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arXiv:cond-mat/0008209 · Physical Review Letters

Comment on “Solution of Classical Stochastic One-Dimensional Many-Body Systems”

Su‐Chan Park, Jeong-Man Park, Doochul Kim

Published Jul 24, 2000• 9 citations
arXivHTMLPDF

Abstract

In a recent Letter, Bares and Mobilia proposed the method to find solutions of the stochastic evolution operator $H=H_0 + {γ\over L} H_1$ with a non-trivial quartic term $H_1$. They claim, ``Because of the conservation of probability, an analog of the Wick theorem applies and all multipoint correlation functions can be computed.'' Using the Wick theorem, they expressed the density correlation functions as solutions of a closed set of integro-differential equations. In this Comment, however, we show that applicability of Wick theorem is restricted to the case $γ= 0$ only.

DOI
10.1103/physrevlett.85.892

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In a recent Letter, Bares and Mobilia proposed the method to find solutions of the stochastic evolution operator H=H0+γLH1H=H_0 + {γ\over L} H_1H=H0​+Lγ​H1​ with a non-trivial quartic term H1H_1H1​. They claim, ``Because of the conservation of probability, an analog of the Wick theorem applies and all multipoint correlation functions can be computed.'' Using the Wick theorem, they expressed the density correlation functions as solutions of a closed set of integro-differential equations. In this Comment, however, we show that applicability of Wick theorem is restricted to the case γ=0γ= 0γ=0 only.
Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.
We develop a highly efficient method to numerically simulate thermal fluctuations and correlations in non-relativistic continuous bosonic one-dimensional systems. We start by noticing the equivalence of their description through the transfer matrix formalism and a Fokker-Planck equation for a distribution evolving in space. The corresponding stochastic differential (Itō) equation is very suitable for computer simulations, allowing the calculation of arbitrary correlation functions. As an illustration, we apply our method to the case of two tunnel-coupled quasicondensates of bosonic atoms.
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The matrix element of a general many-body operator between two Slater determinants is calculated explicitly. For this, a split-and-pair method is introduced that provides a convenient expression of Wick's theorem and simplifies many-body calculations. The same method is used to determine the generating function of the matrix elements of many-body operators. The split-and-pair method allows also for the diagonalization of the density correlation operators $:n({x}_{1})\ensuremath{\cdots}n({x}_{k}):$, where $n(x)={\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(x)\ensuremath{\psi}(x)$ is the density operator. The relation between the split-and-pair method and quantum group theory is clarified.
We consider a system of N interacting bosons in the mean-field scaling regime and construct corrections to the Bogoliubov dynamics that approximate the true N-body dynamics in norm to arbitrary precision. The N-independent corrections are given in terms of the solutions of the Bogoliubov and Hartree equations and satisfy a generalized form of Wick's theorem. We determine the n-point correlation functions of the excitations around the condensate, as well as the reduced densities of the N-body system, to arbitrary accuracy, given only the knowledge of the two-point functions of a quasi-free state and the solution of the Hartree equation. In this way, the complex problem of computing all n-point correlation functions for an interacting N-body system is essentially reduced to the problem of solving the Hartree equation and the PDEs for the Bogoliubov two-point functions.