Transition temperature of the homogeneous and dilute Bose gas in D-dimensions

TL;DR

This paper calculates the phase transition temperature of a homogeneous dilute Bose gas in dimensions 2 to 3 using a mean field statistical approach, deriving new dimension-dependent formulas and comparing with known results.

Contribution

It introduces a generalized dimension-dependent formula for the shift in transition temperature and derives Huang's result for various dimensions, including D=2 and D=3.

Findings

Derived the shift of transition temperature as ΔTc/Tc0 = c * γ^α with α(D)=2(D/2-1)^2.

Showed that the coefficient c(D) is positive across dimensions.

Compared the D=2 transition temperature with Fisher and Hohenberg's KT temperature.

Abstract

The phase transition temperature of the homogeneous and dilute Bose gas in D-dimensions ($2 \le D \le 3$) is calculated by a mean field-based statistical method. The shift of the phase transition temperature is written up to the leading order as $\Delta T_c/T_c^0 = c \gam^{\al}$, where $\gam=n^{1/3}a$. We derived Huang's result of the phase transition temperature in the generalized dimensions. We show that $ c(D)$ is positive and $\al(D)=2(D/2-1)^2$ in the short-wavelength range. The origin of the difference between $\al=1/2$ and $\al=1$ at D=3 is discussed. The $T_c$ at D=2 is calculated in the same scheme. The result is compared with Fisher and Hohenberg's KT temperature.