Self-consistent Hartree-Fock-Bogoliubov approach for bosons: self-eliminating divergence and pure pair condensate

TL;DR

The paper develops a self-consistent, divergence-free Hartree-Fock-Bogoliubov framework for a weakly interacting Bose gas at finite temperature by incorporating pair correlations through an energy-functional approach. For a contact interaction, it derives a self-eliminating divergence (SSED) solution in which a pure pair condensate forms with zero zero-temperature pressure, but thermodynamic stability requires a mixed state of single-particle and pair condensates. In the Popov limit, the spectrum is gapless, while the SSED state exhibits a finite gap associated with pair correlations; the transition temperature is shifted upward by interactions, and the isothermal compressibility analysis indicates instability of the pure pair condensate. Overall, the work provides a consistent divergence-free mean-field description that captures pair correlations, predicts a first-order normal-to-degenerate transition, and highlights the necessity of condensate mixtures beyond pure single-particle or pair condensates. The framework offers a route to reconcile finite-temperature thermodynamics of Bose gases with experimental observations and motivates extensions beyond mean-field near criticality.

Abstract

We investigate the thermodynamic properties of an interacting Bose gas with a condensate within the energy-functional formulation of the Hartree-Fock-Bogoliubov (HFB) approach. For a contact interaction, we derive a self-consistent solution to the HFB equations that intrinsically eliminates divergence. This solution characterizes the equilibrium state featuring a condensate of correlated pairs of particles. We analyze the temperature dependence of key thermodynamic quantities such as condensate density, chemical potential, entropy, pressure, specific heat capacity at constant volume, and isothermal compressibility and compare them with predictions from the Popov approximation (PA). We predict that the transition temperature shifts to higher values due interactions, with the HFB approach yielding a larger shift than the PA. Analysis of the compressibility indicates that a pure pair condensate is unstable, and the stable equilibrium corresponds to only a mixture of single-particle and pair condensates.

Figures (6)

• Figure 1: (Color online) Condensate fraction vs temperature for total density $n=1.6\cdot10^{14}$ cm$^{-3}$ and scattering length $a=4.9$ nm. The curves (from left to right) refer to: ideal Bose gas (IBG), Popov approximation (PA) and solution with self-eliminating divergence (SSED). Magnified sections of the SSED and PA curves with retrograde behaviour are shown in the inset. The scatter plot (MC) illustrates the predictions of the universal relations, calculated using the Monte-Carlo method Prokof'ev_PRA_2004.
• Figure 2: (Color online) Dependencies on gas parameter of: (a) reversal temperature, (b) condensate fraction at reversal temperature. The curves refer to: solution with self-eliminating divergence (SSED) and Popov approximation (PA).
• Figure 3: (Color online) Temperature dependencies of: (a) chemical potential; (b) entropy; (c) pressure; and (d) pressure relative to that of an ideal Bose gas --- for the total density of degenerate state $n^{\textrm{deg}}=1.6\cdot10^{14}$ cm$^{-3}$, the scattering length $a=4.9$ nm, $\mu_c=\mu^{\textrm{NS}}(T=T^{\textrm{IBG}})=2nU$, $S_c=S^{\textrm{NS}}(T=T^{\textrm{IBG}})=5/2\zeta(5/2)/\zeta(3/2)Nk_B$ and $P_c=P^{\textrm{NS}}(T=T^{\textrm{IBG}})=n^2U+\zeta(5/2)/\zeta(3/2)nk_BT^{\textrm{IBG}}$. The solid curves refer to: ideal Bose gas (IBG), solution with self-eliminating divergence (SSED), Popov approximation (PA) and normal state (NS) --- for total density $n=n^{\textrm{deg}}$, i.e. relative density rd$=n/n^{\textrm{deg}}=1$. The dashed curves refer to the normal state for rd$=0.96$. Magnified sections of the foregoing curves with retrograde behaviour are shown in the insets.
• Figure 4: (Color online) Specific heat capacity at constant volume vs temperature for total density $n=1.6\cdot10^{14}$ cm$^{-3}$, scattering length $a=4.9$ nm and $C_c=C^{\textrm{NS}}(T=T^{\textrm{IBG}})=15/4\zeta(5/2)/\zeta(3/2)Nk_B$. The curves refer to: ideal Bose gas (IBG), solution with self-eliminating divergence (SSED), Popov approximation (PA) and normal state (NS). IBG and NS curves coincide for $T>T^{\textrm{IBG}}$.
• Figure 5: (Color online) Compressibility vs temperature for total density $n=1.6\cdot10^{14}$ cm$^{-3}$ and scattering length $a=4.9$ nm and $\kappa_c=\kappa^{\textrm{NS}}(T=T^{\textrm{IBG}})=(2n^2U)^{-1}$. The curves refer to: solution with self-eliminating divergence (SSED), Popov approximation (PA) and normal state (NS). Magnified sections of the foregoing curves with retrograde behaviour are shown in the inset.