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arXiv:1210.3976 · Journal of Statistical Mechanics Theory and Experiment

A perturbative probabilistic approach to quantum many-body systems

Andrea Di Stefano, Massimo Ostilli, Carlo Presilla A

Published Apr 2, 2013• 1 citation
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Abstract

In the probabilistic approach to quantum many-body systems, the ground-state energy is the solution of a nonlinear scalar equation written either as a cumulant expansion or as an expectation with respect to a probability distribution of the potential and hopping (amplitude and phase) values recorded during an infinitely lengthy evolution. We introduce a perturbative expansion of this probability distribution which conserves, at any order, a multinomial-like structure, typical of uncorrelated systems, but includes, order by order, the statistical correlations provided by the cumulant expansion. The proposed perturbative scheme is successfully tested in the case of pseudo spin 1/2 hard-core boson Hubbard models also when affected by a phase problem due to an applied magnetic field.

DOI
10.1088/1742-5468/2013/04/p04002

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In the probabilistic approach to quantum many-body systems, the ground-state energy is the solution of a nonlinear scalar equation written either as a cumulant expansion or as an expectation with respect to a probability distribution of the potential and hopping (amplitude and phase) values recorded during an infinitely lengthy evolution. We introduce a perturbative expansion of this probability distribution which conserves, at any order, a multinomial-like structure, typical of uncorrelated systems, but includes, order by order, the statistical correlations provided by the cumulant expansion. The proposed perturbative scheme is successfully tested in the case of pseudo spin 1/2 hard-core boson Hubbard models also when affected by a phase problem due to an applied magnetic field.
We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy as well as the correlation functions in the ground state are exactly determined as series expansions in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of the analytic character of the present approach is discussed.
Abstract A systematic nonperturbative scheme for calculating the ground state energy is adapted for studying systems of strongly correlated electrons on a lattice. It includes a method for calculating the cumulants of the Hamiltonian and a method, using the t-expansion, for constructing successive approximations to the ground state energy by these cumulants. The scheme is applied to spinless fermion and Hubbard models, and a method is proposed to overcome the problems found in previous attempts to use this scheme to study the Hubbard model.
The ground-state wave function and energy of a finite system of interacting fermions are expanded in terms of multiple-particle excitations on an uncorrelated zero-order state. The resulting set of coupled equations constitutes a systematic variational generalization of Hartree-Fock theory. Comparison is made with many-body perturbation theory and it is shown that to any order the theory incorporates an infinite number of perturbation theory terms. Solutions of the equations for ground-state atomic systems are discussed and related to previous work using many-body perturbation theory. It is shown that the sums of perturbation terms necessary for convergence are automatically included in the equations for two-particle excitations. Application of the equations to open-shell atoms is described.
We propose in this paper an effective mean-field theory, which makes the calculation for any observable being performed in a general form of cumulant expansion. The effective mean-field equation with reasonable physical annotation, which is obtained from the lower bound extreme value condition of free energy, can be applied to general statistic mechanics and Lagrangian quantum theory. By applying it to lattices, our theory, for various models containing nonuniform system and nonequilibrium phenomenon, can give a realistic scheme to study the nonperturbative effect order by order. As a testing example to two-dimensional Ising model, we obtain a result approaching the Onsager solution order by order.
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.
A method for treating nonlinear stochastic systems is described which it is hoped will be useful in both the quantum-mechanical many-body problem and the theory of turbulence. In this method the true problem is replaced by models that lead to closed equations for correlation functions and averaged Green's functions. The model solutions are exact descriptions of possible dynamical systems, and, as a result, they display certain consistency properties. For example, spectral components of Green's functions which must be positive-definite in the true problem automatically are so for the models. The models involve a new stochastic element: Random couplings are introduced among an infinite collection of similar systems, the true problem corresponding to the limit where these couplings vanish. The method is first applied to a linear oscillator with random frequency parameter. The mean impulse-response function of the oscillator is obtained explicitly for two successive models. The results suggest the existence of a sequence of model solutions which converges rapidly to the exact solution of the true problem. Applications then are made to the Schrödinger equation of a particle in a random potential and to Burgers' analog for turbulence dynamics. For both problems, closed model equations are obtained which determine the average Green's function, the amplitude of the mean field, and the covariance of the fluctuating field. The model solutions can be expressed as sums of infinite classes of terms from the formal perturbation expansions of the solutions to the true problems. It is suggested that correspondence to stochastic models may be a useful criterion to help judge the validity of partial summations of perturbation series.