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Journal of the Physical Society of Japan

Solution of the Bloch Equation for Many-Particle Systems in Terms of the Path Integral

Tohru Morita

Published Oct 1, 1973• 16 citations
Publisher

Abstract

The solution of the Bloch equation for many-particle systems enclosed in a box is given in terms of the path integral. The boundary conditions we consider are “zero on the boundary” and “the periodic boundary condition”. The obtained expression under the periodic boundary condition is appropriate for the Monte Carlo calculation of the density matrix and the ground state energy of Bose fluids and quantum crystals.

DOI
10.1143/jpsj.35.980

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The solution of the Bloch equation for many-particle systems enclosed in a box is given in terms of the path integral. The boundary conditions we consider are “zero on the boundary” and “the periodic boundary condition”. The obtained expression under the periodic boundary condition is appropriate for the Monte Carlo calculation of the density matrix and the ground state energy of Bose fluids and quantum crystals.
The solution of the Bloch equation for many-particle systems on a finite lattice is given in terms of the “path summation”. The boundary conditions we consider are “zero on the boundary” and “the periodic boundary condition”. The obtained expression under the periodic boundary condition is appropriate for the Monte Carlo calculation of the density matrix and the ground state energy of Bose lattice systems and the Heisenberg magnet with a ferromagnetic interaction. For the antiferromagnetic Heisenberg magnet, the present formalism is useful only for the lattices in which each of the closed paths has an even number of edges.
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.
The equilibrium properties of a single quantum particle (qp) interacting with a classical gas for a wide range of temperatures that explore the system's behavior in the classical as well as in the quantum regime is investigated. Both the qp and atoms are restricted to the sites of a one-dimensional lattice. A path integral formalism is developed within the context of the canonical ensemble in which the qp is represented by a closed, variable-step random walk on the lattice. Monte Carlo methods are employed to determine the system's properties. For the case of a free particle, analytical expressions for the energy, its fluctuations, and the qp-qp correlation function are derived and compared with the Monte Carlo simulations. To test the usefulness of the path integral formalism, the Metropolis algorithm is employed to determine the equilibrium properties of the qp for a periodic interaction potential, forcing the qp to occupy extended states. We consider a striped potential in one dimension, where every other lattice site is occupied by an atom with potential ε, and every other lattice site is empty. This potential serves as a stress test for the path integral formalism because of its rapid site-to-site variation. An analytical solution was determined in this case by utilizing Bloch's theorem due to the periodicity of the potential. Comparisons of the potential energy, the total energy, the energy fluctuations, and the correlation function are made between the results of the Monte Carlo simulations and the analytical calculations.
By the superposition of Bethe's wave functions, using the Lieb's solution for the system of identical bosons interacting in one dimension via a $\ensuremath{\delta}$-function potential, we construct the wave function of the corresponding system enclosed in a box by imposing the boundary condition that the wave function must vanish at the two ends of an interval. Coupled equations for the energy levels are derived, and approximately solved in the thermodynamic limit in order to calculate the boundary energy of this Bose gas in its ground state. The method of superposition is also applied to the analogous problem of the Heisenberg-Ising chain (not the ring).
We develop an algorithm for bosonic path integral molecular dynamics (PIMD) simulations with periodic boundary conditions (PBC) that scales quadratically with the number of particles. Path integral methods are powerful tools for simulating bosonic condensed phases, which exhibit fundamental physical phenomena such as Bose-Einstein condensation and superfluidity. Recently, we developed a quadratic-scaling algorithm for bosonic PIMD but employed an ad hoc treatment of PBC. Here, we rigorously enforce PBC in bosonic PIMD. This requires summing over the spring energies of all periodic images in the partition function, and a naïve implementation scales exponentially with system size. We present an algorithm for bosonic PIMD simulations of periodic systems that scales only quadratically. We benchmark our implementation on the free Bose gas and a model system of cold atoms in optical lattices. We also study an approximate treatment of PBC based on the minimum-image convention and derive a numerical criterion to determine when it is valid.
A formula is derived that relates the ground-state dispersion of a many-body system with the end-to-end distribution of paths with open boundary conditions in imaginary time. The formula does not involve the energy estimator. It allows direct measurement of the ground-state dispersion by quantum Monte Carlo methods without analytical continuation or auxiliary fitting. The formula is applied to the lattice polaron problem. The exact polaron spectrum and density of states are calculated for several models in one, two, and three dimensions. In the adiabatic regime of the Holstein model, the polaron density of states deviates spectacularly from the free-particle shape.