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Physical Review Letters

Stochastic Simulation of Quantum Systems and Critical Dynamics

T. Schneider, Marco Zannetti, R. Badii, Hans Rudolf Jauslin

Published Dec 3, 1984• 28 citations
Publisher

Abstract

We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical $d$-dimensional system is mapped onto the statics of an associated quantum system and of its classical ($d+1$)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent $\ensuremath{\Delta}$ and the static susceptibility exponents of three models.

DOI
10.1103/physrevlett.53.2191

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We propose the transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy eigenvalues of one-particle systems. In the case of many degrees of freedom, the critical dynamics of a classical ddd-dimensional system is mapped onto the statics of an associated quantum system and of its classical (d+1d+1d+1)-dimensional counterpart. These mappings yield an equality relating the dynamical critical exponent \ensuremathΔ\ensuremath{\Delta}\ensuremathΔ and the static susceptibility exponents of three models.
We propose transformation of the Langevin equation into an eigenvalue problem as the framework of the stochastic simulation of quantum systems. The potential of the method is illustrated by numerical results for the low-lying energy levels of one-particle systems and coupled harmonic oscillators. Another application is the relationship between the dynamics of a d-dimensional system evolving according to the Langevin equation, an associated d-dimensional quantum system, and its (d+1)-dimensional classical counterpart. On this basis we relate the critical dynamics to the static properties of an associated quantum system and its (d+1)-dimensional classical analog. This then leads to equalities and scaling relations between the static and dynamic critical exponents. As a specific example, we treat the time-dependent Ginzburg-Landau model without conservation laws. In the large-n limit, the critical dynamics is traced back to Gaussian (tricritical Lifshitz) behavior of the associated (d+1)-dimensional classical model. The static exponents describing the critical dynamics are determined, however, by the crossover from Lifshitz to Gaussian (tricritical Lifshitz) behavior.
General discussion of stochastic models In the hydrodynamic theories, one writes equations which describe the system at hand on long time and length scales, using local conservation laws and the presence of broken symmetries to determine the identity of these slowly varying quantities. However, one would like a way to take account of the effects of the more rapidly varying degrees of freedom in such theories. Implicitly, these more rapid degrees of freedom are present in the integrals which determine the transport coefficients in the hydrodynamic equations by way of Kubo relations, but no method is presented to calculate the required integrands. One way to take account of the other degrees of freedom is to take on the entire many body problem all the way down to the atomic or electronic level, as one does in molecular dynamics simulations of various sorts. However, it is useful to have some approximate analytical ways to attack this problem as well. Here we review the basis for the most common such approach, the Langevin equation. For most of this discussion, we will assume that the identity of the slow variable or variables is known and that the separation of time scales is extreme so that the faster variables are essentially instantaneous in a sense we will discuss. These assumptions are not often particularly well justified. However, the resulting formulation has yielded very useful insights and is an important part of the subject.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
We establish a novel correspondence between the spacetime correlators of one-dimensional (1D) N-particle classical stochastic models described by a Langevin equation with that of the ground-state dynamics of a class of integrable 1D interacting many-body quantum models of the supersymmetric elliptic type. We show that these seemingly different concepts of stochastic systems, supersymmetry and quantum integrability can be viewed in a unified framework. Starting with an Lax matrix, we show that row (column) sums driven by a Gaussian noise term may be interpreted as a set of forward (backward) Langevin equations. Then, following functional path integral methods of stochastic quantization, we straightforwardly find an associated supersymmetric 1D quantum Hamiltonian. If, further, the classical stochastic system consists of two-body interactions only and we also want the quantum interactions to be of two-body type, we find that the only class of interactions permissible for the quantum models corresponds to the elliptic models. The algebraic structure that emerges very naturally reproduces the proof of integrability and allows the identification of the ground-state wavefunction of these quantum models.
The authors propose a transformation of the Langevin equation into an eigenvalue problem as a method to construct systems with soluble ground-state properties. As examples, they discuss quantum systems resulting from the Toda and sine-Gordon chains evolving according to the Langevin equation. Some ground-state properties are then evaluated with methods originally devised to calculate the partition function of one-dimensional classical systems. They also present numerical results for the two-phonon bound-state frequency and its coupling constant dependence by simulating a generalised quantum sine-Gordon system.