Imaginary chemical potential and finite fermion density on the lattice

TL;DR

This paper tackles the lattice finite-density sign problem by proposing the use of an imaginary chemical potential, which preserves a positive measure and enables Monte Carlo sampling. It develops a framework in which the imaginary-μ partition function $Z(i\nu)$ is sampled and then converted to canonical partitions $Z_N$ via a Fourier transform; the authors show how to manage statistical overlap by employing multiple patches and ratio observables. A feasibility study on the 2D Hubbard model demonstrates reconstruction of $Z_N$ up to $N=6$ from $Z(i\nu)$, supporting the method's viability. The work suggests that imaginary μ can be a practical route to study finite-density effects in QCD (including isospin-density cases) and related phase phenomena, offering an alternative to the Glasgow approach, particularly at high temperature where fluctuations are tractable.

Abstract

Standard lattice fermion algorithms run into the well-known sign problem at real chemical potential. In this paper we investigate the possibility of using imaginary chemical potential, and argue that it has advantages over other methods, particularly for probing the physics at finite temperature as well as density. As a feasibility study, we present numerical results for the partition function of the two-dimensional Hubbard model with imaginary chemical potential. We also note that systems with a net imbalance of isospin may be simulated using a real chemical potential that couples to I_3 without suffering from the sign problem.

Figures (3)

• Figure 1: Free energy as a function of imaginary chemical potential $\nu$ (left) and log of the canonical partition function as a function of particle number (right) for our illustrative toy model (\ref{['im:Ftot']}), with $\beta M_b = 1$ and $\beta M_f = 5$.
• Figure 2: Partition function for 2D Hubbard model as a function of imaginary chemical potential $\nu$, in units of $\pi/\beta$. This is on a $4^2\times 10$ lattice, with $\beta=1.5$, hopping term $K=1$, and interaction strength $U=0.1$, following the conventions of Creutz Creutz.
• Figure 3: Log of canonical partition function $Z_N$ for 2D Hubbard model, obtained by Fourier transform of $Z(i\nu)$ (Fig. \ref{['fig:lnZhub']}).