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Journal of Physics A Mathematical and Theoretical

Interacting electrons in one dimension: a path integral Monte Carlo study

Alexander P. Lyubartsev

Published Jun 12, 2007• 6 citations
Publisher

Abstract

Path integral Monte Carlo simulations have been carried out for a system of interacting fermionic particles in one dimension. Due to the fact that the sign problem can be completely eliminated for one-dimensional systems, such simulations allow one to obtain accurate energies and particle densities in arbitrary external potentials. Two cases were considered: electrons in the field of a uniform neutralizing background and in an effective field of atoms on a lattice. In the latter case, a clear asymmetry of the charge density distribution of conducting electrons and holes has been observed.

DOI
10.1088/1751-8113/40/26/003

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Path integral Monte Carlo simulations have been carried out for a system of interacting fermionic particles in one dimension. Due to the fact that the sign problem can be completely eliminated for one-dimensional systems, such simulations allow one to obtain accurate energies and particle densities in arbitrary external potentials. Two cases were considered: electrons in the field of a uniform neutralizing background and in an effective field of atoms on a lattice. In the latter case, a clear asymmetry of the charge density distribution of conducting electrons and holes has been observed.
An approach is presented to compute properties of excited states in path integral Monte Carlo simulations of quantum systems. The approach is based on the introduction of several images of the studied system which have the total wavefunction antisymmetric over permutations of these images, and a simulation of the whole system at low enough temperature. The success of the approach relies, however, on the solution of the sign problem for the corresponding system. In the cases when the sign problem may be resolved (as for fermions in one dimension), properties of excited states can be computed with high precision. This was demonstrated by a simulation of excited states of a single particle in one- and two-dimensional harmonic oscillators. An example of the simulation of excited states of several interacting electrons in one dimension is also given.
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 formulate a path-integral Monte Carlo algorithm for simulating lattice systems consisting of fictitious particles governed by a generalized exchange statistics. This method, initially proposed for continuum systems, introduces a continuous parameter ξξξ in the partition function that interpolates between bosonic (ξ=1ξ= 1ξ=1) and fermionic (ξ=−1ξ= -1ξ=−1) statistics. We generalize this approach to discrete lattice models and apply it to the two-dimensional Hubbard model of fictitious particles, including the Bose- and Fermi-Hubbard models as special cases. By combining reweighting and ξξξ-extrapolation techniques, we access both half-filled and doped regimes. In particular, we demonstrate that the method remains effective even in strongly correlated, doped systems where the fermion sign problem hinders conventional quantum Monte Carlo approaches. Our results validate the applicability of the fictitious particle framework on lattice models and establish it as a promising tool for sign-problem mitigation in strongly interacting fermionic systems.
High Resolution Image Download MS PowerPoint Slide We combine the recent η-ensemble path integral Monte Carlo approach to the free energy [Dornheim et al. Phys. Rev. B 2025 111, L041114] with a recent fictitious partition function technique based on inserting a continuous variable that interpolates between the bosonic and Fermionic limits [Xiong and Xiong J. Chem. Phys. 2022 157, 094112] to deal with the Fermion sign problem. As a practical example, we apply our setup to the warm, dense, uniform electron gas over a broad range of densities and temperatures. We obtain accurate results for the exchange–correlation free energy down to half the Fermi temperature and find excellent agreement with the state-of-the-art parametrization by Groth et al. [ Phys. Rev. Lett. 2017 119, 135001]. Our work opens up new avenues for the future study of a host of interacting Fermi systems, including warm dense matter, ultracold atoms, and electrons in quantum dots, and for Fermionic free energy calculations with unprecedented system size.
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.
Expanded-ensemble Monte Carlo method with Wang-Landau algorithm was used for calculations of the ratio of partition functions for classes of permutations in the problem of several interacting quantum particles (fermions) in an external field. Simulations for systems consisting of 2 up to 7 interacting particles in harmonic or Coulombic field were performed. The presented approach allows one to carry out calculations for low enough temperatures that makes it possible to extract data for the ground-state energy and low-temperature thermodynamics.