Cluster expansion of operator averages for systems of many particles

A. Foerster, H. A. M. Leymann, Jan Wiersig

Виктор Павлович Маслов

S. Guha, Goutam Sanyal, Seikh Hannan Mandal +1 more

We have developed in this paper a nonperturbative cluster-expansion strategy for generating the stochastically averaged time-evolution operator for quantum systems driven by Ornstein-Uhlenbeck (OU) colored noise. The method induces a boson mapping of the (real or complex) stochastic variable f, and interprets the stochastic average of a pair of variables f at two different times as the expectation value of the time-ordered product of the associated bosons with respect to the boson vacuum \ensuremath{\Vert}${0}_{\mathit{B}}$〉. The stochastic evolution is thus mapped onto a deterministic evolution in an expanded Fock space. The evolution of the system from the groups of state of interest is monitored using our recently developed nonperturbative time-dependent multireference coupled-cluster (TDMRCC) method. In this, the evolution operator U is written in a factorized form ${\mathit{U}}_{\mathrm{ex}}$${\mathit{U}}_{\mathit{M}}$, where ${\mathit{U}}_{\mathit{M}}$ evolves in the space of the starting functions (model space) and ${\mathit{U}}_{\mathrm{ex}}$ brings in the virtual functions. ${\mathit{U}}_{\mathrm{ex}}$ and ${\mathit{U}}_{\mathit{M}}$ are both written as normal ordered exponentials involving cluster operators. In the present context, the TDMRCC method translates into one for generating the evolution operator ${\mathit{U}}_{\mathit{B}}$ for the Hamiltonian containing the additional boson variables, and the stochastic averaging of ${\mathit{U}}_{\mathit{B}}$ is realized as the expectation value 〈${0}_{\mathit{B}}$\ensuremath{\Vert}${\mathit{U}}_{\mathit{B}}$\ensuremath{\Vert}${0}_{\mathit{B}}$〉.We call the TDMRCC method involving the expanded Fock space a cluster cumulant method. We have analyzed the relation of the cluster cumulant approach with the Fox-Kubo operator cumulant expansion and the method of marginal averages involving the Fokker-Planck operator of the OU process. It has been shown that an order by order expansion of our equations in the power of stochastic coupling generates the perturbative cumulant results of Fox and Kubo. It is also demonstrated that the traditional Fokker-Planck method of marginal averages uses a Kubo-Schr\"odinger operator which is related to our boson mapped Hamiltonian by a transformation which converts the Fokker-Planck operator \ensuremath{\Gamma} to a manifestly Hermitian form. However, the method of marginal averages involves a linear expansion of ${\mathit{U}}_{\mathit{B}}$ in terms of eigenfunctions of \ensuremath{\Gamma}, while the cluster cumulant method uses a cluster expansion, which is nonlinear in the expansion parameters. The former is thus analogous to a configuration-interaction expansion, while the latter is analogous to a coupled-cluster strategy. The method is illustrated by applying it to a simple yet nontrivial problem, viz., a harmonic oscillator linearly perturbed by a (real or complex) stochastic variable. Survivalities of both the ground and the first excited states have been considered for a wide range of coupling strengths and noise correlation times. Our method provides exact results in this case; these have been used as benchmarks for assessing performance of the second- and fourth-order cumulant results.

D. J. Thouless

In order to go over from a perturbation expansion of the grand partition function (the unlinked-cluster expansion) to an expansion of the thermodynamic potential (the linked-cluster expansion) in powers of the interaction, it is necessary to treat carefully those terms in which creation (or annihilation) operators for the same state occur twice or more. The unlinked- and linked-cluster expansions for a system of fermions are here shown to be equivalent by a direct comparison of the terms which occur in each. The relation between the two expansions is illustrated by the example of a system of fermions interacting only with a single-particle potential.

Hiroshi Ueyama

Ihor Gapyak, Viktor Gerasimenko

UDC 530.145:517.98 We present a method of cluster expansions for groups of operators associated with the von-Neumann equations for states and the Heisenberg equations for observables aimed at construction of generating operators for the solutions to the Cauchy problem for hierarchies of evolution equations of many-particle quantum systems.