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International Journal of Modern Physics C

Relativistic path integral Monte Carlo: Relativistic oscillator problem

Aleksandr Ivanov, Alexander Novoselov, Oleg Pavlovsky

Published Apr 5, 2016• 3 citations
Publisher

Abstract

Relativistic generalization of path integral Monte Carlo (PIMC) method has been proposed. The problem of relativistic oscillator has been studied in the framework of this approach. Ultra-relativistic and nonrelativistic limits have been discussed. We show that PIMC method can be effectively used for investigation of relativistic systems.
DOI
10.1142/s0129183116501333

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Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic quantum spin systems by sampling from a classical Gibbs distribution using Markov chain Monte Carlo. The PIMC method has been widely used to study the physics of materials and for simulated quantum annealing, but these successful applications are rarely accompanied by formal proofs that the Markov chains underlying PIMC rapidly converge to the desired equilibrium distribution. In this work we analyze the mixing time of PIMC for 1D stoquastic Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well as disordered XY spin chains with nearest-neighbor interactions. By bounding the convergence time to the equilibrium distribution we rigorously justify the use of PIMC to approximate partition functions and expectations of observables for these models at inverse temperatures that scale at most logarithmically with the number of qubits. The mixing time analysis is based on the canonical paths method applied to the single-site Metropolis Markov chain for the Gibbs distribution of 2D classical spin models with couplings related to the interactions in the quantum Hamiltonian. Since the system has strongly nonisotropic couplings that grow with system size, it does not fall into the known cases where 2D classical spin models are known to mix rapidly.