Condensate density of interacting bosons: a functional renormalization group approach

TL;DR

This paper uses the functional renormalization group to compute the temperature-dependent condensate density of interacting bosons in three and two dimensions, providing new insights into critical behavior and the effects of higher-order interactions.

Contribution

It introduces a detailed FRG approach to calculate condensate density across temperatures and dimensions, including higher-order terms, and examines the validity of polynomial approximations.

Findings

Determined the critical exponent β ≈ 0.32 for 3D bosons, close to the expected 0.345.

Calculated condensate density in 2D at zero temperature with higher-order effective potential terms.

Found that cubic and quartic couplings increase condensate density slightly but flow to large values, questioning low-order polynomial approximations.

Abstract

We calculate the temperature dependent condensate density $\rho^0 (T)$ of interacting bosons in three dimensions using the functional renormalization group (FRG). From the numerical solution of suitably truncated FRG flow equations for the irreducible vertices we obtain $\rho^0 (T)$ for arbitrary temperatures. We carefully extrapolate our numerical results to the critical point and determine the order parameter exponent $\beta \approx 0.32$, in reasonable agreement with the expected value $ 0.345$ associated with the XY-universality class. We also calculate the condensate density in two dimensions at zero temperature using a truncation of the FRG flow equations based on the derivative expansion including cubic and quartic terms in the expansion of the effective potential in powers of the density. As compared with the widely used quadratic approximation for the effective potential, the coupling constants associated with the cubic and quartic terms increase the result for the condensate density by a few percent. However, the cubic and quartic coupling constants flow to rather large values, which sheds some doubt on FRG calculations based on a low order polynomial approximation for the effective potential.