Exact Stochastic Mean-Field Approach to the Fermionic Many-Body Problem

Abstract

We investigate a reformulation of the dynamics of interacting fermion systems in terms of a stochastic extension of time-dependent Hartree-Fock equations. From a path-integral representation of the evolution operator, we show that the exact N-body state can be interpreted as a coherent average over Slater determinants evolving in a random mean-field. The imaginary time propagation is also presented and gives a similar scheme which converges to the exact ground state. In addition, the growth of statistical errors is examined to show the stability of this stochastic formulation.

Quantum jumps and entropy production

“…With the help of this property it is easy to demonstrate that the entropy production is non-negative [7]. In fact, applying the definition of convexity to the functional ρ → −tr {D(ρ) ln ρ} one is led to the following inequality which holds for all λ ∈ We divide by λ and perform the limit λ → 0 to arrive at −tr {D(ρ) ln(ρ th )} ≤ −tr {D(ρ) ln(ρ)} , which, by virtue of (26), is equivalent to inequality (19).…”

“…The inequality (19) can now be shown in two alternative ways. One way is to apply a theorem by Lindblad [24] to the dynamical semigroup Λ t = exp(Dt) whose generator is identical to the dissipator D(ρ) of the master equation, and to relate the functional (26) to the time-derivative of the relative entropy with respect to the Gibbs state ρ th .…”

“…This means that the stochastic evolution consists of smooth, deterministic parts which are interrupted by instantaneous quantum jumps. Recently, it has been shown that the stochastic wave function method can be generalized to enable the treatment of non-Markovian quantum processes [17] and of Bosonic [18] and of Fermionic [19] many-body systems by means of an appropriate stochastic representation in the doubled Hilbert space H ⊕ H.…”

Exact and approximate many-body dynamics with stochastic one-body density matrix evolution

“…Indeed the stochastic part is driven either by the kinetic energy part of the Hamiltonian or by a fixed one-body potential in the case of shell model Monte-Carlo calculations [20]. Recently a new formulation [1,21] has been proposed that combines the advantages of the Monte-Carlo methods and of the mean field theories. Application of functional integral theories are of great interest since they pave the way to a full implementation of the nuclear static and dynamical many-body problem using mean field theories in a well defined theoretical framework.…”

“…In this theory, jumps occur between densities formed of pairs of Slater determinants, D ab = |Φa Φ b |, where each state evolves according to the Stochastic Schrödinger Equation (SSE) given in ref. [1]. A stochastic Liouville-von Neumann equation is derived as well as the associated Bogolyubov-Born-Green-Kirwood-Yvon (BBGKY) hierarchy.…”