Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode
(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
The proof of Jasnow and Fisher that no spontaneous ordering can occur in one- and two-dimensional systems with continuous symmetry is shown to fail for nonlocal potentials of sufficiently long range. The occurrence of Bose-Einstein condensation for finite temperature (in the infinite-volume limit) is, therefore, not forbidden for such potentials. This result also follows from the postulate of the completeness of the noninteracting equilibrium states. Thus, the completeness postulate represents a weaker assumption on the nature of an interacting equilibrium state (or, equivalently, on the basic interactions between particles) than is usually found in the literature.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.
Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\ensuremath{\simeq}{T}_{c}$, is approached from above, the volume of the system becomes subextensive,
Vc=O(lnNN), which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume
V(=Vc), the condensate fraction gradually builds up and becomes nonnegligible as
T approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below
T0, it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant
P its volume, over a thermodynamically negligible range of temperatures, would reduce to values
O(N21) which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat
CP, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.
Universal dynamics of spontaneous symmetry breaking is central to understanding the universal behavior of spontaneous defect formation in various system from the early universe, condensed-matter systems to ultracold atomic systems. We explore the universal real-time dynamics in an array of coupled binary atomic Bose-Einstein condensates in optical lattices, which undergo a spontaneous symmetry breaking from the symmetric Rabi oscillation to the broken-symmetry self-trapping. In addition to Goldstone modes, there exist gapped Higgs mode whose excitation gap vanishes at the critical point. In the slow passage through the critical point, we analytically find that the symmetry-breaking dynamics obeys the Kibble-Zurek mechanism. From the scalings of bifurcation delay and domain formation, we numerically extract two Kibble-Zurek exponents
b1=ν/(1+νz) and
b2=1/(1+νz), which give the static correlation-length critical exponent
ν and the dynamic critical exponent
z. Our approach provides an efficient way to simultaneous determination of the critical exponents
ν and
z for a continuous phase transition.
We have carried out a rigorous asymptotic analysis of the thermodynamic behavior of an ideal Bose gas confined to an arbitrary, finite cuboidal geometry under periodic boundary conditions. Our investigation, which is based on the grand canonical ensemble, leads to the construction of an abstract, thermogeometric space in which the process of Bose-Einstein condensation appears as a "collapse" of the lattice points of the space towards its origin. In an infinite geometry, the collapse is accomplished in an infinitesimally small interval of temperature; this results in the appearance of mathematical singularities in the thermodynamic functions of the system. In a finite geometry, the "collapse" proceeds gradually and is spread over a temperature range $\ensuremath{\Delta}T$ such that $\frac{\ensuremath{\Delta}T}{{T}_{0}}=O(\frac{\overline{l}}{{L}_{<}})$, where
l is the mean interatomic distance while
L< is the length of the shortest side of the assembly; accordingly, the thermodynamic functions of the system remain smooth throughout. Special events, such as the specific-heat maximum, occur when the "lattice parameters" of our thermogeometric space acquire certain characteristic values which depend only on the shape of the system and not on its actual size. This leads to a new law of corresponding states for Bose-Einstein systems of finite size.