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Journal of Mathematical Physics

Critical exponents in the imperfect and the free Bose gas

M. Broidioi

Published May 1, 1994• 5 citations
Publisher

Abstract

The field fluctuations are studied for the free and imperfect Bose gas. In the phase where there is condensation, these field fluctuations are abnormal, with critical exponents depending on the way the symmetry-breaking external field tends to zero in the thermodynamic limit. On the critical line there is also abnormal behavior, but the dependence on the way the field disappears, is slightly different. For densities below the critical density, the fluctuations are normal and independent of the way the external field goes to zero.

DOI
10.1063/1.530558

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The field fluctuations are studied for the free and imperfect Bose gas. In the phase where there is condensation, these field fluctuations are abnormal, with critical exponents depending on the way the symmetry-breaking external field tends to zero in the thermodynamic limit. On the critical line there is also abnormal behavior, but the dependence on the way the field disappears, is slightly different. For densities below the critical density, the fluctuations are normal and independent of the way the external field goes to zero.
Abstract Some properties of an ideal gas of massive bosons in a mean field potential and, confined between two infinite parallel slabs in a d-dimensional configuration space are investigated systematically. Here, one particle density of states approach is employed to study the critical temperature, shift of density, Casimir effects and critical exponents, starting from the evaluation of the grand canonical free energy in d-dimension. We have found that, the shift of density, Casimir force and the critical temperature depend on the space dimensionality. But the Casimir force decays as an inverse power law of the distance between two slabs in the condensate and, decays exponentially in the non-condensed state situated very close to the point of phase transition. Specifically, this study enabled us to predict the shift of the density of boson in the excited states due to mean field potential, and also the dimensional dependence of the critical exponent. The form of the critical exponent is found to be 1 d − 2 for the imperfect Bose gas. This leads to a value of critical exponent 1 for d = 3. Most interestingly, in the limit when the mean field potential goes to zero the resulting expressions for the above properties coincide with those of the ideal Bose gas.
We study the low-temperature limit of the d -dimensional imperfect Bose gas. Relying on an exact analysis of the microscopic model, we establish the existence of a second-order quantum phase transition to a phase involving the Bose–Einstein condensate. The transition is triggered by varying the chemical potential and persists at non-zero temperatures T for d > 2. We extract the exact phase diagram and identify the scaling regimes in the vicinity of the quantum-critical point focusing on the behavior of the correlation length ξ. The length ξ develops an essential singularity exclusively for d = 2. We follow the evolution of the phase diagram varying d . For d > 2 our results agree with renormalization-group-based analysis of the effective bosonic order-parameter models with the dynamical exponent z = 2.
We study an anisotropic variant of the ddd-dimensional imperfect Bose gas on a lattice, where the asymptotic behavior of the dispersion ϵk\epsilon_{\bf k}ϵk​ at vanishing momentum k\bf{k}k may differ from the standard quadratic form. We derive a general expression for the lower critical dimension dLd_LdL​ of the model and show that it may be pushed to arbitrarily high values by engineering the hopping parameters. The same applies to the upper critical dimension dUd_UdU​. For d>dLd>d_Ld>dL​ we find a general, exact expression for the Bose-Einstein condensation temperature Tc(μ)T_c(\mu)Tc​(μ) as a function of the chemical potential μ\muμ. The asymptotic behavior of Tc(μ)T_c(\mu)Tc​(μ) at Tc→0T_c\to 0Tc​→0 is governed by the shift exponent ψ\psiψ, whose value turns out to fully determine whether the system is below or above the upper critical dimension. The critical exponents at condensation as well as the exponent ψ\psiψ may attain non-standard values for specific choices of the lattice kinetic couplings. Finally, we study the mechanism leading to the affinity between the imperfect Bose gas and the classical spherical model of spins in the critical regime.
Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.