Path integral Monte Carlo study of tunneling in quantum many-particle systems

Carlos P. Herrero, Gerd Schön, Andrei D. Zaikin

Abstract Strong electron tunneling effects in the single-electron box are investigated by means of a path integral Monte Carlo analysis. Our results for the effective capacitance of the box are compared to previous theoretical predictions and Monte Carlo simulations. We discuss the accuracy of our simulations and convergence of our results as a function of the Trotter number used in the discretization of the path integrals.

Sergio Boixo, Sergei V. Isakov, Guglielmo Mazzola +4 more

William G. Dawkins, Alexandros Gezerlis

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 $N$ distinguishable particles at finite temperature, which can be thought of as an ultracold atomic system containing $N$ 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.

Markus Holzmann

Shiwei Zhang

Estelle M. Inack, Giuliano Giudici, Tommaso Parolini +2 more

In simple ferromagnetic quantum Ising models characterized by an effective double-well energy landscape the characteristic tunneling time of path-integral Monte Carlo (PIMC) simulations has been shown to scale as the incoherent quantum-tunneling time, i.e., as $1/{\mathrm{\ensuremath{\Delta}}}^{2}$, where $\mathrm{\ensuremath{\Delta}}$ is the tunneling gap. Since incoherent quantum tunneling is employed by quantum annealers (QAs) to solve optimization problems, this result suggests that there is no quantum advantage in using QAs with respect to quantum Monte Carlo (QMC) simulations. A counterexample is the recently introduced shamrock model (Andriyash and Amin, arXiv:1703.09277), where topological obstructions cause an exponential slowdown of the PIMC tunneling dynamics with respect to incoherent quantum tunneling, leaving open the possibility for potential quantum speedup, even for stoquastic models. In this work we investigate the tunneling time of projective QMC simulations based on the diffusion Monte Carlo (DMC) algorithm without guiding functions, showing that it scales as $1/\mathrm{\ensuremath{\Delta}}$, i.e., even more favorably than the incoherent quantum-tunneling time, both in a simple ferromagnetic system and in the more challenging shamrock model. However, a careful comparison between the DMC ground-state energies and the exact solution available for the transverse-field Ising chain indicates an exponential scaling of the computational cost required to keep a fixed relative error as the system size increases.