$3d$ fermion-boson map with imaginary chemical potential

TL;DR

This paper explores the phase structure and free energy relations of 3D $U(N)$ models with fixed $U(1)$ charge under imaginary chemical potential, revealing a precise mapping involving special functions and connections to 3D bosonization.

Contribution

It demonstrates a detailed large-$N$ analysis showing a novel free energy mapping between fermionic and bosonic models with imaginary chemical potential, involving Bloch-Wigner functions.

Findings

The phase structures of the models are remarkably similar under imaginary chemical potential.

The free energy densities of the models are exactly mapped into each other.

The free energy map involves special functions like the Bloch-Wigner function.

Abstract

We study the three-dimensional $U(N)$ Gross-Neveu and CP$^{N-1}$ models in the canonical formalism with fixed $U(1)$ charge. For large-$N$ this is closely related to coupling the models to abelian Chern-Simons in a monopole background. We show that the presence of the imaginary chemical potential for the $U(1)$ charge makes the phase structure of the models remarkably similar. We calculate their respective large-$N$ free energy densities and show that they are mapped into each other in a precise way. Intriguingly, the free energy map involves the Bloch-Wigner function and its generalizations introduced by Zagier. We expect that our results are connected to the recently discussed $3d$ bosonization.