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arXiv:1912.05218

Nontrivially Topological Phase Structure of Ideal Bose Gas System within Different Boundary Conditions

Yonghui Xia, Hong-tao Feng, Hong-Shi Zong

Published Dec 11, 2019• 0 citations
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Abstract

The phase structure of ideal Bose gas system within different boundary conditions, i.e., the periodic boundary condition and Dirichlet boundary condition in this work, in an infinite volume, is investigated. It is found that the ground states of ideal Bose gas within those two boundary conditions are both topologically nontrivial, which can not be classified by the traditional symmetry breaking theory. The ground states are different topological phases corresponding to those two boundary conditions, which can be distinguished by the off--diagonal particle number susceptibility. Moreover, this result is universal. The boundary condition may play an important role in pining the critical endpoint of QCD diagram on the approach of the lattice simulations and the computation of some solvable statistical models .

DOI
10.48550/arxiv.1912.05218

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We investigate the probability distribution that governs the number of ground-state particles in a partially condensed ideal Bose gas confined to a cubic volume within the canonical ensemble. Imposing either periodic or Dirichlet boundary conditions, we derive asymptotic expressions for all its cumulants. Whereas the condensation temperature becomes independent of the boundary conditions in the large-system limit, as implied by Weyl's theorem, the fluctuation of the number of condensate particles and all higher cumulants remain sensitive to the boundary conditions even in that limit. The implications of these findings for weakly interacting Bose gases are briefly discussed.
We conduct a rigorous investigation into the thermodynamic instability of an ideal Bose gas confined in a cubic box, without assuming a thermodynamic limit or a continuous approximation. Based on the exact expression of the canonical partition function, we perform numerical computations up to 106{10}^{6}106 particles. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be 761676167616, the ideal Bose gas subject to the Dirichlet boundary condition reveals a thermodynamic instability. Accordingly, we demonstrate that a system consisting of a finite number of particles can exhibit a discontinuous phase transition that features a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, 7616,7616,7616, can be regarded as a characteristic number of a ``cube,'' which is the geometric shape of the box.
It is shown that the lack of condensation in partially finite geometries is not dependent on the use of periodic boundary conditions or the symmetry-breaking technique for a class of independent-particle Bose-gas models. Condensation in an ideal Bose gas in a finite geometry is studied for one-, two-, and three-dimensional systems. Explicit expressions for the chemical potential in terms of the number of particles, the temperature, and the dimensions of the system are obtained by using a contour-integration technique to express sums as integrals plus corrections. Even though there is no true condensation, the range of the reduced density matrix for an ideal Bose gas in a thick slab (${L}_{1}\ifmmode\times\else\texttimes\fi{}{L}_{2}\ifmmode\times\else\texttimes\fi{}{L}_{3}$ with L1{L}_{1}L1​, ${L}_{2}\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\ensuremath{\infty}>{L}_{3}\ensuremath{\gg}{\ensuremath{\lambda}}_{T}=\mathrm{the}\mathrm{thermal}\mathrm{wavelength}$) is found to go from order $\frac{{\ensuremath{\lambda}}_{T}{T}_{\mathrm{cB}}}{(T\ensuremath{-}{T}_{\mathrm{cB}})}$ to order ${L}_{3}\mathrm{exp}{\frac{1}{2}\sqrt{\ensuremath{\pi}}{\ensuremath{\zeta}}_{\frac{3}{2}}(\frac{{L}_{3}}{{\ensuremath{\lambda}}_{T}})[{(\frac{{T}_{\mathrm{cB}}}{T})}^{\frac{3}{2}}\ensuremath{-}1]}$ as the temperature is lowered through the bulk critical temperature ${T}_{\mathrm{cB}}\ifmmode\cdot\else\textperiodcentered\fi{}({\ensuremath{\zeta}}_{\ensuremath{\chi}} \mathrm{is}\mathrm{the}\mathrm{Riemann} \ensuremath{\zeta} \mathrm{function}.)$ The range exhibits a spatial asymmetry. The onset of this extremely long-range order is found to be reflected in PVT diagrams and the specific heat in a manner very similar to that in bulk condensation. It is conjectured that in calculations for infinite systems the existence of a long- but finite-range order may be a more relevant criterion for superfluidity and superconductivity than the usual criterion of an infinitely long-range order.
We classify the ground states and topological defects of a rotating two-component condensate when varying several parameters: the intracomponent coupling strengths, the intercomponent coupling strength, and the particle numbers. No restriction is placed on the masses or trapping frequencies of the individual components. We present numerical phase diagrams which show the boundaries between the regions of coexistence, spatial separation, and symmetry breaking. Defects such as triangular coreless vortex lattices, square coreless vortex lattices, and giant skyrmions are classified. Various aspects of the phase diagrams are analytically justified thanks to a nonlinear $\ensuremath{\sigma}$ model that describes the condensate in terms of the total density and a pseudo-spin representation.
The analytical theory of Bose–Einstein condensation of an ideal gas in mesoscopic systems has been briefly reviewed in application to traps with arbitrary shapes and dimension. This theory describes the phases of the classical gas and the formed Bose–Einstein condensate, as well as the entire vicinity of the phase transition point. The statistics and thermodynamics of Bose–Einstein condensation have been studied in detail, including their self-similar structure in the critical region, transition to the thermodynamic limit, effect of boundary conditions on the properties of a system, and nonequivalence of the description of Bose–Einstein condensation in different statistical ensembles. The complete classification of universality classes of Bose–Einstein condensation has been given.