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Condensation of bosons with several degrees of freedom

Rafael Delgado López

Published Jun 1, 2013• 0 citations
OpenAlex

Abstract

The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited as a first step to tackle the more complex case of a molecular condensate, where several degrees of freedom have to be taken into account. \nIn the charged bose gas, the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to (1/2) NkB and a gradual condensation at lower temperatures. \nIn the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs.

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Journal of the Physical Society of Japan•Sep 12, 2013•0 citations
The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited as a first step to tackle the more complex case of a molecular condensate, where several degrees of freedom have to be taken into account. \nIn the charged bose gas, the conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to (1/2) NkB and a gradual condensation at lower temperatures. \nIn the molecular case, which is currently in progress, we have studied the condensation of rotational levels in a two–dimensional trap within the Bogoliubov approximation, showing that multi–step condensation also occurs.
The condensation of the spinless ideal charged Bose gas in the presence of a magnetic field is revisited. The conventional approach is extended to include the macroscopic occupation of excited kinetic states lying in the lowest Landau level, which plays an essential role in the case of large magnetic fields. In that limit, signatures of two diffuse phase transitions (crossovers) appear in the specific heat. In particular, at temperatures lower than the cyclotron frequency, the system behaves as an effectively one-dimensional free boson system, with the specific heat equal to 1/2Nk(B) and a gradual condensation at lower temperatures.
Abstract We study the effects of a uniform and constant magnetic field on the Bose–Einstein condensation (BEC) of relativistic charged scalar bosons. The condensation of magnetized charged bosons is usually discussed in the weak or strong field regimes separately, and in each case, the properties predicted for the condensate are different. Here, we develop a low‐temperature analysis suitable for any field that allows us to observe how the gas behaves for intermediate values of the magnetic field. We find that BEC may occur at finite magnetic fields, despite the gas becoming effectively one‐dimensional due to Landau quantization of the momentum perpendicular to the magnetic axis.
The statistical mechanics of a system of non-relativistic charged particles in a constant magnetic field is discussed. The spatial dimension DDD is arbitrary with D≥3D\geq 3D≥3 assumed. Calculations are presented from first principles using the effective action method. For D≥5D\geq 5D≥5 the system has a phase transition with a Bose condensate. We show how the effective action method method deals with in a very natural way with the condensate and study it's rôle in the magnetization of the gas. For large values of the magnetic field we show how the magnetized gas in DDD spatial dimensions behaves like the free Bose gas in (D−2)(D-2)(D−2) spatial dimensions. Even though for D=3 the magnetized gas does not have a phase transition for any non-zero value of the magnetic field, we show how the specific heat starts to resemble the result for the free gas as the magnetic field is reduced. A number of analytical approximations for the magnetization and specific heat are given and compared with numerical results. In this way we are able to study in precise detail how the B→0B\to 0B→0 limit of the magnetized gas is achieved.
We present closed-form, analytical expressions for the thermodynamic properties of an ideal, two-dimensional (2D) charged Fermi or Bose gas in the presence of a uniform magnetic field of arbitrary strength. We consider both the homogeneous quantum gas (in which case our expressions are exact) and the inhomogeneous gas within the local-density approximation. Our results for the Fermi gas are relevant to the current-density-functional theory of low-dimensional electronic systems in magnetic fields. For a 2D charged Bose gas (CBG) in a homogeneous magnetic field, we show that the uniform system undergoes a sharp transition at a critical temperature ${T}_{c}^{\ensuremath{\star}},$ below which there is a macroscopic occupation of the lowest Landau level. An examination of the one-body density matrix, however, reveals the absence of long-range order, thereby indicating that the transition cannot be interpreted to a Bose-Einstein condensate. Nevertheless, for $T<{T}_{c}^{\ensuremath{\star}}$ and weak magnetic fields, the system still exhibits magnetic properties which are practically indistinguishable from those of a condensed, superconducting CBG. We therefore conclude that while a condensate is a sufficient condition for the ideal CBG to exhibit a superconducting state, it may not be a necessary condition.
The ideal Bose gas in a highly anisotropic harmonic potential is studied. It is found that Bose-Einstein condensation occurs in two distinct steps as the temperature is lowered. In the first step the specific heat shows a sharp feature, but the system still occupies many one-dimensional quantum states. In the second step, at a significantly lower temperature, the ground state becomes macroscopically occupied. It should be possible to verify these predictions using present-day atom traps. The two-step behavior can occur in a rather general class of anisotropic traps, including the box potential.
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.