Cluster Extended Dynamical Mean Field Approach and Unconventional Superconductivity
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Cluster Extended Dynamical Mean Field Approach and UnconventionalSuperconductivity
J. H. Pixley ∗ , Ang Cai, and Qimiao Si Department of Physics and Astronomy, Rice University, Houston, Texas, 77005, USA (Dated: October 21, 2018)The extended dynamical mean field theory has played an important role in the study of quantumphase transitions in heavy fermion systems. In order to incorporate the physics of unconventionalsuperconductivity, we develop a cluster version of the extended dynamical mean field theory. In thisapproach, we show how magnetic order and superconductivity develop as a result of inter-site spinexchange interactions, and analyze in some detail the form of correlation functions. We also discussthe methods that can be used to solve the dynamical equations associated with this approach.Finally, we consider different settings in which our approach can be applied, including the periodicAnderson model for heavy fermion systems.
PACS numbers: 71.10.Hf, 71.27.+a, 75.20.Hr
I. INTRODUCTION
Unconventional superconductivity in heavy fermionmetals often develops in the vicinity of antiferromagnetic(AF) order . Its understanding is intimately connectedwith that of the AF quantum critical points (QCPs) .Traditional descriptions of heavy fermion quantum crit-icality are based on those for purely itinerant mag-netism, in terms of the fluctuations of the spin-density-wave (SDW) order parameter . Studies in the recentpast have emphasized the “beyond Landau” physics ofKondo destruction . In these studies, considerableprogress has been made based on the extended dynam-ical mean field theory (EDMFT) solution of Kondo lat-tice models . One of the important questions along thisdirection concerns the implications of these theoreticalstudies on the understanding of superconductivity.The EDMFT approach builds on the dynamical meanfield theory (DMFT) , which was developed throughthe infinite dimensional limit of the Hubbard model .DMFT maps an interacting lattice problem to a sin-gle quantum impurity model coupled to a self consis-tent Weiss field. The latter captures the environmentas seen at the single-particle level, and is represented bya fermionic bath. DMFT has made significant contribu-tions to a variety of strongly correlated problems , andhas in particular provided significant new insights on theMott transition.The EDMFT approach treats inter-site density-densityor spin-spin interactions, leading to a single impuritymodel coupled to self consistent fermionic and bosonicbaths . It has been extensively applied to the studyof AF quantum critical heavy fermion metals . Fora Kondo lattice, the EDMFT approach yields a Bose-Fermi Kondo model (BFKM) with self consistent bosonic ∗ Present Address: Condensed Matter Theory Center and JointQuantum Institute, Department of Physics, University of Mary-land, College Park, Maryland 20742-4111, USA and fermionic baths. Kondo destruction arises from thisapproach. This may be seen already at the level of theBFKM in the absence of self consistency. The couplingof the local moment to the bosonic bath competes withthe Kondo effect, i.e. the tendency of singlet formationdue to the AF exchange coupling between the local mo-ment and the fermionic bath. When the spectrum ofthe bosonic bath is sufficiently soft, corresponding tothe spectral function being “sub-ohmic”, this competi-tion gives rise to the destruction of the Kondo effect,in a way that is associated with the criticality of theBFKM . Studying this type of criticality in a varietyof quantum impurity models has led to a number of newinsights regarding Kondo destruction QCPs. For the lat-tice case, through the EDMFT equations, the bosonicspectrum is particularly soft near the AF QCP due tothe critical slowing down, and one consistent solutionis that the Kondo destruction occurs at the AF QCP.There is by now extensive experimental evidence for thistype of beyond-Landau QCP from experiments in heavyfermion metals, both in terms of a unusual scaling ofthe dynamical spin susceptibility in the quantum criticalregime and a sudden jump of the Fermi surface acrossthe QCP . However, in order to study the impor-tant problem of the interplay between this unconven-tional quantum criticality and superconductivity, a clus-ter generalization of the EDMFT is called for. In thismanuscript, we develop such a formalism.In DMFT based approaches, incorporating real spacecorrelations beyond a single site have naturally been donewith the development of quantum cluster theories . Inthis case, strongly correlated problems can be mapped toa quantum cluster model with self consistent fermionicbaths and the interactions within the cluster are treatedexactly. Importantly, dynamical cluster theories incor-porate non-perturbative corrections to DMFT withoutintroducing a non-causal self energy . This can beformulated in real space which leads to cluster DMFT(CDMFT) , or in momentum space which is known asthe dynamical cluster approximation (DCA) ; there areother cluster embedding schemes possible such as thevariational cluster approximation (VCA) . When theWeiss fields are neglected these cluster schemes are nolonger self consistent and reduce to cluster perturbationtheory (CPT), which approximates lattice quantities byexpanding about the isolated cluster limit . A mainadvantage of dynamical cluster theories is the ability toaccount for various types of order not possible withinDMFT . For example, a four site cluster can treat a d-wave superconducting order parameter as well as stripecharge or spin order. Such a pairing mechanism is ex-pected to be appropriate for, e.g. , the cuprates and heavyfermion materials. In this case, any superconductingground state will have cooper pairs formed between siteswhich can lead to a variety of different pairing symme-tries, such as extended s-wave, p-wave or d-wave.In this manuscript, we present a cluster extendeddynamical mean field theory scheme that we dub C-EDMFT. We derive the equations by generalizing ref. 15to the cluster case using a locator expansion about adressed cluster limit. We formulate the equations in bothreal and momentum space. We introduce magnetic orderin the same fashion as EDMFT in refs. 18–20 distinctfrom DMFT, and then generalize this approach to in-clude superconductivity as well. We also construct thepairing correlation functions induced by magnetic inter-actions in the normal state within this approach. Lastly,we use the formalism to derive effective impurity modelsassociated with strongly correlated problems of centralinterest.We note that cluster generalizations of the EDMFThave been carried out in various forms in the past .Where there is overlap, our approach is consistent withthese formulations. We will make the specific compar-isons as we go through the derivation of our approach.In short, the C-EDMFT formalism developed here bringsout two new aspects (for definiteness, we will describethese with Eq. (1) in mind). First, it is the inter-site J ij interactions which underlie both the magnetic and super-conducting orders. Such orders develop through decou-pling the J ij interaction term into the appropriate chan-nels. Second, the approach avoids double-counting theinter-site interaction by suppressing the induced inter-site interactions associated with the polarization of thefermonic bath by the order parameter. As a result, the q -dependence of the dynamical spin susceptibility arisesthrough the J ij interaction term, instead of the bareparticle-hole bubble at the “special- q ” (as opposed to the“generic- q ”, see section III). These two aspects are in thesame spirit as discussed for the case of the EDMFT . A. Development of Cluster EDMFT
For illustration purposes, we consider a one band Hub-bard model with two body inter-site interactions on a generic lattice. H = X h i,j i ,σ t ij ( c † iσ c jσ + h . c) + U X i n i ↑ n i ↓ + X h i,j i ,α J αij S αi S αj (1)where c iσ destroys an electron of spin σ at site i , n i = P σ n iσ , and n iσ = c † iσ c iσ . The index α runs over0 , , ,
3, where for α = 1 , , α = x, y, z ) theoperator S αi ≡ c † iµ ( σ αµν / c iν , is the spin operator, where σ αµν is the α -Pauli matrix for α = x, y, z . In additionwe consider the charge channel with α = 0, where theoperator S i ≡ : n i : denotes the normal ordered density: n i := n i − h n i i . We denote nearest neighbors by h i, j i and only consider nearest neighbor hopping t ij and twobody exchange interaction J αij . For J αij = 0, the modelreduces to the standard Hubbard model with an onsiteCoulomb repulsion of strength U . It is natural to extendthese techniques to multi-band models and longer rangeinteractions.A main focus of this work is a self consistent solu-tion of the single particle Greens function G ijσ ( τ ) = −h T τ c iσ ( τ ) c † jσ i as well as the spin and charge suscep-tibilities χ αij ( τ ) = h T τ : S αi ( τ ) :: S αj : i . In general, for thesingle particle Greens function, a perturbative expansionabout the non-interacting limit yields the Dyson equation G ( k , iω n ) = 1 iω n − µ − t k − Σ lat ( k , iω n ) , (2)where t k is the Fourier transform of t ij =1 /N P k e i k · ( r i − r j ) t k , µ is the chemical potential,Σ lat ( k , iω n ) is the single particle self energy, and we de-note fermionic Matsubara frequencies as ω n . Analogousto the single particle Greens function, we introduce aspin and charge self energy M α lat ( q , iν ) which is definedin terms of each susceptibility as χ α ( q , iν n ) = 1 M α lat ( q , iν n ) + J α q (3)where J α q is the Fourier transform of J αij and ν n is abosonic Matsubara frequency. The spin/charge self en-ergies specify how much their corresponding susceptibil-ities differ from a Gaussian model where χ αij ∝ /J αij (ref. 34). In the following we will derive a self consistentC-EDMFT approach to approximate the lattice quanti-ties Σ lat ( k , iω n ) and M α lat ( q , iν ) and in turn the singleparticle Greens function and spin/charge susceptibilities.The remainder of the paper is organized as follows. Wefocus on the C-EDMFT equations in the absence of anyorder in section II, and those in the presence of magneticorder in section III. We then apply the approach to super-conducting order and correlations in section IV. We usethe formalism to derive an effective cluster model in sec-tion V. Finally, we outline the relevant solution methodsin section VI, discuss several pertinent points in sectionVII, before concluding the paper in section VIII. II. NORMAL STATE IN THE ABSENCE OFBROKEN SYMMETRY
We begin by dividing the lattice of N sites into clustersof size N c , where each lattice site is now labeled by x = r + R , where r labels the cluster and R labels the siteswithin the cluster (see figure 1). In the following wewill use upper case latin letters to denote cluster indices(i.e. indices within the cluster) and lower case letters tolabel each cluster. This is then Fourier transformed to k = ˜ k + K , where K is the intra cluster momentum and ˜ k the inter cluster momentum. With this notation t ij and J αij can be written as A R i R j ( r i − r j ) = A ( r i − r j ), wherethe bold A denotes a matrix in cluster indices. We thenseparate t ij and J αij into intra and inter cluster parts t ( r i − r j ) = t c δ r i , r j + δ t ( r i − r j ) , J α ( r i − r j ) = J αc δ r i , r j + δ J α ( r i − r j ) , (4)where t c and J αc are the interactions within the cluster,whereas δ t and δ J α are the interactions between clusters,note that by construction δ t (0) and δ J α (0) vanish. Rr Jc δ J LcLc
FIG. 1: (Color online) Division of the lattice into
N/N c clus-ters, of size N c = L d c c , where d c is the dimensionality of thecluster. The vector r labels each cluster, while sites withinthe cluster are labelled by R . Interactions are divided intowithin the cluster J c and between clusters δJ , (which is alsodone for the hopping elements t c and δt .) We have omittedthe channel index α for clarity. A. Real Space Formulation
We will first derive the equations in real space. Wefirst focus on a ground state with no broken sym-metry, and will then generalize the equations to thecase of magnetic order and superconductivity in sec-tions III and IV. We perform a locator expansion in δt and δJ about the cluster limit . The isolated clus-ter single particle Greens function and susceptibilitiesare defined by C G ( X, Y ; r , τ ) = −h T τ c r Xσ ( τ ) c † r Y σ i H c and C χ α ( X, Y ; r , τ ) = h T τ S α r Xσ ( τ ) S α r Y σ i H c respectively,where H c is the isolated cluster Hamiltonian at cluster r .In the following we consider problems that have trans-lational invariance between clusters which implies eachcluster correlation function is identical and we can dropthe label r .We now generalize the effective cumulant expansion ofMetzner for the Greens function and Smith and Si forthe susceptibilities from a single site to a cluster, whichleads to matrix quantities. Along these lines, we intro-duce the effective cluster Greens function C G ( X, Y ; τ )and spin/charge susceptibilities C χ α ( X, Y ; τ ) which aredefined as the isolated cluster Greens function and sus-ceptibility (in the α channel) with all local decorationsthat are irreducible by cutting a single δ t and δ J α line re-spectively (see figure 2). The effective cluster correlation δ J = + + + ...++C χ r i r j δ J ik = - χ r i r j r k r j r i r i r i r i (a)(b) C χ C χ FIG. 2: (Color online) (a) Real space diagrammatic repre-sentation of the effective cluster spin susceptibility C χ (solidsquare), as an expansion in bare cumulants C χ (empty square)with all local decorations that are completely irreducible bycutting a single δJ line. (b) Diagrammatic representation ofthe spin susceptibility in equation (6), expanding about theeffective cluster limit. Each term is for a specific two bodyinteraction channel index α , (with the index omitted for clar-ity). functions can be regarded as “dressed” cluster correla-tion function, generalizing the dressed atom picture tothe cluster case. Retaining this class of diagrams canbe formally justified in the large dimensional limit afterrescaling δ t and δ J α by the square root of the coordina-tion raised to the manhattan distance between clusterswhile keeping the dimension and number of sites in thecluster fixed . Performing the locator expansion aboutthe effective cluster correlation functions we arrive at thefollowing Dyson like equations G r i r j ( iω n ) = C G ( iω n ) δ r i , r j + C G ( iω n ) X r l δ t ( r l − r j ) G r l r j ( iω n ) (5) χ α r i r j ( iν n ) = C χ α ( iν n ) δ r i , r j − C χ α ( iν n ) X r l δ J α ( r l − r j ) χ α r l r j ( iν n ) . (6)We note that these are matrix equations and we areusing a bold notation to denote matrices in cluster in-dices. Fourier transforming equations (5) and (6) tointer-cluster momentum ˜ k and ˜ q we arrive at for theGreens function G (˜ k , iω n ) = C G ( iω n ) + C G ( iω n ) δ t (˜ k ) G (˜ k , iω n ) (7)= h C G ( iω n ) − − δ t (˜ k ) i − (8)and for the susceptibility χ α (˜ q , iν n ) = C χ α ( iν n ) − C χ α ( iν n ) δ J α ˜ q χ α (˜ q , iν n )(9)= (cid:2) C χ α ( iν n ) − + δ J α ˜ q (cid:3) − . (10)Rewriting the Dyson equations for G and χ α in equations(2) and (3) in real space cluster indices we find the selfenergy and spin/charge self energies are ˜ k and ˜ q indepen-dent respectively, and only depend on cluster indices. Wearrive at the following equations for the effective clustercorrelation functions C G ( iω n ) − = g c ( iω n ) − − Σ ( iω n ) (11) C χ α ( iν n ) − = J αc + M α ( iν n ) , (12)where the free isolated cluster Greens function is g c ( iω n ) − = ( iω n + µ ) − t c and is the identity matrixin cluster indices.The fact that both self energies are ˜ k and ˜ q indepen-dent implies they can be calculated by an effective clus-ter model. The effective cluster model can be obtainedthrough a generalized cavity method , by expanding thepartition function in terms of δ t and δ J α about a par-ticular cluster o and effectively integrating out all otherdegrees of freedom. This leads to the cluster action S C = S C − Z β dτ dτ ′ X X,Y,σ c † Xσ ( τ ) G − ,XY ( τ − τ ′ ) c Y σ ( τ ′ ) − Z β dτ dτ ′ X X,Y,α S αX ( τ ) χ − α,XY ( τ − τ ′ ) S αY ( τ ′ ) . (13)We have dropped the cluster label o , defined the isolatedcluster action as S C = Z β dτ U X X ∈ C n X ↑ ( τ ) n X ↓ ( τ )+ X h X,Y i ,α J αc,XY S αX ( τ ) S αY ( τ ) , (14) and introduced the effective Weiss fields G − ,XY and χ − α,XY that are related to lattice quantities by G − ( iω n ) = g c ( iω n ) − − X r i , r j δ t or i G ( o ) r i r j ( iω n ) δ t r j o (15) χ − α ( iν n ) = X r i , r j δ J α or i χ α ( o ) r i r j ( iν n ) δ J α r j o . (16)Where G ( o ) and χ α ( o ) are the Greens function and spinsusceptibility of the lattice with the cluster o removed,and we have taken the cluster o to be at the origin. Gen-eralizing the arguments of ref. 15 to the case of matrixcluster quantities we can relate G ( o ) and χ α ( o ) to thefull Greens function and spin/charge susceptibilities toobtain (omitting the frequency labels) G ( o ) r i r j = G r i r j − G r i o ( G oo ) − G or j (17) χ α ( o ) r i r j = χ α r i r j − χ α r i o ( χ α oo ) − χ α or j . (18)With these relations, equations (5) and (6), as well as theself consistency conditions G loc ( iω n ) = N c N X ˜ k G (˜ k , iω n ) , (19) χ α loc ( iν n ) = N c N X ˜ q χ α (˜ q , iν n ) , (20)the Weiss fields are completely determined by the selfenergies and the local correlation functions where G − ( iω n ) = Σ ( iω n ) + G loc ( iω n ) − , (21) χ − α ( iν n ) = M α ( iν n ) + J αc − χ α loc ( iν n ) − . (22)We have used equations (5) and (6) to eliminate the de-pendence on the effective cumulants and use the subscript“loc” denote averages calculated with the effective clusteraction in equation (13), which also corresponds to the lat-tice quantities within the cluster as enforced via the selfconsistent equations (19) and (20). One can generalizethe arguments of refs. 27,28 to prove that the approachhere has manifestly causal self energies for both Σ and M α .It is useful to consider a few limiting cases of theabove equations. First, we note that setting G − ( iω n ) = g c ( iω n ) − and χ − α ( iν n ) = 0 implies the effective clus-ter correlation functions reduce to the isolated clusterquantities, and the self energies are then completely de-termined by solving the isolated cluster problem. There-fore, in the absence of Weiss fields this approach reducesto CPT for both G ij and χ αij and is no longer self con-sistent. This clarifies the meaning of keeping all localdecorations for C G and C χ α and is necessary to prop-erly introduce the dynamical Weiss fields. In the limit ofone site in the cluster N c = 1, the equations reduce toEDMFT; this underscores the fact that the cluster theo-ries incorporate spatial fluctuations beyond standard dy-namical mean field theories. Lastly, in the limit of largecluster sizes, N c → ∞ the theory becomes exact. In thissense, extended dynamical cluster theories interpolatesbetween the EDMFT and the exact answer as the clus-ter size is increased.After self consistency has been reached it is possible torestore translational invariance to the self energies andthereby the correlation functions by interpolating thecluster quantities. Since the self energies are only de-fined for sites within the cluster (or cluster momentum)the interpolation scheme must respect the symmetry ofthe original lattice. Following ref. 26, after the self consis-tent solution has been reached we interpolate the clusterself energies to obtain the lattice quantities with the es-timationΣ lat ( k , iω n ) = 1 N c X X , Y e − i k · ( X − Y ) Σ( X , Y , iω n ) , (23) M α lat ( q , iν n ) = 1 N c X X , Y e − i q · ( X − Y ) M α ( X , Y , iν n ) . (24)We then use these to determine the lattice Greens func-tion and spin susceptibility in equations (2) and (3).Other interpolation schemes are possible as described inref. 36, where each scheme preserves the symmetry of thelattice.Our equations without restoring the translational in-variance have some similarities with those of ref. 33,which invoked a different procedure. Namely, a Hubbard-Stratonovich transformation decouples the inter site twobody interaction term, and the Hubbard-Stratonovichfield becomes a self consistent dynamic bosonic Weissfield in the cluster limit. We also note that, in the ab-sence of conduction electrons similar equations for theself consistent bosonic bath appeared in the constructionfor spin only models in ref. 34. B. Momentum Space Formulation
The momentum space construction parallels the DCAformulation, which restores translation symmetry by giv-ing the cluster periodic boundary conditions . This isachieved by modifying the cluster Fourier transform to[ J α DCA (˜ q )] X i , X j ≡ J α (˜ q ) X i , X j e − i ˜ q · ( X i − X j ) (25)= 1 N c X Q e i Q · ( X i − X j ) J α Q +˜ q (26)In the following we will refer to this as the DCA Fouriertransform. This leads to periodic boundary conditionsin the cluster and a coarse graining of the cluster quan-tities ¯ t K = N c /N P ˜ k t ˜ k + K and ¯ J α Q = N c /N P ˜ q J α ˜ q + Q and in turn, the inter-cluster quantities become δt ˜ k + K = t ˜ k + K − ¯ t K and δJ α ˜ q + Q = J α ˜ q + Q − ¯ J α Q . Applying the DCAFourier transform to equations (5) and (6) leads to matrixequations that are diagonal in momentum space. Here wespecify the equations for the spin susceptibility, with the equations for the Greens function being identical in formto those of DCA. We have a Dyson like equation χ α (˜ q + Q , iν n ) = 11 /C χ α ( Q , iν n ) + δJ α ˜ q + Q , (27)with an effective spin cumulant1 /C χ α ( Q , iν n ) = M α ( Q , iν n ) + ¯ J α Q . (28)The Weiss field is specified by χ − α ( Q , iν n ) = M α ( Q , iν n ) + ¯ J α Q − /χ α loc ( Q , iν n ) , (29)with the self consistent equation χ α loc ( Q , iν n ) = N c N X ˜ q χ α (˜ q + Q , iν n ) . (30)Lastly, the cluster action is now diagonal in cluster mo-mentum, which leads to S C = S C − Z β dτ dτ ′ X K ,σ c † K σ ( τ ) G − , K ( τ − τ ′ ) c K σ ( τ ′ ) − Z β dτ dτ ′ X Q ,α S α Q ( τ ) χ − α, Q ( τ − τ ′ ) S α − Q ( τ ′ ) , (31)where the isolated cluster action [in equation(14)] is written in cluster momentum S C = R β dτ [ U P Q n Q ↑ ( τ ) n − Q ↓ ( τ ) + P Q ,α ¯ J α Q S α Q ( τ ) S α − Q ( τ )].These equations are similar to those of ref. 32, wherethe EDMFT approach was adopted through a Hubbard-Stratonovich transformation that decouples the inter-siteinteraction terms, and then the DCA approach is applied. C. Momentum dependence of the latticesusceptibility
In both the real space and momentum space formula-tions, it is seen that the momentum dependence of thedynamical lattice spin susceptibility reflects the momen-tum dependence associated with the inter-site interac-tion J ij . This feature is similar to what happens in theEDMFT , with the advantage that the inter-siteinteractions that give rise to the momentum dependenceof the dynamical susceptibility also appear in the self-consistent dynamical equations.It is useful to stress that the momentum dependence ofthe lattice susceptibility in the paramagnetic phase doesnot reflect that of the bare particle-hole bubble . Thisis to be contrasted with the standard DMFT and clustergeneralizations, where the bare particle-hole bubble isresponsible for the momentum dependence of the latticesusceptibility, which we discuss in detail in the followingsection. III. MAGNETIC ORDER
Before we discuss the cluster generalization of symme-try broken phases within EDMFT, we find it very usefulto review the different schemes used to introduce mag-netic order in the single site case. In the context of theEDMFT, there are two ways to introduce magnetic or-der into the system, namely whether or not the magneticorder parameter polarizes the single particle Weiss field G − ,σ (see ref. 37 for details). It has been shown thatallowing G − ,σ to polarize amounts to keeping the particlehole bubble contribution, [ χ ph ( q , ω )] − − [ χ ph , loc ( ω )] − ,to the spin susceptibility (where the particle hole bub-bles are constructed using the full lattice and localGreens function respectively obtained with DMFT andthe brackets [ . . . ] denote a matrix form ). Within thecontext of DMFT, such a term will exist due to the dis-tinction between “normal” and “special” q ’s; cf. ref. 12.However, due to promoting J ij to the same level as t ij within the EDMFT, there are no special q ’s allowed,since this would make J ( q ) ∼ O ( √ d ) which would di-verge in the large d limit. The absence of any special q ’simplies [ χ ph ( q , ω )] − = [ χ ph , loc ( ω )] − , and the particlehole bubble contribution vanishes . Keeping the parti-cle hole bubble within the EDMFT amounts to doublecounting contributions from the spin-spin interaction .In the following section, we focus on the clusterEDMFT case. This is to be contrasted with DCA,which parallels DMFT and therefore retains the distinc-tion between special and generic ˜ q ’s and a similar par-ticle hole contribution to the spin susceptibility, namely[ χ ph (˜ q , ω )] − − [ χ ph , loc ( ω )] − (here the particle hole bub-ble is constructed with the full single particle latticeand local cluster Greens function respectively, obtainedwithin DCA and the bold denotes matrices in clustermomentum) . As in EDMFT, the cluster EDMFT pro-motes δJ to the same level as δt . Generalizing theEDMFT argument to the cluster case we conclude thatthere should only be generic ˜ q ’s. This amounts to not al-lowing the single particle Weiss field G − σ to polarize (i.e.is σ independent), and is equivalent to the suppressionof the particle hole bubble contribution.We now consider magnetic order with an ordering wavevector q = q or ≡ ˜ q or + Q or within the channel α = λ with λ = 0. The cluster chosen must be large enoughto accommodate the type of magnetic order under con-sideration, for example a four site cluster can describethe collinear AF order with q or = (0 , π ) whereas a twosite cluster can only treat either ferro- or AF order. Thisthen implies that the order pattern within each clustermust be the same and therefore ˜ q or = 0.We will consider the equations in real space and mo-mentum space consecutively. After separating J αij intointer- and intra-cluster parts we treat the cluster in-teractions exactly and normal order the interaction be-tween clusters, via S αi =: S αi : + h S αi i . All of the pre-vious steps apply, but now we perform the locator ex- pansion in the normal ordered interaction between clus-ters δJ αXY ( r i − r j ) : S α r i X :: S α r j Y :. This corresponds toadding an additional term to the cluster action S C → S C − Z β dτ X X h X loc S λX ( τ ) (32)and the local magnetic field is determined self consis-tently from h X loc = − X Y h δJ λXY (˜ q or = 0) + χ − λ,XY ( iν n = 0) i M Y , (33)where M Y = h S λY i C and the average is over the clusteraction.In momentum space, this approach amounts to addingto the action S C → S C − Z β dτ h loc S λ − Q or ( τ ) . (34)Now the local field is given by h loc = − (cid:2) δJ λ Q or + χ − λ ( Q or , iν n = 0) (cid:3) M (35)where M = h S λ Q or i C and in this case the inter-clusterinteraction is δJ λ Q or = J λ Q or − ¯ J λ Q or where we have coarsegrained the cluster interaction as described previously.It is useful to note that, in the limit of no dynamicalWeiss field, the mean field equations for the self consis-tent field h loc reduce to that of cluster Weiss mean fieldtheory. Here, the dynamical Weiss field renormalizes thestatic field due to the dynamical interactions mediatedby χ − λ . A. Alternate Derivation of Spin Susceptibility
In the following section, we use the self consistentequations that incorporate magnetic order to provide analternative way of deriving the lattice spin susceptibil-ity at an ordering wave vector Q or , which we define as χ z ( Q or , iν n ) ≡ χ or ( iν n ). This approach will also provideinsight into the way a superconducting long range ordercan be incorporated in C-EDMFT. In order to treat anordering wave vector more naturally, we will consider themomentum space formalism.In order to ease the notation in the following subsectionwe use a subscript to denote each Matsubara frequency,e.g. S αi,n ( iν n ) = S αi,n . First we observe that if we includea dynamic source field for each Matsubara frequency iν n ,we must normal order every frequency modes instead ofthe static mode only, by writing S αi,n =: S αi,n : + h S αi,n i , asa result equation (34) will become S C → S C − β X n h loc ,n S λ − Q or , − n , (36)with h loc ,n = − (cid:2) δJ λ Q or + χ − λ ( Q or , iν n ) (cid:3) h S λ Q or ,n i .We introduce an additional term, S L , to the latticeaction which couples the λ = z component of the spinoperators at wave vector ± Q or to a dynamic source field S L = − β X n (cid:0) h − Q or , − n S z q = Q or ,n + h Q or ,n S z q = − Q or , − n (cid:1) . (37)Mapping this into the cluster action, and assuming thatthe Weiss fields cannot be polarized (due to the absenceof any special ˜ q ’s), we end up adding only one extra term S to equation (34), namely S = − β r N c N X n (cid:0) h − Q or , − n S z Q or ,n + h Q or ,n S z − Q or , − n (cid:1) (38)Here S z Q or , which is defined in the cluster, need to be dis-tinguished from S z q = Q or , which is defined on the lattice,since we have S z Q or = (1 / √ N c ) P K c † K + Q or α ( σ zαβ / c K β and S z q = Q or = (1 / √ N ) P k c † k + Q or α ( σ zαβ / c k β respec-tively.The self consistency between the lattice and the clusterquantities implies the following. h S z Q or ,n i C = r N c N h S z q = Q or ,n i L , (39)Here h . . . i C and h . . . i L denote averaging over clus-ter action of equation (36) and the lattice action withthe additional source field term S L and S respec-tively. To stress that the expectation value h S z Q or ,n i C is calculated self consistently we define h S z Q or ,n i C ≡ f n ( h S z Q or ,n i C , h Q or ,n , h − Q or , − n ).Differentiating both sides of equation (39) with respectto h Q or ,n at h ± Q or , ± n = 0, the right-hand side gives thelattice correlation function, d h S z ( q = Q or , iν n ) i L dh Q or ,n = 1 β h : S z − Q or , − n :: S z Q or ,n : i L = χ or ( iν n ) (40)Whereas the left-hand side gives the cluster correlationfunction, using df n dh Q or ,n = ∂f n ∂ h S z Q or ,n i C d h S z Q or ,n i C dh Q or ,n + ∂f n ∂h Q or ,n (41)and, ∂f n ∂h Q or ,n = r N c N β h : S z − Q or , − n :: S z Q or ,n : i C = r N c N χ loc , or ( iν n ) (42) ∂f n ∂ h S z Q or ,n i C = dh loc ,n d h S z Q or ,n i C ∂f n ∂h loc ,n (43)= − (cid:2) δJ λ Q or + χ − λ ( Q or , iν n ) (cid:3) χ loc , or ( iν n ) where we have defined χ z loc ( Q or , iν n ) ≡ χ loc , or ( iν n ) asthe local spin susceptibility in the ordering channel Q or .We also have d h S z Q or ,n i C /dh Q or ,n = p N c /N χ or ( iν n ) be-cause of eq. (39) and eq. (40). Finally, we can solve for χ or ( iν n ) by combining eq.(39) through eq.(44)1 /χ or ( iν n ) = χ − z ( Q or , iν n ) + δJ z Q or + 1 /χ loc , or ( iν n ) . (44)We immediately recognize this result as the C-EDMFTself-consistent equation for the spin susceptibility at theordering wave vector Q or and spin component z. Mostimportantly, this result indicates that a non-polarizedWeiss field is consistent with the C-EDMFT treatment.We will show in the appendix how the additional particlehole bubble contribution to the expression for χ or , for-bidden by the absence of special ˜ q ’s, is introduced if weallow the Weiss field to be polarized. These considera-tions have important implications for the incorporationof superconductivity into the formalism, which we turnto in the following section. IV. SUPERCONDUCTIVITY
We now apply the approach to the study of supercon-ductivity, with the pairing driven by the spin-spin inter-action in equation (1). We first focus on an AF Isinginteraction with J zij ≡ J ij > J αij to zero. The AF interaction J ij favors pairing of elec-trons with opposite spins; a parallel consideration can bemade for the case of ferromagnetic interactions, whichfavors pairing between electrons of the same spins. Wethen discuss the case of a full AF Heisenberg interaction,where J aij ≡ I ij > a = 1 , , q ’s implies we should not include a contribution fromthe particle particle bubble in the susceptibility. Thisamounts to not allowing the conduction electron bandto become “polarized” by a finite superconducting orderparameter. In the appendix, we also discuss what hap-pens when the conduction electrons are allowed to bepair-polarized. A. Ising Spin Interaction
Conceptually, we would like to keep the strong intersite interactions (that give rise to the dynamic bosonicbath) while promoting a single mode in the static pairingchannel in order to give it the chance to condense. Wedo so, by singling out the static, attractive pairing in-teraction between the paring operators ˆ∆ † iσj ¯ σ and ˆ∆ iσj ¯ σ [defined as ˆ∆ † iσj ¯ σ ( τ ) = c † iσ ( τ ) c † j ¯ σ ( τ )]. We rewrite thespin-spin interaction as Z β dτ X h i,j i J ij S zi ( τ ) S zj ( τ )= − X h i,j i ,σ J ij β X ω ˆ∆ † iσj ¯ σ ( iω ) ˆ∆ iσj ¯ σ ( iω )+ X h i,j i β X ω,ω ,ω J ij − δ α, ¯ γ ) c † iα ( iω − iω ) × σ zαβ c iβ ( iω ) c † jγ ( iω + iω ) σ zγδ c jδ ( iω ) (45)= − X h i,j i ,σ J ij β ˆ∆ † iσj ¯ σ ( iω = 0) ˆ∆ iσj ¯ σ ( iω = 0)+ X h i,j i β X ω,ω ,ω J ij − δ ω , − ω δ α, ¯ γ ) c † iα ( iω − iω ) × σ zαβ c iβ ( iω ) c † jγ ( iω + iω ) σ zγδ c jδ ( iω ) , (46)where the repeated indices α, β, γ, δ are summed over.Here in the first step we separated out the pairing inter-action, and then further separated out the static mode ofthe pairing interaction in the second step. We then intro-duce a Hubbard-Stratonovich transformation to decouplethe attractive interaction in the pairing channel.exp X h i,j i ,σ J ij β ˆ∆ † iσj ¯ σ ( iω = 0) ˆ∆ iσj ¯ σ ( iω = 0) = Z D [∆ , ¯∆] exp − X h i,j i ,σ h β | ∆ iσ,j ¯ σ | J ij + Z β dτ ∆ iσ,j ¯ σ ˆ∆ † iσj ¯ σ ( τ ) + ¯∆ iσ,j ¯ σ ˆ∆ iσj ¯ σ ( τ ) i! . (47)We note that if we choose to decouple the first termof equation (45) – instead of that of equation (46) –and then make the approximation that the Hubbard-Stratonovich field is constant as a function of τ (i.e. keep-ing only the static mode of the Hubbard-Stratonovichfield), we would arrive at exactly the RHS of equation(47). Thus we see that by separating out the static modebefore rather than after the Hubbard-Stratonovich de-coupling, the non-static modes of the pairing interactionwill be absorbed in the second term in equation (46).This term is interpreted as the remaining spin-spin inter-action.Now we take the saddle point approximation of ∆ andfollow the steps in section II A to carry out a generalizedcavity construction. Up to additive constants, we obtain the effective cluster action, S C,I = S C,I − X h X,Y i ,σ Z β dτ (cid:16) ∆ XσY ¯ σ ˆ∆ † XσY ¯ σ ( τ ) + h . c . (cid:17) − Z β dτ dτ ′ X X,Y,σ c † Xσ ( τ ) G − ,XY ( τ − τ ′ ) c Y σ ( τ ′ ) − Z β dτ dτ ′ X X,Y S zX ( τ ) χ − ,XY ( τ − τ ′ ) S zY ( τ ′ )+ δS. (48)The saddle point equation leads to an additional self con-sistent equation for the superconducting order parameter∆ cX i σX j ¯ σ = J cX i X j β Z β dτ h ˆ∆ X i σX j ¯ σ ( τ ) i C , where the average is taken with respect to the effectivecluster model in equation (48). The additional term inthe action, δS represents all the modifications in the ef-fective action caused by separating the zero frequencypairing interaction. The exact expression of δS can befound in the Appendix B. We see from the expressionthere that all terms in δS are suppressed by factors of1 / ( Jβ ); therefore, at sufficiently low temperatures in-cluding the quantum critical regime, δS can be safelyneglected. In the following we only consider the low tem-perature limit and make the approximation that δS ≈ G − ,XY and χ − ,XY remain the same as inthe previous sections.It is useful to note that this approach can also be for-mulated in momentum space. Using the DCA Fouriertransform defined in equation (25) amounts to replac-ing J ij by J DCAij = N c /N P ˜ q e i ˜ q · (˜ x i − ˜ x j ) [ J α DCA (˜ q )] X i , X j in the spin channel. In order to treat the pairing channelon the same footing, we use J DCAij starting in equation(46). This then amounts to replacing J cX i X j in equation(49) with [ J cDCA ] X i X j , which is the coarse grained inter-action in cluster momentum ( ¯ J Q ) that is Fourier trans-formed back to real space cluster variables (see ref. 26)[ J cDCA ] X i X j = 1 /N c P Q e i Q · ( X i − X j ) ¯ J Q . In addition, thisleads to periodic boundary condition on the cluster in realspace. B. Heisenberg Spin Interaction
We now consider the case of an AF Heisenberg spinspin interaction I ij . The derivation proceeds in parallelwith the previous Ising case. Here, we separate out thesinglet term with an attractive interaction Z β dτ X h i,j i I ij ~S i ( τ ) · ~S j ( τ )= − X h i,j i I ij β Z β dτ dτ ˆ∆ † ij ( τ ) ˆ∆ ij ( τ )+ X h i,j i ω,ω ,ω I ij β ( σ αµβ σ αγν − δ ω , − ω (2 δ µδ δ βν − δ µβ δ γν )) × c † iµ ( iω − iω ) c iβ ( iω ) c † jγ ( iω + iω ) c jν ( iω ) , (49)we have defined the singlet creation operator betweensites i and j asˆ∆ † ij = 1 √ (cid:16) c † i ↑ c † j ↓ − c † i ↓ c † j ↑ (cid:17) , (50)and its hermitian conjugate ˆ∆ ij = ( c j ↓ c i ↑ − c j ↑ c i ↓ ) / √ † ij ˆ∆ ij , the saddle point equation for ∆ ij now becomes∆ X i X j = I cX i X j β Z β dτ h ˆ∆ X i X j ( τ ) i C . (51)with an effective action, S C,H = S C,H − X h X,Y i Z β dτ (cid:16) ∆ XY ˆ∆ † XY ( τ ) + h . c . (cid:17) − Z β dτ dτ ′ X X,Y,σ c † Xσ ( τ ) G − ,XY ( τ − τ ′ ) c Y σ ( τ ′ ) − Z β dτ dτ ′ X X,Y a =0 S aX ( τ ) χ − a,XY ( τ − τ ′ ) S aY ( τ ′ ) . (52)Again, we have not allowed the single particle Weiss fieldto become polarized from the finite superconducting or-der parameter. The Heisenberg isolated cluster action S C,H , is equation (14) with J αc = 0 for α = 0 and J αc = I c for α = 1 , ,
3. We have also ignored the additional partof the action δS H that is suppressed by a factor of 1 / ( Jβ )(see the appendix for a discussion of the Ising case). Weremark that it is possible to use this formalism to describestates that have both magnetic order and superconduc-tivity by including a finite h loc as described in section III. C. Pairing Susceptibility
In this section we derive the zero momentum latticepairing susceptibility χ SC ( iν n ) ≡ χ latpair ( q = 0 , iν n ) de- fined as χ SC ( iν n ) = 1 N ( z/ X h i,j i ,σ X h k,l i ,λ f ∗ i,j f k,l g ∗ σ ¯ σ g λ ¯ λ × Z β dτ h T τ : ˆ∆ iσj ¯ σ ( τ ) :: ˆ∆ † kλl ¯ λ : i e iν n τ , (53)where N ( z/
2) is the number of bonds in the lattice, with z being the number of nearest neighbors, and f i,j , g i,j arethe pairing symmetry factors 39 in real and spin spacerespectively. We will focus on the case of an Ising spininteraction J ij but this can be easily generalized to thecase of a Heisenberg interaction. We first project thesuperconducting gap onto a particular symmetry channelassuming the gap amplitude ∆ is uniform across eachbond, i.e. ∆ XσY ¯ σ = f XY g σ ¯ σ ∆ (and complex conjugate∆ ∗ XσY ¯ σ = f ∗ XY g ∗ σ ¯ σ ∆ ∗ ), where the phase factor in realspace is given by f XY and that in spin space is g σ ¯ σ , andthey have the property | f XY | , | g σ ¯ σ | = 1 (see ref. 39).The procedure is similar to that described in section III Afor magnetism, although here we have to project intoa particular symmetry channel (through f ij , g σ ¯ σ ). Wearrive at the following expression for the zero momentumlattice pairing susceptibility. χ SC ( iν n ) = 11 /χ locpair ( iν n ) − J SC , (54)where we have defined the effective pairing interaction1 /J SC = N b P h X i ,X j i ,σ /J cX i X j , and the cluster pairingsusceptibility χ locpair ( iν n ) ≡ χ locpair ( Q = 0 , iν n ), where χ locpair ( iν n ) = 1 N b X h X,X ′ i ,σ X h Y,Y ′ i ,λ f ∗ X,X ′ f Y,Y ′ g ∗ σ ¯ σ g λ ¯ λ × Z β dτ h T τ : ˆ∆ XσX ′ ¯ σ ( τ ) :: ˆ∆ † Y λY ′ ¯ λ : i C e iν n τ (55)where N b = P h X,Y i is the number of individual bonds inthe cluster. Previous treatments of two particle responsefunctions in various cluster theories demands much morecomputational effort because it involves the inversion ofthe Bethe-Salpeter equation, which is in principle a ma-trix equation of infinite dimension in the space of threewave vectors and frequencies. In our approach, the clus-ter susceptibilities completely determine the correspond-ing lattice quantities.We note that this can also be formulated in momentumspace after the symmetry factors are first coarse grainedin momentum space ¯ f ( K ) = N c /N P ˜ k f (˜ k + K ) (where f ( k ) is the Fourier transform of f ij ) and then Fouriertransformed to real cluster space ¯ f X i X j = [ f DCA ] X i X j =1 /N c P K e i K · ( X i − X j ) ¯ f K . In this case, the coarse grainedsymmetry factors no longer have to satisfy | ¯ f X i X j | = 1,and as a result the effective pairing interaction becomes1 /J SC = N b P h X i ,X j i ,σ | ¯ f X i X j | /J cX i X j .0In the appendix, we contrast this expression with theresult of allowing the single particle Weiss field to acquireanomalous terms. In this case, we find additional con-tributions, corresponding to the particle particle bubblewhich only contributes if special ˜ q ’s exist. V. EFFECTIVE CLUSTER MODELS
The formalism we have discussed so far also applies toa variety of strongly correlated problems aside from theHamiltonian we have been considering in equation (1).One such example is a two band model, namely the An-derson lattice Hamiltonian appropriate for the descrip-tion of heavy fermion materials . The model describes aband of conduction electrons hybridized with a band oflocalized, highly correlated f -electrons and is defined as H AL = X h i,j i ,σ t ij ( c † iσ c jσ + h . c) + X i ( ǫ f n fi + U n fi ↑ n fi ↓ )+ X i,σ (cid:16) V c † iσ f iσ + h . c . (cid:17) + X h i,j i J ij S zfi S zfj . (56)As usual, we have explicitly included an Ising RKKYinteraction between the f -electron spins. The RKKYand Kondo interactions compete, and tuning their ratiocan lead to a quantum phase transition between a heavyFermi liquid and an antiferromagnet. In certain cases,the QCP is of the SDW type, where the heavy quasi-particles remain intact across the transition and undergoa SDW transition. In other cases, the SDW descriptionfails, and the physics of critical Kondo destruction comesinto play.Focusing on the normal state properties, applying theextended dynamical cluster theory of section II B in mo-mentum space we arrive at the effective cluster action S AL C = Z β dτ X Q U n f Q ↑ ( τ ) n f − Q ↓ ( τ ) + ǫ f n f Q ( τ ) (57)+ Z β dτ X Q ¯ J Q S zf Q ( τ ) S zf − Q ( τ ) − Z β dτ dτ ′ X K ,σ f † K σ ( τ ) G − , K ( τ − τ ′ ) f K σ ( τ ′ ) − Z β dτ dτ ′ X Q S zf Q ( τ ) χ − , Q ( τ − τ ′ ) S zf − Q ( τ ′ ) . For the case of 2-d AF exchange fluctuations, the di-vergence of the spin susceptibility at the ordering wavevector implies [through the self consistent equation (30)]that the local spin susceptibility with cluster momentum Q or is also (logarithmically) divergent . This leads to aspin Weiss field associated with the critical momentumchannel χ ( Q AF , iν n ) that develops a sub-ohmic spectraldensity Im χ ( Q AF , ω + 0 + ) ∼ ω s . Based on universal-ity, we can regard the effective cluster model in equation (58) with a sub-ohmic density of states for the orderedchannel, as an effective model that contains both Kondodestruction and pairing correlations induced from AF ex-change interactions. For the simplest case of N c = 2, andkeeping only the critical degrees of freedom [i.e. only theWeiss field in the ordered channel χ ( Q AF , iν n )] we ar-rive at a simplified model to study pairing correlationsnear a Kondo destroyed QCP. This model was proposedand solved in ref. 40 using a combination of continuoustime quantum Monte Carlo and the numerical renormal-ization group. It was shown that the cluster pairing sus-ceptibility χ locpair is enhanced at the Kondo destructionQCP. It will be important to consider the full self consis-tent solution and determine if local quantum criticalitywith Kondo destruction survives finite size cluster cor-rections and if so, how large χ SC is near the local QCP.Another strongly correlated problem of central interestis the extended Hubbard model, which adds to the stan-dard Hubbard model an inter-site density-density inter-action. We note in passing that incorporating a spin or-bit coupling can lead to topologically non-trivial groundstates in the presence of interactions . The extendedHubbard model is thought to be the appropriate model todescribe certain types of organic superconductors, stripecharge order in the cuprates and different types of Motttransitions. In addition, when the inter-site interactionis attractive ( V ij < . It is defined as H = X i,j,α,β t αβij ( c † iα c jβ + h . c) + U X i n i ↑ n i ↓ + X h i,j i V ij : n i :: n j : , (58)where we have generalized the tight-binding hoppingterms to a spin dependent hopping matrix t αβij ; this formis sufficiently general to allow for a spin orbit couplingterm and keep i, j that are not necessarily nearest neigh-bors. Tuning the ratio of U/V can lead to a varietyof quantum phase transitions between a Fermi liquid, aband insulator, and a Mott insulator. Whereas tuningthe strength of the spin orbit coupling can lead to topo-logical transitions between a band and topological insu-lator. Applying the formalism of section II A leads to the1effective cluster action S EH C = Z β dτ X X ∈ C U n X ↑ ( τ ) n X ↓ ( τ )+ Z β dτ X h X,Y i V c : n X ( τ ) :: n Y ( τ ) : − Z β dτ dτ ′ X X,Y,µ,ν c † Xµ ( τ ) G − ,XY µν ( τ − τ ′ ) c Y ν ( τ ′ ) − Z β dτ dτ ′ X X,Y : n X ( τ ) : χ − ,XY ( τ − τ ′ ) : n Y ( τ ′ ) :(59)Studying the different types of quantum phase transi-tions in the effective cluster model alone [i.e., withoutimplementing the self-consistency] can lead to significantnew insights regarding behavior of the QCPs in the lat-tice problem. In other words, the effective cluster modelof equation (59) serves as a simplified model that canprovide insights into different types of Mott transitions,superconductivity and even interacting topological phasetransitions. VI. C-EDMFT SOLUTION METHODS
The self consistent cluster dynamical mean field equa-tions form a set of highly non-linear equations. Theirsolution requires an accurate and reliable solution of thecluster impurity model, which is iteratively solved. Thereare analytical tools and computational methods that aresuitable for studying the cluster impurity models. From acomputational perspective, solving the cluster impuritymodels represents the most extensive efforts of solvingthe self consistent equations. In the presence of a phasetransition, the number of iterations necessary to solve theequations can become quite large due to a “critical slow-ing down”. In this case, it is very useful to use mixingtechniques that are well known in the context of den-sity functional theory, in order to reduce the number ofiterations that are needed to achieve self consistency .The cluster model can be solved using, for examplethe exact diagonalization (ED), the numerical renormal-ization group (NRG), the density matrix renormalizationgroup (DMRG) and quantum Monte Carlo methods.Including bosonic baths in diagonalization based tech-niques can be done, but it requires a truncation of theinfinite bosonic Hilbert space. In addition, the presenceof numerous bosonic baths can make such an approachquite computationally demanding. However, this doesnot rule out ED and NRG techniques, provided there issome physical intuition of which bosonic bath is goingto drive the system through a quantum phase transition.Then retaining this single bosonic bath is necessary tocapture the critical universal properties, while the other baths serve to renormalize the effective model parame-ters.The recently developed continuous time quantumMonte Carlo (CT-QMC) has the advantage that thebosons are traced out and the algorithm is numeri-cally exact. These methods have been adapted to treatscalar bosonic baths that interact with the impuritiescharge or spin degrees of freedom. The case of avector bosonic bath has also been studied.Recently, the CT-QMC has been generalized to a twoimpurity model in the presence of a single bosonicbath. As the additional bosonic baths that arise inthe C-EDMFT treatment commute with each other, themethod in Ref. 40 can naturally be used to solve the C-EDMFT equations to high accuracy. Generalizations tolarger cluster is still possible within such a framework,provided the algorithm doesn’t suffer from a sign prob-lem arising from the fermionic degrees of freedom. Asdescribed in section V, for the minimal case of the two-impurity C-EDMFT approach to the periodic Andersonmodel, in the vicinity of an antiferromagnetic QCP, itwould be adequate to keep only a single bosonic baththat is coupled to the staggered combination of the local-moment spins in the cluster. The method of Ref. 40 thenmakes it feasible to study the quantum critical behaviorof the periodic Anderson model. VII. DISCUSSION
In this work we have derived a cluster EDMFT for-mulation by a locator expansion about a dressed clusterlimit. An alternative derivation can also be done using aBaym-Kadanoff functional . In this case the generat-ing functional of the grand potential is (focusing on oneof the two particle channels)Γ BK [ G , χ ] = Tr [log( G )] − Tr (cid:2) G ( G − − G − ) (cid:3) −
12 Tr [log( χ )] + Tr (cid:2) χ ( χ − − χ − ) (cid:3) + Φ[ G , χ ] , (60)with the stationary conditions δ Γ /δ G = δ Γ /δ χ = 0.The self energies are then given by δ Φ[ G , χ ] /δ G = Σ , δ Φ[ G , χ ] /δ χ = M . Within the cluster approximation,the self energies are calculated from an effective clustermodel. Therefore, the approximation that the functionalΦ is only a functional of G loc and χ loc leads to the clusterEDMFT equations. Therefore, analogous to the EDMFTapproach , the C-EDMFT approach is also conserv-ing .As we have discussed in the introduction, a main suc-cess of EDMFT compared to DMFT has been in its so-lution to the Kondo lattice Hamiltonian and the theoryof local quantum criticality . This rests ontreating the RKKY interactions between the local mo-ments in a dynamical fashion, through the self-consistentbosonic bath. The self-consistent solution yields an un-conventional QCP with a dynamical spin susceptibility2satisfying E/T scaling and anomalous dynamical criticalexponent, similar to what has been observed in quantum-critical heavy-fermion metals . For this result, it isimportant that the mapped Bose-Fermi Kondo model it-self contains a Kondo-destruction QCP . Studies ofthe two impurity Bose-Fermi Anderson model in Ref. 40have shown that a Kondo-destruction QCP persists, withsimilar scaling properties for the staggered dynamicalspin susceptibility. This raises the prospect that the C-EDMFT approach discussed here will be able to studythe interplay between the unconventional quantum crit-ical normal state and superconductivity. As discussedearlier in the paper, such an interplay is important to theunderstanding of heavy fermion superconductivity. Wenote that the dynamical treatment of the RKKY interac-tions differentiates EDMFT and C-EDMFT from DMFT,which does not treat the RKKY interactions beyond astatic mean field approximation, as well as the cluster-DMFT , which treats the RKKY interactions within thecluster in a dynamical fashion, but does not do so forthose interactions outside of a chosen cluster. VIII. CONCLUSIONS
In this manuscript we have presented a new clusterextended dynamical mean field approach. We have de-veloped the equations in both real and momentum space,incorporating magnetic order and superconductivity. Wehave also determined the form of the superconductingcorrelation functions in the normal state. We have thenused the formalism to derive effective cluster models thatare relevant to heavy fermion metals and Mott-Hubbardsystems. In particular, this formulation for unconven-tional superconductivity is expected to play a central roleto the study of quantum critical heavy fermion metals.
Acknowledgements.
We would like to thank KevinIngersent and Lili Deng for useful discussions. Thiswork has been supported by the NSF Grant No. DMR-1309531, the Robert A. Welch Foundation Grant No. C-1411 (J.H.P., A.C. & Q.S.), the Alexander von Hum-boldt Foundation (Q.S.), and the East-DeMarco fellow-ship (J.H.P.). One of us (Q. S.) acknowledges the hospi-tality of the Aspen Center for Physics (NSF Grant No.1066293), the Institute of Physics of Chinese Academy ofSciences, and the Karlsruhe Institute of Technology.
IX. APPENDIX A
In this appendix we derive the expression for δS , de-fined in equation (48). We use the generalized cavityapproach to derive the effective action S C , S C = S o − ∞ X n =0 n ! h ( − ∆ S ) n i ( o )connected (61)We have defined ∆ S as the part of the action that con-nects cluster o to its neighbors. The expectation value h . . . i ( o ) is taken with respect to the action S ( o ) , whichis defined as the action with cluster o removed, and S o corresponds to the action of the isolated cluster o . Sim-ilar to DMFT, after rescaling δt and δJ , in the limit oflarge coordination, only the n = 1 and n = 2 termscontribute. In the remainder of this appendix, we omitthe “connected” label and take cluster o to be equal to r = o .We separate ∆ S into three pieces∆ S = Z β dτ X r i ,XY δJ ( r − r i ) XY S z r X ( τ ) S z r i Y ( τ )+ Z β dτ dτ X r i ,X,Y,σ δJ ( r − r i ) XY β × c † r Xσ ( τ ) c r Xσ ( τ ) c † r i Y ¯ σ ( τ ) c r i Y ¯ σ ( τ )+ Z β dτ X iσ c † r Xσ ( τ ) δt ( r − r i ) c r i Y σ ( τ ) ≡ ∆ S + ∆ S + ∆ S , (62)Since there is no interference between the one particlesector and two particle sector in the expansion, we canignore ∆ S in the calculation of δS . We first note that h ∆ S i ( o ) vanishes, since we are considering a case withno magnetic order. The expectation value h ∆( S ) i ( o ) gives the standard expression for the spin Weiss field (seeequation (16) of the main text). The rest of the termscombined give δS . δS = h ∆ S i ( o ) − h ∆ S ∆ S i ( o ) − h (∆ S ) i ( o ) (63)The expression for each term is listed below, h ∆ S i ( o ) = − X r i ,X,Y,σ δJ ( r − r i ) XY β Z β dτ dτ c † r Xσ ( τ ) h c r i Y ¯ σ ( τ ) c † r i Y ¯ σ ( τ ) i ( o ) c r Xσ ( τ ) , (64) h T τ ∆ S ∆ S i ( o ) = X r i , r j X X,Y,Z,W X σ,α δJ ( r − r i ) XY δJ ( r − r j ) ZW β Z β dτ dτ dτ S z r X ( τ ) c † r Zσ ( τ ) c r Zσ ( τ ) ×h T τ c † r i Y α ( τ ) c r i Y α ( τ ) c † r j W ¯ σ ( τ ) c r j W ¯ σ ( τ ) i ( o ) σ zαα (65)3 h (∆ S ) i ( o ) = X r i , r j X X,Y,Z,W X α,β δJ ( r − r i ) XY δJ ( r − r j ) ZW β Z β dτ dτ dτ dτ c † r Xα ( τ ) c r Xα ( τ ) c † r Zβ ( τ ) c r Zβ ( τ ) × h T τ c † r i Y ¯ α ( τ ) c r i Y ¯ α ( τ ) c † r j W ¯ β ( τ ) c r j W ¯ β ( τ ) i ( o ) (66)The effect of h ∆ S i ( o ) is to modify the one particle Weissfield. The other two terms modify the interactions bygenerating a general two particle interaction which isnonlocal in time. In contrast to the standard Weiss fields[as in equation (13)], each term contains an additionalfactor of 1 /β , and the last term carries a factor of 1 /β .This implies each term is suppressed by at least a fac-tor of 1 / ( Jβ ) relative to the standard Weiss fields, and1 / ( Jβ ) can serve as a small parameter for sufficiently lowtemperatures. In the zero temperature limit, all threeterms vanish and do not affect any of the quantum criti-cal properties. X. APPENDIX B
Here we discuss when happens when the single particleWeiss fields become polarized by a finite magnetic or su-perconducting order parameter. In the following, we fo-cus on the momentum space formulation. We show thatthe additional terms (particle hole or particle particlebubble contributions) come from a polarized single par-ticle Weiss field. Our considerations here parallel thosefor EDMFT discussed in ref. 37.
A. Magnetism
We first discuss the case of magnetic order, with anorder parameter finite in the z -direction. Allowing themagnetic order parameter to polarize the single particleWeiss field we obtain an effective cluster action S C = S C − Z β dτ h loc S z Q or ( τ ) − Z β dτ dτ ′ X K ,σ c † K σ ( τ ) G − , K σ ( τ − τ ′ ) c K σ ( τ ′ ) − Z β dτ dτ ′ X Q ,α S α Q ( τ ) χ − α, Q ( τ − τ ′ ) S α − Q ( τ ′ ) , (67)where now the single particle Weiss field G − , K σ is differ-ent for different spin components and h loc is given byequation (35). As a result, the expression for the latticespin susceptibility has changed, focusing on the staticspin susceptibility we obtain˜ χ ( Q or , iν n = 0) = 1 + δI h /χ or ( iν n = 0) − δI M (68) where δI M is δI M = Z β dτ dτ ′ X K ,σ h T τ c † K σ ( τ ) c K σ ( τ ′ ) S z Q or i C × ∂∂M G − , K σ ( τ − τ ′ )[ χ loc ( iν n = 0)] − . (69)In addition, we have defined δI h , which is equivalent to δI M , with M replaced with p N c /N h Q or ,n =0 and thederivative is evaluated at h Q or = 0.We can compare our result to that of DCA , and con-clude that the additional contribution δI comes from theparticle hole contribution associated with the special ˜ q ’s. B. Superconductivity
We now consider allowing the finite superconductingorder parameter to “polarize” the single particle Weissfields, i.e. introduce anomalous terms. We take the sad-dle point approximation of ∆ and carry out a generalizedcavity construction . Up to additional constants, we ob-tain the effective cluster action S C,I = S C,I − Z β dτ dτ ′ Ψ † ( τ ) G − ( τ − τ ′ )Ψ( τ ′ ) − Z β dτ dτ ′ X X,Y S zX ( τ ) χ − ,XY ( τ − τ ′ ) S zY ( τ ′ )+ δS (70)where the Ising isolated cluster action S C,I is defined inequation (14) with J αc = 0 for α = z . We have definedthe Nambu spinor Ψ † = ( c † X ↑ , ...c † X Nc ↑ , c X ↓ , ..., c X Nc ↓ ) , and we adopt the Nambu-Gorkov formalism: G Ψloc ( τ ) = −h T τ Ψ( τ )Ψ † i . We use the subscript Ψ to label a 2x2matrix in Nambu space where each element is a matrixin cluster indices. Now the single particle Weiss field hasadditional anomalous terms G − ( iω n ) = G − ( iω n ) F − ( iω n ) F ∗− ( iω n ) − G − T ( − iω n ) ! . (71)The spin Weiss field χ − ,XY ( τ − τ ′ ) assumes the same formas equation (16) when expressed in terms of the cavitycorrelation function. As we have discussed in the maintext, the additional term in the action, δS representsall the modifications in the effective action caused byseparating the zero frequency pairing interaction. Again,4in the following, we only consider the low temperaturelimit and make the approximation that δS ≈ G − , weperform a cumulant expansion in inter cluster interac-tions. Similar to the inter-site interactions, we begin byseparating ∆ iσ,j ¯ σ also into intra and inter cluster parts ∆ σ ¯ σ ( r i − r j ) = ∆ cσ ¯ σ δ r i , r j + δ ∆ σ ¯ σ ( r i − r j ). This natu-rally arises from a locator expansion in δ J after we rescale δ ∆ σ ¯ σ ( r i − r j ) by δ J ( r i − r j ), which leads to a single par-ticle Greens function in Nambu space G Ψ (˜ k , iω n ) = h C − G Ψ ( iω n ) − δ T Ψ (˜ k ) i − . (72)The single particle cumulant is now C − G Ψ ( iω n ) = ( iω n + µ ) τ ⊗ − T c Ψ − Σ Ψ ( iω n ) (73)where τ is the z -Pauli matrix in Nambu space, togetherwith the following definition for the generalized intracluster hopping matrix T c Ψ = (cid:18) t c − ∆ c ↑↓ − ¯ ∆ c ↓↑ − t Tc (cid:19) , (74)and the inter cluster hopping matrix δ T Ψ (˜ k ) = (cid:18) δ t (˜ k ) − δ ∆ ↑↓ (˜ k ) − δ ¯ ∆ ↓↑ (˜ k ) − δ t T (˜ k ) (cid:19) . (75)Now the generalized one particle Weiss field takes theform G − ( iω n ) = Σ Ψ ( iω n ) + G − ( iω n ) (76)and the self consistency condition becomes G Ψloc ( iω n ) = N c N X ˜ k G Ψ (˜ k , iω n ) . (77)The superconducting order parameter is then determinedself consistently from the saddle point value:∆ cX i σX j ¯ σ = J c,X i X j β Z β dτ h ˆ∆ cX i σX j ¯ σ ( τ ) i C (78)In principle, for a real space cluster scheme such asCDMFT, the translation invariance inside the cluster is broken since the couplings on the boundary are nowtreated different than those inside the cluster. Thus, theorder parameter ∆ cX i σX j ¯ σ may in principle take differentvalue on different bonds. We could obtain an estimateof the pairing amplitude ∆ by averaging ∆ cX i σX j ¯ σ overeach bond (note that the 2 × δ ∆ σ ¯ σ ( r i − r j ) is thenconstructed using the translation invariance of the lat-tice. Two major differences between these self consistentequations and those in CDMFT are the explicit ap-pearance of the order parameter and the fact that theinter-site magnetic interaction is driving the supercon-ducting pairing. C. Pairing Susceptibility
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