Cluster Dynamical Mean Field Theories: Some Formal Aspects

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

2. References • Dynamical Mean Field Theory and a cluster extension, CDMFT: G.. Kotliar,S. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) • Cluser 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)

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 D , Weiss Field

7. Effective Action point of view. • Identify observable, A. Construct a free energy functional of <A>=a, G [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=H0+l H1. G [a,J0]=F0[J0 ]–a J0 _ + Ghxc [a] • Functional of two variables, a ,J0. • H0 +A J0 Reference system to think about H. • J0 [a] Is the functional of a with the property <A>0 =a < >0 computed with H0+A J0 • Many choices for H0 and for A • Extremize a to get G [J0]=exta G [a,J0]

8. Ex: Baym Kadanoff functional,a= G, H0 = free electrons. Viewing it as a functional of J0, 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. 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) ] U/t=4.

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/L2. (L linear size of the cluster). • In DCA Weiss field is uniform in the cluster and is of order 1/L2. • 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+d delta dnearest 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. 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. 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,j i,k

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

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.

40. 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. References • Model for kappa organics. [O. Parcollet, G. Biroli and G. Kotliar PRL, 92, 226402. (2004)) ] • Model 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

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. <b> coherence order parameter. k, D singlet formation order parameters.

45. RVB phase diagram of the Cuprate Superconductors. Superexchange. G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988) • 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. 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. COHERENCE INCOHERENCE CROSSOVER 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). .

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.