A Comparison of Two Quantum Cluster Approximations

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A Comparison of Two Quantum Cluster Approximations

A Comparison of Two Quantum Cluster ApproximationsTh. A. Maier and M. JarrellUniversity of Cincinnati, Cincinnati OH 45221, USAWe provide microscopic diagrammatic derivations of thethe Molecular Coherent Potential Approximation (MCA) andDynamical Cluster Approximation (DCA) and show thatboth are Φ-derivable. The MCA (DCA) maps the latticeonto a self-consistently embedded cluster with open (periodic)boundary conditions, and therefore violates (preserves)the translational symmetry of the original lattice. As a consequenceof the boundary conditions, the MCA (DCA) convergesslowly (quickly) with corrections O(1/L c) (O(1/L 2 c)),where L c is the linear size of the cluster. These analyticalresults are demonstrated numerically for the one-dimensionalsymmetric Falicov-Kimball model.Introduction One of the most active areas in condensedmatter physics is the search for new methods totreat disordered and correlated systems. In these systems,especially in three dimensions or higher, approximationswhich neglect long ranged correlations are generallythought to provide a reasonable first approximationfor many properties.Perhaps the most successful of these methods are theCoherent Potential Approximation (CPA) [1] and theDynamical Mean Field Approximation (DMFA) [2–5],for disordered and correlated systems, respectively. Althoughthese approximations have different origins, theyshare a common microscopic definition. Both the DMFA[3] and the CPA [6] may be defined as theories which completelyneglect momentum conservation at all internaldiagrammatic vertices. When this principle is applied,the diagrammatic expansion for the irreducible quantitiesin each approximation collapse onto that of a selfconsistentlyembedded impurity problem.Many researchers have actively searched for a techniqueto restore non-local corrections to these approaches.Here, we discuss just two approaches which are fullycausal and self-consistent: the Molecular Coherent PotentialApproximation (MCA) [7,8] and the DynamicalCluster Approximation (DCA) [10–12,6]. Recently theCellular Dynamical Mean Field Approach [9] was proposedfor ordered correlated systems, while the MolecularCoherent Potential Approximation has traditionallybeen applied to disordered systems. Since both methodsshare a common microscopic definition we use the termMCA to refer to both techniques in the following.While the MCA is traditionally defined in the real spaceof the lattice, the DCA is traditionally defined in its reciprocalspace. In the MCA, the system lattice is splitinto a series of identical molecules. Interactions betweenthe molecules are treated in a mean-field approximation,while interactions within the molecule are explicitly accountedfor. In the DCA, the reciprocal space of thelattice is split into cells, and momentum conservation isneglected for momentum transfers within each cell whileit is (partially) conserved for transfers between the cells.These approximations share many features in common:they both map the lattice problem onto that of a selfconsistentlyembedded cluster problem. Both recover thesingle site approximation (CPA or DMFA) when the clustersize reduces to one and become exact as the clustersize diverges. Both are fully causal [7,11], and providedthat the clusters are chosen correctly [6], they maintainthe point group symmetry of the original lattice problem.Here, we provide a microscopic diagrammatic derivationof both the MCA and the DCA, and explore their convergencewith increasing cluster size.Formalism For simple Hubbard-like models, momentumconservation at each vertex is completely describedby the Laue function∆ = ∑ xe ix·(k1+k2−k3−k4) = Nδ k1+k 2,k 3+k 4, (1)where k 1 , k 2 (k 3 , k 4 ) are the momenta entering (leaving)the vertex. Müller-Hartmann [3] showed that theDynamical Mean Field (DMF) theory may be derived bycompletely ignoring momentum conservation at each internalvertex by setting ∆ = 1. Then, one may freelysum over all of the internal momentum labels, and thegraphs for the generating functional Φ and its irreduciblederivatives, contain only local propagators.The DCA and MCA techniques may also be defined bytheir respective Laue functions. Since our object is todefine cluster methods we divide the original lattice of Nsites into N/N c clusters (molecules), each composed ofN c = L D c sites, where D is the dimensionality. We usethe coordinate ˜x to label the origin of the clusters andX to label the N c sites within a cluster, so that the siteindices of the original lattice x = X + ˜x. The points ˜xform a lattice with a reciprocal space labeled by ˜k. Thereciprocal space corresponding to the sites X within acluster shall be labeled K, with K α = n α · 2π/L c andinteger n α . Then k = K + ˜k. Note that e iK·˜x = 1 sincea component of ˜x must take the form m α L c with integerm α .In the MCA, we approximate the Laue function by∆ MC = ∑ Xe iX·(K1+K2−K3−K4+˜k 1+˜k 2−˜k 3−˜k 4) . (2)Thus, the MCA omits the phase factors e i˜k·˜x resultingfrom the position of the cluster in the original latticebut retains the (far less important) phase factors e i˜k·Xassociated with the position within a cluster. In the DCAwe also omit the phase factors e i˜k·X , so that1


∆ DC = N c δ K1+K 2,K 3+K 4. (3)Both the MCA and DCA Laue functions recover the exactresult when N c → ∞ and the DMFA result, ∆ = 1,when N c = 1.If we apply the MCA Laue function Eq. 2 to diagramsin Φ, then each Green function leg is replaced by theMCA coarse-grained Green functionḠ(X 1 , X 2 ; ˜x = 0) =1N 2 ∑K 1 ,K 2˜k 1,˜k 2e i(K1+˜k 1)·X 1G(K 1 , K 2 ; ˜k 1 − ˜k 2 )e −i(K2+˜k 2)·X 2=N 2 cN 2∑˜k 1,˜k 2G(X 1 , X 2 , ˜k 1 − ˜k 2 ) , (4)or in matrix notation for the cluster sites X 1 and X 2ˆḠ = N ∑cĜ(˜k) . (5)N˜kThe summations of the cluster sites X remain to be performed.Note that the inclusion of the phase factors e i˜k·Xin the MCA Laue-function Eq. 2 leads directly to a clusterapproach formulated in real space that violates translationalinvariance. Therefore the Green function is afunction of two cluster momenta K 1 , K 2 or two sites X 1 ,X 2 respectively.If we apply the DCA Laue function Eq. 3, Green functionlegs in Φ are replaced by the DCA coarse grainedGreen functionḠ(K) = N cN∑G(K, ˜k) , (6)˜ksince Green functions can be freely summed over the ˜kvectors within a cell about the cluster momentum K.(We have dropped the frequency dependence for notationalconvenience.) As a result, Φ is a functional of thecoarse grained Green function Ḡ(K) and thus dependson the cluster momenta K only.To establish a connection between the cluster and thelattice we minimize the lattice free energyF = −k B T (Φ c − tr [ΣG] + tr ln [G]) (7)where Φ c is the generating functional calculated with thecoarse-grained propagators, Σ is the lattice self-energyand G is the full lattice Green function. The trace indicatessummation over frequency, momentum and spin.As we have discussed elsewhere, only the compact partof the free energy, Φ, is coarse-grained. F is stationarywith respect to G when δFδG= 0. This happens for theMCA if we estimate the lattice self energy asΣ(K 1 , K 2 ; ˜k 1 − ˜k 2 ) =∑X 1,X 2e −i(K1+˜k 1)·X 1Σ MC (X 1 , X 2 )e i(K2+˜k 2)·X 2. (8)Thus, the corresponding lattice single-particle propagatorreads in matrix notation[Ĝ(˜k, z) = zI − ˆɛ(˜k) − ˆΣ−1MC (z)], (9)where the dispersion ˆɛ(˜k) and self-energy ˆΣ MC (z) arematrices in cluster real space with[ˆɛ(˜k)] X1X 2= ɛ(X 1 − X 2 , ˜k) (10)= 1 ∑e −K·(X1−X2) ɛN K+˜kcKbeing the intracluster Fourier transform of the dispersion.For the DCA, Σ(k) = Σ DC (K) is the proper approximationfor the lattice self energy corresponding toΦ DC . The corresponding lattice single-particle propagatoris then given byG(K, ˜k; z) =1z − ɛ K+˜k− Σ DC (K, z) . (11)Both the MCA and DCA are optimized when we equatethe lattice and cluster self energies. A similar relationholds for two-particle quantities. Thus, with few exceptions[13], only the irreducible quantities on the clusterand lattice correspond one-to-one.The MCA (DCA) algorithm, with steps A→D, followsdirectly: A we first make an initial guess for the clusterself-energy matrix Σ. B This is used with Eqs. 5 and 9(6 and 11) to calculate the coarse-grained Green functionḠ. C The cluster excluded Green function Ĝ = [ ˆḠ −1 +ˆΣ MC ] −1 (G(K) = [Ḡ(K)−1 + Σ DC (K)] −1 ) is defined toavoid overcounting self energy corrections on the cluster.It is used to D compute a new estimate for the clusterself-energy which is used to reinitialize the process. Onceconvergence is reached, the irreducible quantities on thecluster may be used to calculate the corresponding latticequantities.In order to compare the character of the two differentcluster approaches as a function of the cluster size N c itis instructive to rewrite the corresponding coarse grainedGreen-functions Eqs. 5 and 6 to suitable forms by makinguse of the independence of the self-energy Σ on the integrationvariable ˜k. For the MCA coarse grained Greenfunction we find[ˆḠ(z) = zI − ˆɛ o − ˆΣ MC (z) − ˆΓ−1MC (z)], (12)with the “cluster-local” energy ˆɛ o = N c /N ∑˜k ˆɛ(˜k). Forthe DCA we obtain a similar expressionḠ(K, z) = [z − ¯ɛ K − Σ DC (K, z) − Γ DC (K, z)] −1 , (13)with the coarse grained average ¯ɛ K = N c /N ∑˜k ɛ(K, ˜k).The hybridization functions ˆΓ MC/DC (z) describe thecoupling of the cluster to the mean field representing theremainder of the system.2


The behavior of Γ for large N c is important. For theMCA, Γ averaged over the cluster and frequency¯Γ MC = 1 ∑( ) 2DΓ MC (X 1 , X 2 ) ∼ O , (14)N c L cX 1,X 2where L c = Nc1/D is the linear cluster size. A detailedderivation of this form will be published elsewhere. However,since in the MCA the cluster is defined in real spacewith open boundary conditions, this form is evident sinceonly the sites on the surface ∝ 2D · L D−1c of the clustercouple to the effective medium. For the DCA we havepreviously shown that Γ(K) ∼ O(1/Nc2/D ) [12] so thatwe obtain for the average hybridization of the DCA clusterto the effective medium¯Γ DC =1 ∑( ) 1Γ DC (K) ∼ ON c L 2 . (15)cKThe DCA coarse graining results in a cluster in K-space;thus, the corresponding real space cluster has periodicboundary conditions, and each site in the cluster has thesame hybridization strength ¯Γ with the host.In both the MCA and the DCA, the average hybridizationstrength acts as the small parameter. The approximationperformed by the MCA (DCA) is to replace thelattice Green function Ĝ(˜k) = [zI − ˆɛ(˜k) − ˆΣ(˜k, z)] −1(G(K, ˜k, z) = [z − ɛ K+˜k− Σ(K, ˜k, z)] −1 ) by its coarsegrained quantity ˆḠ (Ḡ(K)) in diagrams for the selfenergyΣ. Once the sums over ˜k are performed, all termswhich are lower order in 1/L c than Γ vanish. Thus theMCA (DCA) is an approximation with corrections of order¯Γ ∼ O(1/L c ) (∼ O(1/L 2 c)).Numerical Results To illustrate the differences in convergencewith cluster size N c we performed MCA andDCA simulations for the symmetric one-dimensional(1D) Falicov-Kimball model (FKM). At half filling theFKM Hamiltonian readsH = −t ∑ i(d † i d i+1 + h.c.) + U ∑ i(n d i − 1/2)(n f i − 1/2) ,(16)with the number operators n d i = d† i d i and nf i = f † i f i andthe Coulomb repulsion U between d and f electrons residingon the same site. The FKM can be considered as asimplified Hubbard model with only one spin-species (d)being allowed to hop. However it still shows a complexphase diagram including a Mott transition and Ising-likecharge ordering with the corresponding transition temperatureT c being zero in 1D. The dispersion in (1D)ɛ k = 2t cos k; thus for t = 1/4 the bandwidth W = 1which we use as unit of energy. To simulate the effectivecluster models of the MCA an the DCA we use a quantumMonte Carlo (QMC) approach described in [11].To check the scaling relations Eqs. 14 and 15, we showin Fig.1 the average hybridization functions ¯Γ MC and¯Γ DC for the MCA and DCA respectively at the inversetemperature β = 17 for U = W = 1. For N c = 1 bothapproaches are equivalent to the DMFA and thus ¯Γ MC =¯Γ DC . For increasing N c ¯ΓMC can be fitted by 0.3361/N cand ¯Γ DC by 1.1946/N 2 c when N c > 2. Cluster quantities,such as the self energy and cluster susceptibilities, areexpected to converge with increasing N c like ¯Γ. This isillustrated in the inset for the staggered (Q = π) chargesusceptibility χ c (Q) of the cluster.Γ0.40.30.20.1DCAMCAW=1.0, U=1.0T=0.059, n=1χ c(Q,N c=∞) - χ c(Q,N c)10.500 0.05 0.1 0.15 0.2 0.251/N c00 10 20 30 40N cFIG. 1. The average integrated hybridization strengths ¯Γof the MCA (squares) and DCA (circles) versus the clustersize N c when β = 17 and U = W = 1. The solid and dashedlines represent the fits 1.1946/Nc 2 and 0.3361/N c respectively.Inset: Convergence of the cluster charge susceptibility forQ = π. The solid and dashed lines are quadratic and linearfits, respectively.Since only the compact parts represented by Φ of thelattice free energy (Eq. 7) are coarse-grained, this scalingis expected to break down when lattice quantities, suchas the lattice charge susceptibility, are calculated. Thesusceptibility of the cluster χ c (Q) cannot diverge for anyfinite N c ; whereas the lattice χ(Q) diverges at the transitiontemperature T c to the charge ordered phase. Notethat the residual mean field character of both methodscan result in finite transition temperatures T c > 0 forfinite N c < ∞. However as N c increases, this residualmean field character decreases gradually and thusincreased fluctuations should drive the solution to theexact result T c = 0.In the DCA [11], χ(Q) is calculated by first extractingthe corresponding vertex function from the clustersimulation. This is then used in a Bethe-Salpeter equationto calculate χ(Q). T c is calculated by extrapolatingχ(Q) −1 to zero using the function χ(Q) −1 ∝ (T − T c ) γ(see inset to Fig.2). This procedure is difficult, if not impossible,in the MCA due to the lack of translationalinvariance. Here, we calculate the the order parameterm(T ) = 1/N c∑i (−1)i 〈n d i 〉 in the symmetry brokenphase. T c is then obtained from extrapolating m(T ) tozero using the function m(T ) ∝ (T c − T ) β . For the DCAthis extrapolation is shown by the solid line in the insetto Fig.2 for N c = 4. The values for T c obtained from the3


calculation in the symmetry broken phase and in the unbrokenphase must agree, since as we have shown above,both the DCA and MCA are Φ-derivable. This is illustratedin Fig.2 for the DCA.T c0.030.02U=W=1DCAMCADCA: U=W=1, N c =40.4T0.4c =0.28m0.010.2 0.2m(T)χ(Q,T) -10 00 0.05 0.1T00 0.1 0.2 0.3 0.4 0.51/N cFIG. 2. The transition temperature T c for the DCA (circles)and MCA (squares) when U = W = 1 versus thecluster size N c. For all values of N c the DCA prediction iscloser to the exact result (T c = 0). Inset: Order parameterm(T ) and inverse charge susceptibility χ(Q) −1 versus temperature.The solid (dashed) line represents a fit to the functionsm(T ) ∝ (T c − T ) β with β = 0.245 (χ(T ) ∝ (T − T c) −γ withγ = 1.07).A comparison of the DCA and MCA estimate of T c ispresented in Fig. 2. T c obtained from MCA (squares)is larger than T c obtained from DCA (circles). Moreoverwe find that the DCA result seems to scale to zero almostlinearly in 1/N c (for large enough N c ), whereas the MCAdoes not show any scaling form and in fact seems to tendto a finite value for T c as N c → ∞. This striking differenceof the two methods can be attributed to the differentboundary conditions. The open boundary conditions ofthe MCA cluster result in a large surface contribution sothat ¯Γ MC > ¯Γ DC . This engenders pronounced mean fieldbehavior that stabilizes the finite temperature transitionfor the cluster sizes treated here. For larger clusters weexpect the bulk contribution to the MCA free energy todominate so that T c should fall to zero.Complementary results are found in simulations offinite-sized systems. In general, systems with openboundary conditions are expected to have a surface contributionin the free energy of order O(1/L c ) [14]. Thisterm is absent in systems with periodic boundary conditions.As a result, simulations of finite-sized systemswith periodic boundary conditions converge much morequickly than those with open boundary conditions [15].Summary. By defining appropriate Laue functions,we provide microscopic diagrammatic derivations of theMCA and DCA. We show that they are Φ-derivable, andthat the lattice free energy is optimized by equating theirreducible quantities on the lattice to those on the cluster.The MCA maps the lattice to a cluster with openχ(Q) -1boundaries and consequently, the cluster violates translationalinvariance. In contrast, the DCA cluster hasperiodic boundary conditions, and therefore preservesthe translational invariance of the lattice. This differencein the boundary conditions translates directly todifferent asymptotic behaviors for large clusters N c . Aswe find analytically as well as numerically, the surfacecontributions in the MCA lead to an average hybridization¯Γ of the cluster to the mean field that scales like1/L c as compared to the 1/L 2 c scaling of the DCA. Since¯Γ acts as the small parameter for these approximationschemes, the DCA converges much more quickly thanthe MCA. These effects are more pronounced near a transition,where the large surface contribution of the MCAstabilizes the mean-field character of the transition. Consequently,the DCA result for the transition temperatureT c of the 1D symmetric FKM model scales almost like1/N c to the exact result T c = 0, whereas the MCA resultconverges very slowly. Since the origin of this differencelies in the different boundary conditions we expect thisprimacy of the DCA over the MCA to hold generally forany model of electrons moving on a lattice.Acknowledgements We acknowledge useful conversationswith N. Blümer, A. Gonis, M. Hettler, H.R. Krishnamurthy,D.P. Landau, Th. Pruschke, Th. Schulthess,W. Shelton, and A. Voigt. This work was supported byNSF grant DMR-0073308.[1] D.W. Taylor, Phys. Rev. 156 1017 (1967); P. Soven Phys.Rev. 156 809 (1967); P.L. Leath and B. Goodman, Phys.Rev. 148 968 (1966).[2] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324(1989).[3] E. Müller-Hartmann: Z. Phys. B 74, 507–512 (1989).[4] T. Pruschke, M. Jarrell and J.K. Freericks, Adv. in Phys.42, 187 (1995).[5] A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg,Rev. Mod. Phys. 68, 13 (1996).[6] M. Jarrell and H.R. Krishnamurthy, Phys. Rev. B, 63125102 (2001).[7] F. Ducastelle, J. Phys. C. Sol. State. Phys. 7 1795 (1974).[8] A. Gonis, Green functions for ordered and disordered systems,in the series Studies in Mathematical Physics Eds.E. van Groesen and E.M. DeJager, (North Holland, Amsterdam,1992).[9] G. Kotliar, S.Y. Savrasov, G. Palsson, preprint condmat/0010328.This method reduces to the MCA whennon-overlapping clusters are chosen.[10] M.H. Hettler et al., Phys. Rev. B 58, 7475 (1998).[11] M.H. Hettler, M. Mukherjee, M. Jarrell, and H.R. Krishnamurthy,Phys. Rev. B 61, 12739 (2000).[12] Th. Maier, M. Jarrell, Th. Pruschke and J. Keller, Eur.Phys. J. B 13, 613 (2000).[13] In both the DCA and MCA, on-site lattice and cluster4


quantities correspond one-to-one. In the MCA, the singleparticlegreen functions within the cluster and the latticealso correspond.[14] M.E. Fisher and M.N. Barber, Phys. Rev. Lett. 28, 1516(1972).[15] D.P. Landau, Phys. Rev. B 13, 2997 (1976).5

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