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The thermal density matrix of a charged particle in an external magnetic field in twisted periodic boundary conditions

Tamás Haidekker Galambos, Csaba Tőke

Published Feb 6, 2017• 0 citations
OpenAlex

Abstract

We provide closed-form expressions for the thermal density matrix (or the Euclidean propagator) of a charged particle in a uniform external magnetic field on the flat torus under very general conditions. We propose this formula for use in phase-fixed path integral Monte Carlo simulation of time-reversal symmetry breaking systems, including quantum Hall systems and rotating Bose-Einstein condensates.

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We provide closed-form expressions for the thermal density matrix (or the Euclidean propagator) of a charged particle in a uniform external magnetic field on the flat torus under very general conditions. We propose this formula for use in phase-fixed path integral Monte Carlo simulation of time-reversal symmetry breaking systems, including quantum Hall systems and rotating Bose-Einstein condensates.
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 propose an efficient stochastic method to implement numerically the Bogolubov approach to study finite-temperature Bose-Einstein condensates. Our method is based on the Wigner representation of the density matrix describing the non-condensed modes and a Brownian motion simulation to sample the Wigner distribution at thermal equilibrium. Allowing it to sample any density operator Gaussian in the field variables, our method is very general and it applies both to the Bogolubov and to the Hartree-Fock Bogolubov approach, in the equilibrium case as well as in the time-dependent case. We think that our method can be useful to study trapped Bose-Einstein condensates in two or three spatial dimensions without rotational symmetry properties, as in the case of condensates with vortices, where the traditional Bogolubov approach is difficult to implement numerically due to the need to diagonalize very big matrices.
The relationship between the two different ordering procedures taking place in an ideal charged relativistic Bose gas in the presence of a uniform external magnetic field B is investigated in an arbitrary number of spatial dimensions d. In particular the analytical form of the charge-density condensate qc{q}_{c}qc​ and the magnetization density \ensuremath{\rho} are calculated as a function of B and the inverse temperature \ensuremath{\beta} for relativistic bosons without spin. In the small-field limit (B\ensuremath{\rightarrow}0) this system exhibits the complete restoration of the symmetry of the ideal relativistic Bose gas for all d at any finite temperature.