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Institute of Physics eBooks

Path integral Monte Carlo

V. Apaja

Published Aug 1, 2025• 0 citations
Publisher

Abstract

Using imaginary time, quantum statistical mechanics of a finite-temperature system in thermodynamical equilibrium turns into a problem of classical ring polymers made of particle paths. We need to consider all possible ring polymers at a given temperature, which again is where Monte Carlo methods are effective, especially using a so-called worm algorithm. We show how paths can be sampled using Monte Carlo, and how the physical properties, such as energy and radial distribution, are computed from the paths. Identical particles can have their paths exchanged at will, and we see how boson superfluidity comes out of this picture. Finally, the stochastic series expansion method is introduced to show finite temperature Hubbard model can be simulated.

DOI
10.1088/978-0-7503-6310-5ch5

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It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.

Cambridge University Press eBooks•May 31, 2016•1 citation
Using imaginary time, quantum statistical mechanics of a finite-temperature system in thermodynamical equilibrium turns into a problem of classical ring polymers made of particle paths. We need to consider all possible ring polymers at a given temperature, which again is where Monte Carlo methods are effective, especially using a so-called worm algorithm. We show how paths can be sampled using Monte Carlo, and how the physical properties, such as energy and radial distribution, are computed from the paths. Identical particles can have their paths exchanged at will, and we see how boson superfluidity comes out of this picture. Finally, the stochastic series expansion method is introduced to show finite temperature Hubbard model can be simulated.
We review the path integral method wherein quantum systems are mapped with Feynman's path integrals onto a classical system of "ring-polymers" and then simulated with the Monte Carlo technique. Bose or Fermi statistics correspond to possible "cross-linking" of polymers. As proposed by Feynman, superfluidity and Bose condensation result from macroscopic exchange of bosons. To map fermions onto a positive probability distribution, one must restrict the paths to lie in regions where the fermion density matrix is positive. We discuss a recent application to the two-component electron-hole plasma. At low temperature excitons and bi-excitons form. We have used nodal surfaces incorporating paired fermions and see evidence of a Bose condensation in the energy, specific heat and superfluid density. In the restricted path integral picture, pairing appears as intertwined electron-hole paths. Bose condensation occurs when these intertwined paths wind around the periodic boundaries.
In 1953, Richard Feynman showed how superfluidity can be understood from the point of view of imaginary-time path integrals. In this picture, each atom becomes a cyclic polymer and the superfluid transition occurs when the atomic paths join up to make larger units which stretch a macroscopic distance. Boson statistics imply that all possible ways of connecting or "permuting" the "polymers" have equal weight. Feynman's path integrals are an exact isomorphism between quantum statistical mechanics of bosons and the classical statistical mechanics of these special "polymers". We have implemented Feynman's path integral method on a computer. Efficient methods, based on the Metropolis algorithm have been developed to move the paths quickly through path configuration space and permutation space. In addition, accurate approximations to the many-body off-diagonal density matrix involving pairs and triplets of atoms have been calculated.
Path integral Monte Carlo is paths in imaginary time, so it can be re-formulated to give the ground state using the principles of diffusion Monte Carlo imaginary time evolution. Instead of following paths that make up ring polymers, one can start the paths from points given by an analytical wave function. Unlike diffusion Monte Carlo, which can simulate imaginary time evolution for an arbitrarily long time, the path integral ground state method can evolve for a finite imaginary time. This evolution can still be long enough to project out the ground state, with the benefit that the process is very stable.
Trapped bosons exhibit fundamental physical phenomena and are at the core of emerging quantum technologies. We present a method for simulating bosons using path integral molecular dynamics. The main difficulty in performing such simulations is enumerating all ring-polymer configurations, which arise due to permutations of identical particles. We show that the potential and forces at each time step can be evaluated by using a recurrence relation which avoids enumerating all permutations, while providing the correct thermal expectation values. The resulting algorithm scales cubically with system size. The method is tested and applied to bosons in a 2-dimensional (2D) trap and agrees with analytical results and numerical diagonalization of the many-body Hamiltonian. An analysis of the role of exchange effects at different temperatures, through the relative probability of different ring-polymer configurations, is also presented.
The goal of this work is to lay the groundwork to construct and characterize a quantum device; which we refer to as a superfluid ring lattice; that could serve as a multi-qubit system in the future. Accordingly, a mathematical framework, called the Integer Lattice Method (ILM), is exploited to construct a two-dimensional optical landscape which could facilitate a superfluid ring qubit. The Integer Lattice Method allows one to design and explore both periodic and quasi-periodic structured coherent wave interference patterns. Furthermore, the formalism allows for a direct link to experimental realization. The lattices obtained from ILM can be investigated using WA-PIMC. In particular, the spatial and superfluid density at equilibrium are observables of interest to obtain our objective. In this thesis, the ab-initio Path Integral Monte Carlo Worm algorithm is implemented with the future aim of investigating the construction of superfluid ring lattices. It is written in independent and interchangeable modules in the novel programming language Julia. To ensure scalability, a custom nearest-neighbour algorithm is added. The simulation software is benchmarked by comparing it with exact analytical results from the non-interacting Bose gas model. Emerging issues with our implementation of the Worm algorithm are identified and thoroughly investigated. Finally, some preliminary results on the equilibrium density distribution of the found ILM lattices are presented and discussed.
It is shown from first principles that, in spite of the large interatomic forces, liquid 4 He should exhibit a transition analogous to the transition in an ideal Bose–Einstein gas. The exact partition function is written as an integral over trajectories, using the space-time approach to quantum mechanics. R.P. Feynman, 1953 Summary In this chapter we discuss imaginary-time path integrals and the path-integral Monte Carlo method for the calculation of properties of quantum systems at non-zero temperature. We discuss how Fermi and Bose statistics enter, and how to generalize the fixed-node procedure to non-zero temperature. We then discuss an auxiliary-field method for the Hubbard model. The path-integral method can be used to perform ground-state calculations, allowing calculations of properties with less bias than the projector Monte Carlo method. We also discuss the problem of estimating real-time response functions using information from imaginary-time correlation functions. In previous chapters we described two QMC methods, namely variational QMC and projector (diffusion) QMC. Both of these methods are zero temperature, or, more properly, are formulated for single states. In this chapter we discuss the path-integral Monte Carlo (PIMC) method, which is explicitly formulated at non-zero temperature. Directly including temperature is important because many, if not most, measurements and practical applications involve significant thermal effects. One might think that to do calculations at a non-zero temperature we would have to explicitly sum over excited states. Such a summation would be difficult to accomplish once the temperature is above the energy gap, because there are so many possible many-body excitations. In addition, the properties for each excitation are more difficult to calculate than those for the ground state. As we will see, path-integral methods do not require an explicit sum over excitations. As an added bonus, they provide an interesting and enlightening window through which to view quantum systems. However, the sign problem, introduced in the previous chapter, is still present for fermion systems. The fixed-node approximation is used again. An advantage of PIMC is the absence of a trial wavefunction. As a result, quantum expectation values, including ones not involving the energy, can be computed directly. For the expert, the lack of an importance function may seem a disadvantage; without it one cannot push the simulation in a preferred direction.