Connected diagram expansion for lattice hamiltonian models

Debashis Mukherjee, Werner Kutzelnigg, Sigurd Koch

We generalize here the formalism of the preceeding paper to encompass the case of the general incomplete model space. The classification of operators as diagonal or nondiagonal depends in this case upon the specific m-valence model space. It is stressed that even then one has to work in Fock space in order to get connected-diagram expansions, since connectedness is a Fock space property. Two choices of separable normalization of the wave operator W leading to a connected Heff are discussed. It is shown that the intermediate normalization is not separable in general and hence not compatible with a connected-diagram expansion. We also discuss how to generate ‘‘subduced’’ incomplete model spaces of lower particle rank such that Heff remains a valid effective Hamiltonian for these subduced model spaces as well. We discuss the nature of the various disconnected diagrams encountered in many-body formalisms and point out which of these are really worth worrying about. We finally comment on the question of the separability of the wave function into proper fragments.

Weihong Zheng, Chris J. Hamer, J. Oitmaa

Edward Perepelitsky

We develop a diagrammatic approach for calculating the high temperature expansion of dynamic correlation functions, such as the electron Green's function and the time-dependent density-density and spin-spin correlation functions, for the infinite-U Hubbard Model with any number of spin species. The formalism relies on the use of restricted lattice sums, in which distinct vertices of the diagram represent distinct sites on the lattice. We derive a new formula for the restricted lattice sum of a disconnected diagram consisting of several connected components, and use it to prove the linked cluster theorem with respect to "generalized connected diagrams", formed by overlapping the original connected components on the lattice. This enables us to express all quantities as a sum over these generalized connected diagrams. We compute the Green's function to 4th order in βt for the case of m spin species on a d-dimensional hypercube by hand. We take the m\to\infty limit, enabling us to obtain expressions for the Dyson-Mori self-energy to 4th order in βt for the case of an infinite number of spin species. This may have connections to slave boson techniques used for the study of this model. Our approach is computationally more efficient than any used previously for the calculation of the high temperature expansion of dynamic correlation functions, and high order results for both the Green's function and the time-dependent density-density and spin-spin correlation functions shall be presented in a separate paper [25].

Andreas Hehn

Ada Altieri, Maria Chiara Angelini, Carlo Lucibello +3 more

Abstract For every physical model defined on a generic graph or factor graph, the Bethe M -layer construction allows building a different model for which the Bethe approximation is exact in the large M limit, and coincides with the original model for M = 1 . The 1 / M perturbative series is then expressed by a diagrammatic loop expansion in terms of so-called fat diagrams. Our motivation is to study some important second-order phase transitions that do exist on the Bethe lattice, but are either qualitatively different or absent in the corresponding fully connected case. In this case, the standard approach based on a perturbative expansion around the naive mean field theory (essentially a fully connected model) fails. On physical grounds, we expect that when the construction is applied to a lattice in finite dimension there is a small region of the external parameters, close to the Bethe critical point, where strong deviations from mean-field behavior will be observed. In this region, the 1 / M expansion for the corrections diverges, and can be the starting point for determining the correct non-mean-field critical exponents using renormalization group arguments. In the end, we will show that the critical series for the generic observable can be expressed as a sum of Feynman diagrams with the same numerical prefactors of field theories. However, the contribution of a given diagram is not evaluated by associating Gaussian propagators to its lines, as in field theories: one has to consider the graph as a portion of the original lattice, replacing the internal lines with appropriate one-dimensional chains, and attaching to the internal points the appropriate number of infinite-size Bethe trees to restore the correct local connectivity of the original model. The actual contribution of each (fat) diagram is the so-called line-connected observable, which also includes contributions from sub-diagrams with appropriate prefactors. In order to compute the corrections near to the critical point, Feynman diagrams (with their symmetry factors) can be read directly from the appropriate field-theoretical literature; the computation of momentum integrals is also quite similar; the extra work consists of computing the line-connected observable of the associated fat diagram in the limit of all lines becoming infinitely long.

Martin P. Gelfand