Bose–Einstein Condensation in Layered Lattices under Magnetic Field

Abstract

Since the experimental realization of Bose–Einstein condensation of ultra cold atoms, extensive studies have been conducted on boson systems. One of recent epochmaking technologies is the synthesis of a fictitious gauge field acting on neutral atoms, which opened another wide field of studies on bosons and/or fermions in a synthetic magnetic field. Charged bosons were studied more than half a century ago by Schafroth. Charged bosons in a magnetic field were then studied in some respects and a rigorous proof was given by Angelescu and Corviovei, who showed absence of Bose– Einstein condensation of free charged bosons in a uniform magnetic field. This is due to the divergence of the density of states at the lowest band energy, i.e., the lowest Landau level. However the situation changes in the presence of a periodic potential. Namely some external periodic potentials are able to restore the Bose–Einstein condensation of the free charge bosons in a magnetic field in threedimensions (3D), although there has been no actual calculation of the condensation temperature of bosons in a typical lattice structure under a magnetic field. According to these previous studies, it is expected that free cold bosonic atoms under a synthetic magnetic field, if realized experimentally, would not condensate in a continuum space but would have a finite Bose–Einstein condensation temperature in the presence of an optical lattice potential. The aim of this short note is to demonstrate the restoration of the Bose–Einstein condensation under a magnetic field by the introduction of a periodic potential, taking typical lattice structures as examples and calculating the condensation temperature analytically. In what follows, with focus on bosons in a 3D lattice under a uniform magnetic field, we calculate the condensation temperature for different lattices and different magnitude of the magnetic field. In particular we consider a layered structure of two dimensional lattices such as square, triangular and honeycomb ones, with a magnetic field applied in the direction normal to the two-dimensional (2D) lattices. The energy spectrum of a particle on a two dimensional lattice, which we assume to be a square lattice for a moment, under a uniform magnetic field shows a characteristic pattern, so-called Hofstadter’s diagram. The one-particle energy of this system is described by the following Hamiltonian in the tight-binding approximation.