Current methods to describe the thermodynamic behavior of many-particle systems are often based on perturbation theory with an unperturbed system consisting of free particles. Therefore, only a few methods are able to describe both strongly and weakly correlated systems along the same lines. In this article we propose a cumulant approach which allows for the evaluation of excitation energies and is especially appropriate to account for the thermodynamics at low temperatures. The method is an extension of a cumulant formalism which was recently proposed to study statical and dynamical properties of many-body systems at zero temperature. The present approach merges into the former one for vanishing temperature. As an application we investigate the thermodynamics of the hole-doped antiferromagnetic phase in high-temperature superconductors in the framework of the anisotropic $t$-$J$ model.
Current methods to describe the thermodynamic behavior of many-particle systems are often based on perturbation theory with an unperturbed system consisting of free particles. Therefore, only a few methods are able to describe both strongly and weakly correlated systems along the same lines. In this article we propose a cumulant approach which allows for the evaluation of excitation energies and is especially appropriate to account for the thermodynamics at low temperatures. The method is an extension of a cumulant formalism which was recently proposed to study statical and dynamical properties of many-body systems at zero temperature. The present approach merges into the former one for vanishing temperature. As an application we investigate the thermodynamics of the hole-doped antiferromagnetic phase in high-temperature superconductors in the framework of the anisotropic t-J model.
Cumulants represent a natural language for expressing macroscopic properties of many-body systems. The most important property of cumulants is that of size consistency, i.e. a cumulant expression for an extensive variable scales with the size of the system, independent of possible further approximations used in the evaluation procedure. Cumulants can be considered as a generalization of linked diagrams known from diagrammatic technique of many-body theory. In this paper we outline a recently introduced method based on cumulants in order to derive expressions for zero-temperature properties of many-particle systems, i.e. the ground-state energy, static expectation values and dynamical correlation functions. This cumulant formalism allows one to describe weakly and strongly correlated systems along the same lines. We show that the coupled-cluster method known from quantum chemistry can be derived from our cumulant approach. Finally, we demonstrate the usefulness of the cumulant method by applying it to examples from solid-state physics and quantum chemistry.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
In a large variety of systems, like the high‐temperature superconductors, the dominant parameter of the system is the on‐site Coulomb repulsion that becomes comparable with the kinetic energy in the vicinity of the metal‐insulator transition. This implies that perturbation expansions must include this interaction in the zeroth order Hamiltonian and treat the hopping term as a perturbation. Thus, one gets a situation radically different from the usual one in which the particles are initially free and needs to develop a nonstandard perturbation theory based on a cumulant expansion. In this paper, we revisit the general formalism of the cumulant expansion approach for the single band and two‐band Hubbard model, being interested in introducing a systematic way to perform approximations which take into account cooperative phenomena. As an example we analyze the effect of the charge fluctuations on the single‐particle spectral properties.
We present a finite-temperature extension of the retarded cumulant Green's function for calculations of exited-state, correlation, and thermodynamic properties of electronic systems. The method incorporates a cumulant to leading order in the screened Coulomb interaction W, and improves on the GW approximation of many-body perturbation theory. Results for the homogeneous electron gas are presented for a wide range of densities and temperatures, from cool to warm dense matter regimes, which reveal several hitherto unexpected properties. For example, correlation effects remain strong at high T while the exchange-correlation energy becomes small; also the spectral function broadens and damping increases with temperature, blurring the usual quasiparticle picture. These effects are evident, e.g., in Compton scattering which exhibits many-body corrections that persist at normal densities and intermediate T. The approach also yields exchange-correlation energies and potentials in good agreement with existing methods.
A many-body perturbation theory is developed for the calculation of the antiferromagnetic ground state and lower excited states of one-electron atoms in the insulating state. It is assumed that the interatomic distances are moderately small and that the interactions are well described by exchange coupling, but that distortion of atoms in the lattice is still negligible. The method is essentially the Heitler-London approach: A set of Slater determinants is constructed from nonorthogonal atomic orbitals with all possible spin arrangements. The unperturbed ground state ∣0〉 has an alternating spin configuration, while excited states are generated one by one from ∣0〉 by interchanges of spins and by projecting out the lower excited states. The resulting orthonormal states introduce the concept of quasielectron (or quasihole) states and define creation and destruction operators of those quasiparticles. Although the quasiparticles are correlated with each other through the overlap integrals in the projected wave functions, the energy matrix elements can be decomposed into linked clusters of localized electrons and holes. It is found that the clusters thus generated are cumulants. The expression for the energy matrix is then transformed into the form of an effective Hamiltonian consisting of cumulants and of creation and destruction operators of quasifermions, and the concept of the exchange interaction is defined in terms of those operators. The new formulation not only facilitates the use of the familiar many-body perturbation theory, but also eliminates difficulties in handling spin variables in the Heisenberg theory of magnetism. If only the two-body Coulomb and exchange diagrams are retained and the higher-order diagrams are neglected, however, the effective Hamiltonian takes a form reminiscent of the anisotropic exchange Hamiltonian, ${H}_{\mathrm{spin}}=\ensuremath{-}{\ensuremath{\Sigma}}_{k,h}{J}_{\mathrm{kh}}[{{S}_{k}}^{z}{{S}_{h}}^{z}+\ensuremath{\lambda}({{S}_{k}}^{x}{{S}_{h}}^{x}+{{S}_{k}}^{y}{{S}_{h}}^{y})]$, which is used in the generalized Heisenberg theory and which includes the Ising model in its limit $\ensuremath{\lambda}\ensuremath{\rightarrow}0$. Finally, it is shown that excitations accompanying electron transfers are readily included in the present method as additional diagrams by an extension of the prescriptions for the generation of excited spin states. This yields, in its first approximation, the intra-atomic interaction and the electron-transfer interaction, leading to the Hamiltonian assumed by Hubbard and Anderson.
We present a simple numerical approach to compute dynamic correlation functions in quantum many-body systems at zero temperature. The method is based on a projective technique for the memory-function formalism, and requires as a starting point an independent evaluation of the ground-state energy and wave function. We illustrate the method with a one-dimensional model of spinless fermions and calculate the density-density, current-current, and single-particle correlation functions. The obtained spectral functions show different types of elementary excitations: particle-hole, soliton-antisoliton, exciton, etc. We compare our results with analytical results, and with those obtained through different approaches available in the literature.