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Physical review. B, Condensed matter

Langevin simulations of quantum systems

R. Badii, T. Schneider, Mads Peter Soerensen

Published Jan 1, 1987• 5 citations
Publisher

Abstract

We explore the potential of the Langevin simulation of quantum systems. A generalized quantum sine-Gordon chain, a generalized Toda chain, and impenetrable bosons on a ring are treated specifically to estimate low-lying excitation energies from the long-time behavior of the Langevin process. In the sine-Gordon case, we investigated the dependence of the bound-state frequency on the coupling constant, while in the Toda chain the phonon dispersion curves are obtained for different coupling constants. Finally, for impenetrable bosons the estimated dynamic form factor for density fluctuations is compared with exact results.

DOI
10.1103/physrevb.35.297

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We explore the potential of the Langevin simulation of quantum systems. A generalized quantum sine-Gordon chain, a generalized Toda chain, and impenetrable bosons on a ring are treated specifically to estimate low-lying excitation energies from the long-time behavior of the Langevin process. In the sine-Gordon case, we investigated the dependence of the bound-state frequency on the coupling constant, while in the Toda chain the phonon dispersion curves are obtained for different coupling constants. Finally, for impenetrable bosons the estimated dynamic form factor for density fluctuations is compared with exact results.
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.
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.
Inspired by quantum walks (QW), we propose the concept of classical phonon random walks (PRW). With this we show a correspondence between classical and quantum dynamics. To be specific, we explain how a classical chain of particles with linear interactions can be used to simulate a quantum like wave function ψm(t)\psi_{m}(t)ψm​(t). In particular, the real (imaginary) part of ψm(t)\psi_{m}(t)ψm​(t) is the normalized classical momentum (a new π/2\pi/2π/2 shifted momentum) correlation function of the thermalized many-body system. Energy and heat densities are directly analogous to wave function's modulus squared ∣ψm(t)∣2\left|\psi_{m}(t)\right|^{2}∣ψm​(t)∣2. We also propose a formula predicting the non-universal spreading of these energy packets in terms of the phonon dispersion of the system. Our equilibrium molecular dynamics simulations of several linear chains perfectly verify these relations. The extension of the concept to integrable Toda chain is also discussed.
The sine-Gordon field theory emerges as the low-energy description in a wealth of quantum many-body systems. Recent efforts have been directed towards realizing quantum simulators of the model, by interfering two weakly coupled one-dimensional cold atomic gases. The weak interactions within the atomic clouds provide a sine-Gordon realization in the semiclassical regime. Furthermore, the intricate microscopic dynamics prevents a quantitative understanding of the effective sine-Gordon validity realm. In this work, we focus on a spin-ladder realization and observe the emergent sine-Gordon dynamics deep in the quantum regime. We use matrix-product state techniques to numerically characterize the low-energy sector of the system and compare it with the exact field-theory predictions. From this comparison, we obtain quantitative boundaries for the validity of the sine-Gordon description. We provide encompassing evidence for the emergent field theory by probing its rich spectrum and by observing the signatures of integrable dynamics in scattering events.
The aim of this work is to study the dynamics of one-dimensional Klein-Gordon anharmonic chains of interacting bosons. Especially, we focus our attention on the quantum version of Hamiltonian perturbation theory, which allows us to achieve the Hamiltonian of our bosonic system written in normal form. More precisely we show that Hamiltonian normal form of the quoted system consists in a Bose-Hubbard-like Hamiltonian. We consider both the case of short-range and long-range (all-to-all) interaction. We also treat the case of disordered chains with a distribution of the local potentials.
A path integral simulation algorithm which includes a higher-order Trotter approximation (HOA)is analyzed and compared to an approach which includes the correct quantum mechanical pair interaction (effective Propagator (EPr)). It is found that the HOA algorithmconverges to the quantum limit with increasing Trotter number P as P^{-4}, while the EPr algorithm converges as P^{-2}.The convergence rate of the HOA algorithm is analyzed for various physical systemssuch as a harmonic chain,a particle in a double-well potential, gaseous argon, gaseous helium and crystalline argon. A new expression for the estimator for the pair correlation function in the HOA algorithm is derived. A new path integral algorithm, the hybrid algorithm, is developed.It combines an exact treatment of the quadratic part of the Hamiltonian and thehigher-order Trotter expansion techniques.For the discrete quantum sine-Gordon chain (DQSGC), it is shown that this algorithm works more efficiently than all other improved path integral algorithms discussed in this work. The new simulation techniques developed in this work allow the analysis of theDQSGC and disordered model systems in the highly quantum mechanical regime using path integral molecular dynamics (PIMD)and adiabatic centroid path integral molecular dynamics (ACPIMD).The ground state phonon dispersion relation is calculated for the DQSGC by the ACPIMD method.It is found that the excitation gap at zero wave vector is reduced by quantum fluctuations. Two different phases exist: One phase with a finite excitation gap at zero wave vector, and a gapless phase where the excitation gap vanishes.The reaction of the DQSGC to an external driving force is analyzed at T=0.In the gapless phase the system creeps if a small force is applied, and in the phase with a gap the system is pinned. At a critical force, the systems undergo a depinning transition in both phases and flow is induced. The analysis of the DQSGC is extended to models with disordered substrate potentials. Three different cases are analyzed: Disordered substrate potentials with roughness exponent H=0, H=1/2,and a model with disordered bond length. For all models, the ground state phonon dispersion relation is calculated.