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arXiv:1702.01710 · Physical review. E

Path-integral Monte Carlo simulation of time-reversal noninvariant bulk systems with a case study of rotating Yukawa gases

Tamás Haidekker Galambos, Csaba Tőke

Published Feb 26, 2018• 2 citations
arXivHTMLPDF

Abstract

We elaborate on the methodology to simulate bulk systems in the absence of time-reversal symmetry by the phase-fixed path-integral Monte Carlo method under (possibly twisted) periodic boundary conditions. Such systems include two-dimensional electrons in the quantum Hall regime and rotating ultracold Bose and Fermi gases; time-reversal symmetry is broken by an external magnetic field and the Coriolis force, respectively. We provide closed-form expressions in terms of Jacobi elliptic functions for the thermal density matrix (or the Euclidean propagator) of a single particle on a flat torus under very general conditions. We then modify the multislice sampling method in order to sample paths by the magnitude of the complex-valued thermal density matrix. Finally, we demonstrate that these inventions let us study the vortex melting process of a two-dimensional Yukawa gas in terms of the de Boer interaction strength parameter, temperature, and rotation (Coriolis force). The bosonic case is relevant to ultracold Fermi-Fermi mixtures of widely different masses under rotation.

DOI
10.1103/physreve.97.022140

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We elaborate on the methodology to simulate bulk systems in the absence of time-reversal symmetry by the phase-fixed path-integral Monte Carlo method under (possibly twisted) periodic boundary conditions. Such systems include two-dimensional electrons in the quantum Hall regime and rotating ultracold Bose and Fermi gases; time-reversal symmetry is broken by an external magnetic field and the Coriolis force, respectively. We provide closed-form expressions in terms of Jacobi elliptic functions for the thermal density matrix (or the Euclidean propagator) of a single particle on a flat torus under very general conditions. We then modify the multislice sampling method in order to sample paths by the magnitude of the complex-valued thermal density matrix. Finally, we demonstrate that these inventions let us study the vortex melting process of a two-dimensional Yukawa gas in terms of the de Boer interaction strength parameter, temperature, and rotation (Coriolis force). The bosonic case is relevant to ultracold Fermi-Fermi mixtures of widely different masses under rotation.
We provide a detailed description of the path-integral Monte Carlo worm algorithm used to exactly calculate the thermodynamics of Bose systems in the canonical ensemble. The algorithm is fully consistent with periodic boundary conditions, which are applied to simulate homogeneous phases of bulk systems, and it does not require any limitation in the length of the Monte Carlo moves realizing the sampling of the probability distribution function in the space of path configurations. The result is achieved by adopting a representation of the path coordinates where only the initial point of each path is inside the simulation box, the remaining ones being free to span the entire space. Detailed balance can thereby be ensured for any update of the path configurations without the ambiguity of the selection of the periodic image of the particles involved. We benchmark the algorithm using the non-interacting Bose gas model for which exact results for the partition function at finite number of particles can be derived. Convergence issues and the approach to the thermodynamic limit are also addressed for interacting systems of hard spheres in the regime of high density.
In general, Quantum Monte Carlo methods suffer from a sign problem when simulating fermionic systems. This causes the efficiency of a simulation to decrease exponentially with the number of particles and inverse temperature. To circumvent this issue, a nodal constraint is often implemented, restricting the Monte Carlo procedure from sampling paths that cause the many-body density matrix to change sign. Unfortunately, this high-dimensional nodal surface is not a priori known unless the system is exactly solvable, resulting in uncontrolled errors. We will discuss two possible routes to extend the applicability of finite-temperatue path integral Monte Carlo. First we extend the regime where signful simulations are possible through a novel permutation sampling scheme. Afterwards, we discuss a method to variationally improve the nodal surface by minimizing a free energy during simulation. Applications of these methods will include both free and interacting electron gases, concluding with discussion concerning extension to inhomogeneous systems.
We present diffusion Monte Carlo (DMC) and path-integral Monte Carlo (PIMC) calculations of a one-dimensional Bose system with realistic interparticle interactions in a periodic external potential. Our main aim is to test the predictions of the Luttinger liquid (LL) theory, in particular with respect to the superfluid-Mott insulator transition at both zero and finite temperatures, in the predicted robust and fragile superfluid regimes. For that purpose, we present our results of the superfluid fraction ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$, the one-body density matrix, the two-body correlation functions, and the static structure factor. The DMC and PIMC results in the limit of very low temperature for ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ agree, but the LL model for scaling ${\ensuremath{\rho}}_{s}/{\ensuremath{\rho}}_{0}$ does not fit the data well. The critical depth of the periodic potential is close to the values obtained for ultracold gases with different models of interaction, but with the same value of the bare LL parameter, demonstrating the universality of LL description. Algebraic decay of correlation functions is observed in the superfluid regime and exponential decay in the Mott-insulator one, as well as in all regimes at finite temperature for distances larger than a characteristic length.
Ultracold atomic systems have been of great research interest in the past, with more recent attention being paid to systems of mixed species. In this work, we carry out nonperturbative path-integral Monte Carlo (PIMC) simulations of NNN distinguishable particles at finite temperature, which can be thought of as an ultracold atomic system containing NNN distinct species. We use the PIMC approach to calculate thermodynamic properties of particles interacting via hard-sphere and hard-cavity potentials. The first problem we study is a two-particle system interacting via a hard-sphere and hard-cavity interaction in order to test the effectiveness of two approximations for the thermal density matrix corresponding to these potentials. We then apply the PIMC method to a system of many hard-sphere particles under periodic boundary conditions at varying temperature in order to calculate the energy per particle, pressure, and specific heat of the system. We examine how finite-size effects impact the results of PIMC simulations of hard-sphere particles and when the thermodynamic limit has been reached. Our results provide microscopic benchmarks for a system containing distinguishable particles, which can be thought of as a limiting case for ultracold atomic systems of mixed species.
We present extensive ab initio path-integral Monte Carlo (PIMC) simulations of two-dimensional quantum dipole few-body systems ($2\ensuremath{\le}N\ensuremath{\le}7$) in a harmonic confinement, taking into account both Bose- and Fermi-statistics. This allows us to study the nonclassical rotational inertia, which can lead to a negative superfluid fraction in the case of fermions [Phys. Rev. Lett. 112, 235301 (2014)]. Moreover, we study in detail the structural characteristics of such systems and are able to clearly resolve the impact of quantum statistics on density profiles and the respective shell structure. Further, we present results for a more advanced center-two-particle correlation function [Phys. Rev. E 91, 043104 (2015)], which allows detection of differences between Fermi and Bose systems even when they are almost absent in other observables like the density. Overall, we find that bosonic systems sensitively react to even small values of the dipole-dipole coupling strength, whereas such a weak interaction is effectively masked for fermions by the Pauli exclusion principle. In addition, the abnormal superfluid fraction for fermions is not reflected by the structural properties of the system, which are equal to the bosonic case even though the moments of inertia diverge from each other. Lastly, we have explored the possibility of fermionic PIMC simulations of quantum dipole systems despite the notorious fermion sign problem, which can be further extended in future investigations in this field.