Composite bosons in the 2D BCS-BEC crossover from Gaussian fluctuations

TL;DR

This paper analyzes Gaussian fluctuations in a 2D Fermi gas across the BCS-BEC crossover, revealing their importance for an accurate equation of state and deriving the bosonic scattering length consistent with Monte Carlo results.

Contribution

It provides an analytical derivation of the effective interaction between composite bosons in 2D, matching numerical Monte Carlo findings, and clarifies the role of Gaussian fluctuations in the BEC regime.

Findings

Gaussian fluctuations are crucial for the BEC regime equation of state.

Derived the bosonic scattering length as approximately 0.551 times the fermionic scattering length.

Analytical results agree with recent Monte Carlo calculations.

Abstract

We study Gaussian fluctuations of the zero-temperature attractive Fermi gas in the 2D BCS-BEC crossover showing that they are crucial to get a reliable equation of state in the BEC regime of composite bosons, bound states of fermionic pairs. A low-momentum expansion up to the fourth order of the quadratic action of the fluctuating pairing field gives an ultraviolent divergent contribution of the Gaussian fluctuations to the grand potential. Performing dimensional regularization we evaluate the effective coupling constant in the beyond-mean-field grand potential. Remarkably, in the BEC regime our grand potential gives exactly the Popov's equation of state of 2D interacting bosons, and allows us to identify the scattering length $a_B$ of the interaction between composite bosons as $a_B=a_F/(2^{1/2}e^{1/4})= 0.551... a_F$, with $a_F$ is the scattering length of fermions. Remarkably, the value from our analytical relationship between the two scattering lengths is in full agreement with that obtained by recent Monte Carlo calculations.