1. Cluster Dynamical Mean Field Theories: Some Formal Aspects G. Kotliar Physics Department and Center for Materials Theory Rutgers Sherbrook July 2005

2. oDynamical Mean Field Theory and a cluster extension, CDMFT: G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) oCluser Dynamical Mean Field Theories: Causality and Classical Limit. G. Biroli O. Parcollet G.Kotliar Phys. Rev. B 69 205908 Cluster Dynamical Mean Field Theories a Strong Coupling Perspective. T. Stanescu and G. Kotliar ( 2005) References

3. Outline Important concepts from DMFT. Weiss Field. Functional Derivations. Effective Action Constructions. Weiss Field Functionals. What is the best variable ? Optimal discretizations for ED+DMFT ? Use of Cumulants vs Self Energies. Periodization. What is shorter range ? Convergence Issues. Semiclassical Limit. How fast does CMDFT converge ? Causality Issues and Nested Cluster Schemes Are they any good ? Why is causality violated in NCS ? Does it matter ? Conclusions: Outlook.

4. Outline Concepts from DMFT. Weiss Field. Functional Derivations. Effective Action Constructions. Weiss Field Functionals. What is the best variable to formulate the problem ? Optimal discretizations for ED+DMFT ? Importance of the Weiss field. Use of Cumulants vs Self Energies. Periodization. What is shorter range ? The Cumulant ? Convergence Issues. Semiclassical Limit. How fast does CMDFT converge ? Exponential Convergence. Causality Issues and Nested Cluster Schemes. Are they any good ? Why is causality violated in NCS ? Does it matter ? Likely to be very useful for larger clusters. Conclusions: Outlook.

5. Cluster DMFT schemes Mapping of a lattice model onto a quantum impurity model (degrees of freedom in the presence of a Weiss field, the central concept in DMFT). Contain two elements. 1) Determination of the Weiss field in terms of cluster quantities. 2) Determination of lattice quantities in terms of cluster quantities (periodization).

6. Impurity Model-----Lattice Model  Weiss Field

7. Effective Action point of view. Identify observable, A. Construct a free energy functional of =a,  [a] which is stationary at the physical value of a. Example, density in DFT theory. (Fukuda et. al.). DMFT (R. Chitra and G.K (2000) (2001). H=H 0 + H 1.  [a,J 0 ]=F 0 [J 0 ]–a J 0 _ +  hxc [a] Functional of two variables, a,J 0. H 0 + A J 0 Reference system to think about H. J 0 [a] Is the functional of a with the property 0 =a 0 computed with H 0 + A J 0 Many choices for H 0 and for A Extremize a to get  [J 0 ]= ext a  [a,J 0 ]

8. Ex: Baym Kadanoff functional,a= G, H 0 = free electrons. Viewing it as a functional of J 0, Self Energy functional(Potthoff)

9. CDMFT and NCS as truncations of the Baym Kadanoff functional

10. Weiss Field Functional

11. Example: single site DMFT semicircular density of states. GKotliar EPJB (1999) Extremize Potthoff’s self energy functional. It is hard to find saddles using conjugate gradients. Extremize the Weiss field functional.Analytic for saddle point equations are available Minimize some distance

12. U/t=4. Testing CDMFT (G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) ) with two sites in the Hubbard model in one dimension V. Kancharla C. Bolech and GK PRB 67, 075110 (2003)][[M.Capone M.Civelli V Kancharla C.Castellani and GK PR B 69,195105 (2004) ]

13. CDMFT : Strong Coupling Point of View Using Cumulants Useful to treat constrained models such as the t-J model, or models containing pair hopping terms, or more general Coulomb interactions. Useful to compare CDMFT constructions built from resummations of the perturbative expansion in the interactions or in the hopping or intersite terms. [No difference in single site DMFT ]

15. From cluster back to the lattice. There are several procedures suggested in the literature. Periodize the greens function. D. Senechal, D. Perez, and M. Pioro-Ladriere, Phys. Rev. Lett. 84, 522 (2000). Periodize the self energy. [ G. Kotliar, S.Y. Savrasov, G. Palsson, and G. Biroli, Phys. Rev. Lett. 87. 186401 (2001). Periodize the cumulant. [Stanescu and Kotliar (2005)]

16. Cumulant Periodization: 2X2 cluster

17. Self energy and Greens function Periodization.

18. Comparison of 2 and 4 sites

19. Convergence for large cluster sizes In DCA observables converge as 1/L 2. (L linear size of the cluster). In DCA Weiss field is uniform in the cluster and is of order 1/L 2. In CDMFT local observables converge exponentially at finite temperatures and away from critical points. The Weiss field is of order one, on the boundary of the cluster and zero in the interior of the cluster. Hence

20. To gain intuition into the various cluster schemes it is useful to derive a semiclassical limit of models such as the Falikov Kimball model that maps onto an Ising model. [Nested Cluster Schemes correspond to CVM and converge the best ]

21. Causality Issues. Pair Scheme: restrict the BK functional to Gii and Gi,i+  delta  nearest neighbor. Georges and Kotliar, (1995), Schiller and Ingerstsen, Implementation of the scheme shows its was not causal. But why ?? Zarand et. al. The lack of causality is not intrinsic to the scheme but the result of using approximate solvers.

22. Biroli Parcollet and Kotliar. The causality violations are intrinsic to the scheme and not due to approximate solvers. Lack of closure of the Cutkovski-T’hooft cutting equations. Causality is violated only when the actual range of the self energy exceeds substantially the range of the truncation. Generalized the pair scheme to allow for longer range truncations. Express in terms of a nested set of AIM. Nested Cluster Schemes. The approach is more rapidly convergent, at least in the semiclassical limit (where it reduces to the cluster variation method) than CDMFT.

24. Sufficient condition for causality Sufficient Condition for Causality:All diagrams in the class considered can be obtained once and only once by gluing one right part and one left part.

25. Condition is not satisfied in pair scheme. For example L can contain sites i and j, R can contain sites i and k. But the glued diagram contains i j and k which is not present in the pair scheme. i,ji,k Sufficient Condition for Causality:All diagrams in the class considered can be obtained once and only once by gluing one right part and one left part.

26. Benchmark :1D Hubbard Model M. Capone, M. Civelli, S.S. Kancharla, C. Castellani, G. Kotliar, Phys. Rev B 69, 195105 (2004) H= -t ∑ jσ c † jσ c j+1σ + h.c +U ∑ n j↑ n j↓ - μ N Exact Bethe Ansatz (BA) VS Dynamical Mean Field Theory VS Cellular Dynamical Mean Field Theory (2 site cluster only!) U= 4 t

27. Convergence of the approach n_CMDFT-n_BA vs mu

28. Comparison between cluster quantities and DMRG.

29. Kinetic energy

31. Extraction of lattice quantities: benchmark. Cluster quantities are good to compute G11 but not good to compute G12 or the kinetic energy. (overestimation of Sigma_12 due to the procedure of moving bonds). Lattice Greens function, yields good G12 or kinetic energy. The reduction of the self energy by averaging or by imposing causality is PHYSICAL!. Cumulant method gives nice physical quantities near the Mott insulator.

32. Outline Concepts from DMFT. Weiss Field. Functional Derivations. Effective Action Constructions. Weiss Field Functionals. What is the best variable to formulate the problem ? Optimal discretizations for ED+DMFT ? Importance of the Weiss field. Use of Cumulants vs Self Energies. Periodization. What is shorter range ? The Cumulant ? Convergence Issues. Semiclassical Limit. How fast does CMDFT converge ? Exponential Convergence. Causality Issues and Nested Cluster Schemes. Are they any good ? Why is causality violated in NCS ? Does it matter ? Likely to be very useful for larger clusters. Conclusions: Outlook.

33. CDMFT vs single site DMFT and other cluster methods.

34. Conclusions.

37. Approximate functional, truncate the Weiss field to a finite number of parameters. Is there an optimal choice ? xxx

38. Get the review convergence of Krauth Caffarel. Explain from Marcello’s handout. What we have done. Put the analytical equations for the determination of the mean field equations. Advocate the Projective Self Consistent Method.

39. Conclusions.

41. Cellular Dynamical Mean Field Theories of the Mott Transition. G. Kotliar Physics Department and Center for Materials Theory Rutgers Sherbrook July 2005

42. oModel for kappa organics. [O. Parcollet, G. Biroli and G. Kotliar PRL, 92, 226402. (2004)) ] oModel for cuprates [O. Parcollet (Saclay), M. Capone (U. Rome) M. Civelli (Rutgers) V. Kancharla (Sherbrooke) GK(2005). Cluster Dynamical Mean Field Theories a Strong Coupling Perspective. T. Stanescu and G. Kotliar (in preparation 2005) Talk by B. Kyung et. al.. cond-mat/0502565 Short-Range Correlation Induced Pseudogap in Doped Mott Insulators References

43. Outline Motivation and Objectives.Schematic Phase Diagram(s) of the Mott Transition. Finite temperature study of very frustrated anisotropic model. [O. Parcollet ] Low temperature study of the normal state of the isotropic Hubbard model. [M. Civelli, T. Stanescu ] [See also B. Kyung’s talk] Superconducting state near the Mott transition. [ M. Capone. See also V. Kancharla’s talk ] Conclusions.

44. RVB phase diagram of the Cuprate Superconductors P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987) Connection between the anomalous normal state of a doped Mott insulator and high Tc. Slave boson approach. coherence order parameter.  singlet formation order parameters.

45. RVB phase diagram of the Cuprate Superconductors. Superexchange. The approach to the Mott insulator renormalizes the kinetic energy Trvb increases. The proximity to the Mott insulator reduce the charge stiffness, TBE goes to zero. Superconducting dome. Pseudogap evolves continously into the superconducting state. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) Related approach using wave functions:T. M. Rice M. Randeria N. Trivedi, A. Paramenkanti

46. Problems with the approach. Neel order Stability of the pseudogap state at finite temperature. [Ubbens and Lee] Missing incoherent spectra. [ fluctuations of slave bosons ] Dynamical Mean Field Methods are ideal to remove address these difficulties.

47. T/W Phase diagram of a Hubbard model with partial frustration at integer filling. M. Rozenberg et.al., Phys. Rev. Lett. 75, 105-108 (1995)..Phys. Rev. Lett. 75, 105-108 (1995). COHERENCE INCOHERENCE CROSSOVER

49. Focus of this work Generalize and extend these approaches. Obtain the solution of the 2X 2 plaquette and gain physical understanding of the different CDMFT states. Even if the results are changed by going to larger clusters, the short range physics is general and will teach us important lessons. Furthermore the results can be stabilized by adding further interactions.

51. Finite T Mott tranisiton in CDMFT Parcollet Biroli and GK PRL, 92, 226402. (2004))

52. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

53. Evolution of the k resolved Spectral Function at zero frequency. (QMC study Parcollet Biroli and GK PRL, 92, 226402. (2004)) ) Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W U/D=2 U/D=2.25

54. Momentum Space Differentiation the high temperature story T/W=1/88

55. Physical Interpretation Momentum space differentiation. The Fermi liquid –Bad Metal, and the Bad Insulator - Mott Insulator regime are realized in two different regions of momentum space. Cluster of impurities can have different characteristic temperatures. Coherence along the diagonal incoherence along x and y directions. Connection with slave Boson theory divergence of Sigma_13.

56. Cuprate superconductors and the Hubbard Model. PW Anderson 1987

57. . Allows the investigation of the normal state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition. Earlier studies (Katsnelson and Lichtenstein, M. Jarrell, M Hettler et. al. Phys. Rev. B 58, 7475 (1998). T. Maier et. al. Phys. Rev. Lett 85, 1524 (2000). ) used QMC as an impurity solver and DCA as cluster scheme. We use exact diag ( Krauth Caffarel 1995 with effective temperature 32/t=124/D ) as a solver and Cellular DMFT as the mean field scheme. Connect the solution of the 2X2 plaquette to simpler mean field theories. CDMFT study of cuprates

58. ED and QMC

60. Follow the “normal state” with doping. Evolution of the spectral function at low frequency. If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

61. Hole doped case t’=-.3t, U=16 t n=.71.93.97 Color scale x=.37.15.13

62. K.M. Shen et. al. Science (2005). For a review Damascelli et. al. RMP (2003)

63. Approaching the Mott transition: CDMFT Picture Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! D wave gapping of the single particle spectra as the Mott transition is approached. Similar scenario was encountered in previous study of the kappa organics. O Parcollet G. Biroli and G. Kotliar PRL, 92, 226402. (2004).

64. Results of many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maekawa Phys. Rev. B 67, 092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66, 140513 (2002)

66. Experiments. Armitage et. al. PRL (2001). Momentum dependence of the low-energy Photoemission spectra of NCCO

67. Approaching the Mott transition: CDMFT picture. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, BUT the location of the cold regions depends on parameters. Quasiparticles are now generated from the Mott insulator at ( , 0). Results of many numerical studies of electron hole asymmetry in t-t’ Hubbard models Tohyama Maekawa Phys. Rev. B 67, 092509 (2003) Senechal and Tremblay. PRL 92 126401 (2004) Kusko et. al. Phys. Rev 66, 140513 (2002)

68. Comparison with Experiments in Cuprates: Spectral Function A(k,ω→0)= -1/π G(k, ω →0) vs k hole dopedelectron doped K.M. Shen et.al. 2004 P. Armitage et.al. 2001

69. To test if the formation of the hot and cold regions is the result of the proximity to Antiferromagnetism, we studied various values of t’/t, U=16.

70. Introduce much larger frustration: t’=.9t U=16t n=.69.92.96

71. Approaching the Mott transition: Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, but the location of the cold regions depends on parameters. With the present resolution, t’ =.9 and.3 are similar. However it is perfectly possible that at lower energies further refinements and differentiation will result from the proximity to different ordered states.

72. Understanding the result in terms of cluster self energies (eigenvalues)

73. Cluster Eigenvalues

74. Evolution of the real part of the self energies.

75. Fermi Surface Shape Renormalization ( t eff ) ij =t ij + Re(  ij 

76. Fermi Surface Shape Renormalization Photoemission measured the low energy renormalized Fermi surface. If the high energy (bare ) parameters are doping independent, then the low energy hopping parameters are doping dependent. Another failure of the rigid band picture. Electron doped case, the Fermi surface renormalizes TOWARDS nesting, the hole doped case the Fermi surface renormalizes AWAY from nesting. Enhanced magnetism in the electron doped side.

77. Understanding the location of the hot and cold regions. Interplay of lifetime and fermi surface.

78. oQualitative Difference between the hole doped and the electron doped phase diagram is due to the underlying normal state.” In the hole doped, it has nodal quasiparticles near ( ,  /2) which are ready “to become the superconducting quasiparticles”. Therefore the superconducing state can evolve continuously to the normal state. The superconductivity can appear at very small doping. oElectron doped case, has in the underlying normal state quasiparticles leave in the (  0) region, there is no direct road to the superconducting state (or at least the road is tortuous) since the latter has QP at (  /2,  /2).

79. Systematic Evolution

80. Comparison of periodization methods for A(w=0,k)   

81. Qualitative Difference between the periodization methods. The cumulant periodization, is a non linear interpolation of the self energies (linear interpolation of the cumulants). When the off diagonal elements of the self energy get large, it gives rise to lines of poles of the self energy, in addition to the Fermi lines.

84. How is the Mott insulator approached from the superconducting state ? Work in collaboration with M. Capone

87. Superconductivity in the Hubbard model role of the Mott transition and influence of the super- exchange. (M. Capone V. Kancharla. CDMFT+ED, 4+ 8 sites t’=0).

88. D wave Superconductivity and Antiferromagnetism t’=0 M. Capone V. Kancharla (see also VCPT Senechal and Tremblay ). Antiferromagnetic (left) and d wave superconductor (right) Order Parameters

89. Evolution of the low energy tunneling density of state with doping. Decrease of spectral weight as the insulator is approached. Low energy particle hole symmetry.

90. Alternative view

91. DMFT is a useful mean field tool to study correlated electrons. Provide a zeroth order picture of a physical phenomena. Provide a link between a simple system (“mean field reference frame”) and the physical system of interest. [Sites, Links, and Plaquettes] Formulate the problem in terms of local quantities (which we can usually compute better). Allows to perform quantitative studies and predictions. Focus on the discrepancies between experiments and mean field predictions. Generate useful language and concepts. Follow mean field states as a function of parameters. Conclusions

92. Qualitative effect, momentum space differentiation. Formation of hot –cold regions is an unavoidable consequence of the approach to the Mott insulating state! General phenomena, but the location of the cold regions depends on parameters. Study the “normal state” of the Hubbard model is useful. On the hole doped normal and superconducting state can be connected to each other as in the RVB scenario. High Tc superconductivity may result follow from doping a Mott insulator phase but it is not necessarily follow from it. One may not be able to connect the Mott insulator to the superconductor if the nodes are in the “wrong place”.

93. Unfortunately the Hubbard model does not capture the trend of supra with t’. Need augmentation.

96. Estimates of upper bound for Tc exact diag. M. Capone. U=16t, t’=0, ( t~.35 ev, Tc ~140 K~.005W)

98. Site  Cell. Cellular DMFT. C-DMFT. G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) hopping expressed in the superlattice notations. Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998) Katsnelson and Lichtenstein periodized scheme, Nested Cluster Schemes, causality issues, O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)Phys. Rev. B 69, 205108 (2004)

99. Understanding in terms of cluster self-energies. Civelli et. al.

100. Insulating anion layer  -(ET) 2 X are across Mott transition ET = X -1 [(ET) 2 ] +1 conducting ET layer t’ t modeled to triangular lattice X-X- Ground State U/tt’/t Cu 2 (CN) 3 Mott insulator 8.21.06 Cu[N(CN) 2 ]Cl Mott insulator 7.50.75 Cu[N(CN) 2 ]BrSC7.20.68 Cu(NCS) 2 SC6.80.84 Cu(CN)[N(CN) 2 ] SC6.80.68 Ag(CN) 2 H 2 OSC6.60.60 I3I3 SC6.50.58

101. Electron doped case t’=.9t U=16t n=.69.92.96 Color scale x=.9,.32,.22

102. Two paths for calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions DMFT ideas can be used in both cases.

103. Outline ____________________________________________________________ INTUITIVE NOTIONS OF DMFT AND WEISS FIELD CAVITY CONSTRUCTION. Mapping of lattice onto a cluster in a medium. With a prescription for building the medium from the computation of the cluster quantities. Prescription for reconstructing lattice quantities. Weiss field describe the medium. Cavity Construction is highly desireable. Delta is non zero on the boundary. ------------------------------------------------------- ------------------------------------------------------ EFFECTIVE ACTION CONSTRUCTION. Show how it is done. It is not perturbative. It is general. It includes everything you want to know. It gives you a reference system. Examples. APPROXIMATE WEISS FIELD. -------------------------------------------- --------------------------------------------- SUBTLETIES WITH THE WEISS FIELD, WHAT ABOUT GETTING NOT ONLY Gii but also Gij, and admiting Delta_ij, is the unique solution t_ij ? ----------------------------------------------- ---------------------------------------------- ----------------------------------------------- How to go from lattice to impurity model. CDMFT construction. Picture giving simga_c, M_cluster chi_cluster Gamma_cluster ------------------------------------------- OTHER CLUSTER SCHEMES. CDMFT-PCDMFT. Truncation using U---> and G----> ------------------------------------- ----------------------------------- CPT and VCPT from the point of view of a Weiss field functional. cal{G-0} ----> cal{G_0} atom cal{G_0} -----Cal{G_0}+Anomalous static self energy. ------------------------------------------------ ------------------------------------------------- List of pair schemes. Bethe Peiersls Pair Schemes. DCA. PCDMFT.

104. a) Baym Kadanoff Functional b) Self energy functional. c) DFT. Want to generate good approximations. and their hybrids. ------------------------------------------------ WEIS FIELD. I NEED TO GET THIS RIGHT HOW I DO THE SEPARATION, INTO PIECES WHAT IS THE EXACT AND WHA IS TEH

105. Nested Cluster Schemes. Explicit Cavity constructions. ------------------------------------------------- -------------------------------------------------- CAUSALITY PROBLEMS. Parcollet Biorli KOtliar clausing. ---------------------------------------- ------------------------------------------- CONVERGENCE. 1/L vs e-L. DCA converges as 1/L2 Classical limits. See what olivier ahs done on w-cdmft. -------------------------------------- Impurity Solvers. OCA-QMC-ED. Lessons form the past. Combinations of methods. ---------------------------------------- -------------------------------------- ED discretization Role of distance. Role of functionals. Why not variational is usesless. Few sites. ------------------------------ -------------------------- Things in practice. Examples from one dimension. How to extract physical quantities. ------------------------------ ------------------------------------ CONCLUSIONS:

106. Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

107. Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)

108. Insulating anion layer  -(ET) 2 X are across Mott transition ET = X -1 [(ET) 2 ] +1 conducting ET layer t’ t modeled to triangular lattice t’ t modeled to triangular lattice

109. References

111. CDMFT one electron spectra n=.96 t’/t=.-.3 U=16 t i

112. Other cluster extensions (DCA Jarrell Krishnamurthy, M Hettler et. al. Phys. Rev. B 58, 7475 (1998)Katsnelson and Lichtenstein periodized scheme. Causality issues O. Parcollet, G. Biroli and GK Phys. Rev. B 69, 205108 (2004)Phys. Rev. B 69, 205108 (2004)

113. Medium of free electrons : impurity model. Solve for the medium using Self Consistency G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001)

114. Difference in Periodizations

116. Advantages of the Weiss field functional. Simpler analytic structure near the Mott transition.

117. How to determine the parameters of the bath ? Extremize Potthoff’s self energy functional. It is hard to find saddles using conjugate gradients. Extremize the Weiss field functional.Analytic for saddle point equations are available Minimize some distance.

118. Convergence of Cluster Schemes as a function of cluster size. Aryamanpour et. al. DCA observables converge as 1/L^2. cond-mat/0205460 [abs, ps, pdf, other] :abspspdfother –Title: Two Quantum Cluster Approximations Authors: Th. A. Maier (1), O. Gonzalez (1 and 2), M. Jarrell (1), Th. Schulthess (2) ((1) University of Cincinnati, Cincinnati, USA, (2) Oak Ridge National Laboratory, Oak Ridge, USA)Th. A. MaierO. GonzalezM. JarrellTh. Schulthess Aryamampour et. al. The Weiss field in CDMFT converges as 1/L. Title: The Dynamical Cluster Approximation (DCA) versus the Cellular Dynamical Mean Field Theory (CDMFT) in strongly correlated electrons systems Authors: K. Aryanpour, Th. A. Maier, M. Jarrell Comments: Comment on Phys. Rev. B 65, 155112 (2002). 3 pages, 2 figures Subj-class: Strongly Correlated Electrons Journal-ref: Phys. Rev. B 71, 037101 (2005)K. AryanpourTh. A. MaierM. Jarrell Biroli and Kotliar. Phys. Rev. B 71, 037102 (2005);cond-mat/0404537. Local observables (i.e. observables contained in the cluster ) converge EXPONENTIALLY at finite temperatures away from critical points.Phys. Rev. B 71, 037102 (2005)cond-mat/0404537.