare free and, thus, for almost isolated atoms (when the energy of interatomic effective interactions is much slammerthan intraatomic one) can be considered as almost integrals of motionTo illustrate this statement, let us consider an atomic limit of the single-band Hubbard model. Single-electron

Green’s function of Hubbard atom at the half-filling

gat =−iω

ω2 + (U/2)2(24)

has a time-scale about τU ∝ U−1, which we call fast. Contrary, the spin-spin correlator < sτsτ ′ > is independent, asone can easily check, of time arguments. So, a time-scale for < sτsτ ′ > is the inverse temperature β. It means thatapart from a small domain |τ − τ ′| ≈ U−1, the value of < sτsτ ′ > is determined by its non-Gaussian part, which doesnot fall as |τ −τ ′| increases. Much slower dynamics of γ in comparison with g just reflects the fact that rotation of thespin does not change the energy of the atom, whereas the change of the particle number does. Direct calculation of γfor the Hubbard atom does support this observation. Finite-temperature expressions27 for γ1234 contain a “singular”term proportional to βU2δΩ0. So, the first two time arguments of γ in time-domain must coincide, as well as the lasttwo: γ = γτττ ′τ ′ , but the difference τ − τ ′ can be large.Properties of the impurity problem are different from the isolated atom. However, for the strong-coupling limit the

hybridisation is small. In this case the fast dynamics stay roughly unchanged, so that atomic Green’s function (24)can be used, and four-point vertex obeys the form γτττ ′τ ′ . Hybridization can introduce its own slow timescale τΛ, sothat γ does depend only on τ − τ ′. In frequency domain, it means

γ = γΩ, (25)

and at small frequencies γΩ ∝ τΛU2. Since γ carry a large factor, we can keep only irreducible term in the expressions

for λ and χ. Note that this is completely opposite to the case considered in the previous Section.First, we obtain from Eq. (12) ignoring the small trivial terms proportional to gg the following relation:

χΩ = χ(0)Ω γΩχ

(0)Ω . (26)

In the above equation we introduced a “bare susceptibility” of the impurity problem

χ(0)Ω ≡ −

∑

ωσ

gωgΩ+ω. (27)

Next, we use the relation 14 and keep only the large factor γ:

λΩ = (χ(0)Ω )−1. (28)

Note that in the present approximation λ, like γ, has only single frequency argument.Let us consider a half-filled Hubbard lattice with the nearest-neighbour hopping t, at large U . A low-frequency

behaviour of this system is essentially a dynamics of Heisenberg model. An effective-medium description of spin wavesin this case should result in the expression

χΩ =∑

k

1

χ−1Ω + ΛΩ − Jk

, (29)

with Jij =t2

Ufor nearest neighbors. Despite this expression is rather simple (in our notation, it just Π′ = Jk), it was

not reproduced by any DMFT-like approximation so far. Exchange interaction containig t2 can easily be obtainedfrom a single loop of the fermionic lines. But the RPA and GW+EDMFT denominators contain an inverse of thisquantity (see the previous Section), so the structure of formulas is drastically different.In our formalism, the desired expression appears from the ladder summation, that is the calculation for the diagram

shown in Fig. (7). In this summation, we take into account the “slow” vertex (25) only. Since γ depends only on asingle time argument, the calculation is quite simple. The fermionic ladder with coinsiding arguments at each side(see Fig. (7) ) equals to

X(0)ΩK

1− γΩX(0)ΩK

, (30)

9

...

FIG. 7: (Color online) Bosoinc dual self-energy in the ladder approximation. A triangle represents the λ vertex and squarerepresents the γ vertex

where

X(0)ΩK ≡

∑

ωkσ

GωkGΩ+ω,K+k (31)

is a “bare susceptibility” of the dual system.Now, adding the triangle vertices to the both ends of the ladder and using the exact relation (18), we obtain

Π′ΩK =

(

λX

(0)ΩK

1− γΩX(0)ΩK

λ

)−1

+ χΩ

−1

. (32)

Finally, we substitute formulas (26,28) for χ and λ. This results in cancellation of γ out of the expressions and givesa simple expression

Π′ΩK = IΩX

(0)ΩKIΩ. (33)

where IΩ = (χ(0)Ω )−1 is an effective local Stoner parameter. Note that χ

(0)Ω obey “fast” dynamics only. To determine

the low-energy properties, which we are interested in, it is enough to calculate it at Ω equal zero, and to replace

summation with integration. Straightforward calculation of (27) with g = gat gives χ(0)Ω=0 = 1

2U . To calculate X(0),

we switch to the real-space and use a strong-coupling limit. It gives the Green’s function G = t(g(0))2 for the nearest

neighbors and G = 0 elsewhere. Now, the integration gives XΩ=0 = t2

4U3 for nearest-neighbors, and we end up withthe desired expression (29). This derivation can be considered as a generalization of the effective exchange formulafor magnetically ordered case28,29 to the paramagnetic state.

VIII. CONCLUSIONS

In summary, the general scheme for non-local correlations effects within the dual boson approach is presented.This method gives a direct way to obtain the higher-order diagrammatic contribution beyond the EDMFT schemeand can be very useful for description of a magnon spectrum renormalization in strongly correlated systems. Themethod combined the numerically exact solution of effective single-cite impurity model with frequency-dependentelectron-electron interactions as well as analytical diagrammytic expansion of non-local fermionic and bosonic self-energies. This approach is designed for description correlated systems with the strong local fermion correlations andthe fluctuating non-local bosonic modes on equal footing.This work was supported by DFG Grant No. 436 113/938/0-R and RFFI 11-02-01443. MIK acknowledges a support

from the EU-India FP-7 collaboration under MONAMI. AIL acknowledges a support from SFB 668 and FOR-1346.

APPENDIX: Different choices of dual variables

The formalism presented here has an important difference with the common introduction of the EDMFT: whereaswe introduce dual bosons η by Hubbard-Stratonovich decoupling of (V − Λ)ρ∗ρ quadratic form, the term decouplingV ρ∗ρ is decoupled in standard EDMFT scheme14,30. As one can see, this does not result in any physical consequences,as two approaches yield the identical results for the averages G,X over original variables. However, properties of thedual bosons, of course, do depend on their definition. In particular, the difference is about local properties of dualpropagator: for a present formalism, its local part vanishes in the EDMFT approximation,

∑

k

X = 0, (34)

10

whereas the “effective-medium” condition∑

k

X V = χφ (35)

holds for the decoupling (36). Here X V is the Green’s function of the dual variables φ arising from the decoupling ofV ρ∗ρ:

V ρ∗ρ → α(ρ∗φ+ φ∗ρ)− V −1α2φ∗φ, (36)

and χφ ≡< φφ∗ >imp is defined for the impurity problem with decoupled bosonic hybridization, so that

Simp = Sat +∆ωc†ωcω + α(ρ∗ΩφΩ + φ∗

ΩρΩ)− Λ−1Ω α2φ∗

ΩφΩ. (37)

To maintain a similarity with our formalism, we introduced in the above expressions a scaling factor α, that is absentin the standard EDMFT scheme14. All physical quantities do not depend on a particular value of this factor.Averages X V and χφ can be found from the exact relations similar to (15):

α2X V = V XV + V ;

α2χφ = ΛχΛ + Λ.(38)

We will now consider a general case of some approximation utilizing a decoupling of Wρ∗ρ with an arbitrary W .The approximation can be non-Gaussian, so that there is a correction Π′ to the EDMFT polarization operator:

X =1

χ−1 + Λ− V −Π′. (39)

The Green’s function for dual bosons can be found from the exact relation

α2XW = WXW +W. (40)

One can realise that there are two special cases. First, let us take W = V +Π′ − Λ, and use α = χ−1, as in the mainbody of the paper. For this choice, using (39, 40) one can show straightforwardly that

XV+Π′−Λ =χ(V +Π′ − Λ)

χ−1 + Λ− V −Π′≡ X − χ. (41)

Summing over k and taking Eq.(19) into account we conclude that the condition (34) is fulfilled

∑

k

XV+Π′−Λ = 0. (42)

It turns out that dual propagator vanishes for variables obtained from the Hubbard-Stratonovich decoupling of therenormalized interaction, with the hybridization part excluded, if the condition (19) fulfills. For the EDMFT approx-imation, Π′ is absent and the condition (34) fulfills for the decoupling with V +Π′ used in the present paper.Now, let us consider the case W = V +Π′. It is suitable to take α = χ−1 + Λ in this case. We obtain

XV+Π′

=(χ−1 + Λ)−1(V +Π′)

χ−1 + Λ− V −Π′≡ X − (χ−1 + Λ)−1. (43)

Taking again a summation over k and substituting Eq.(19), we obtain∑

k XV+Π′

= χΛ(

Λ−1 + χ)−1

. The second

line of (38) with α = χ−1 + Λ gives the same answer for χφ, therefore we obtain

∑

k

XV+Π′

= χφ. (44)

The “effective-medium” condition fulfills for variables obtained from the Hubbard-Stratonovich decoupling of the renor-malized interaction, with the hybridization part included., if the condition (19) fulfills. For the EDMFT, it correspondsto the decoupling of purely V ρ∗ρ part used in Ref.14.

11

We stress that, strictly speaking, apart from the simple EDMFT case conditions (34, 35) are not fulfilled simul-taneously with (34). This does not make serious problems for the GW+EDMFT calculations, because one typicallydo self-consistent calculations of the EDMFT problem and then calculate diagram corrections for purely EDMFThybridization. On the other hand, for higher order diagrammatic approximations the difference can be essential.Finally we note that a specific choice of the decoupling procedure is physically irrelevant. As we have seen, the

EDMFT can be constructed using W = V − Λ or W = V , and the result for initial ensemble is the same. It can bechecked that any other decoupling will also work; what matters is the Gaussian approximation for dual variables andthe value of hybridization. The later is defined by the condition (34), that does not include any dual quantities. Weuse the theory based on W = V −Λ decoupling because it allows a simple construction of the dual diagrammatic series.It is also nice that the EDMFT dual Green’s function has a very simple physical meaning in this case: accordingly to(41), it is just Green’s function of initial variables with local part subtracted.

∗ Electronic address: alex@shg.phys.msu.ru1 T. Moriya, Spin fluctuations in itinerant electron magnetism (Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985).2 I. E. Dzyaloshinskii and P. S. Kondratenko, Sov. Phys. JETP 43, 1036 (1976).3 S. V. Vonsovsky, M. I. Katsnelson, and A. V. Trefilov, Phys. Metal. Metallography 76, 247 (1993).4 S. V. Vonsovsky, M. I. Katsnelson, and A. V. Trefilov, Phys. Metal. Metallography 76, 343 (1993).5 A. I. Lichtenstein, M. I. Katsnelson, and G. Kotliar, Phys. Rev. Lett. 87, 067205 (2001).6 A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).7 E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and P. Werner, ArXiv e-prints (2010).8 E. Gorelov, J. Kolorenc, T. Wehling, H. Hafermann, A. B. Shick, A. N. Rubtsov, A. Landa, A. K. McMahan, V. I. Anisimov,M. I. Katsnelson, et al., Phys. Rev. B 82, 085117 (2010).

9 E. Gorelov, T. O. Wehling, A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 80, 155132 (2009).10 G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006).11 A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57, 6884 (1998).12 M. I. Katsnelson, V. Y. Irkhin, L. Chioncel, A. I. Lichtenstein, and R. A. de Groot, Rev. Mod. Phys. 80, 315 (2008).13 R. Chitra and G. Kotliar, Phys. Rev. B 63, 115110 (2001).14 P. Sun and G. Kotliar, Phys. Rev. B 66, 085120 (2002).15 M. I. Katsnelson and A. I. Lichtenstein, Journal of Physics: Condensed Matter 22, 382201 (2010).16 S. Biermann, F. Aryasetiawan, and A. Georges, Phys. Rev. Lett. 90, 086402 (2003).17 S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, 1999).18 T. D. Stanescu and G. Kotliar, Phys. Rev. B 70, 205112 (2004).19 S. Pairault, D. Senechal, and A.-M. S. Tremblay, Phys. Rev. Lett. 80, 5389 (1998).20 A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Pergamon

Press, New York, 1965).21 D. Bohm and D. Pines, Phys. Rev. 92, 609 (1953).22 V. A. Khodel, V. R. Shaginyan, and V. V. Khodel, Physics Reports 249, 1 (1994).23 A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 77, 033101 (2008).24 A. N. Rubtsov, M. I. Katsnelson, A. I. Lichtenstein, and A. Georges, Phys. Rev. B 79, 045133 (2009).25 Rubtsov, Phys. Rev. B 66, 052107 (2002).26 H. Hafermann, G. Li, A. N. Rubtsov, M. I. Katsnelson, A. I. Lichtenstein, and H. Monien, Phys. Rev. Lett. 102, 206401

(2009).27 H. Hafermann, C. Jung, S. Brener, M. I. Katsnelson, A. N. Rubtsov, and A. I. Lichtenstein, Europhys. Lett. 85, 27007

(2009).28 A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, Journal of Magnetism and Magnetic Materials

67, 65 (1987).29 M. I. Katsnelson and A. I. Liechtenstein, J. Phys. Cond. Mat. 16, 7439 (2004).30 S. Florens, Ph.D. thesis (University Pierre and Marie-Curie, Paris, 2003).