On the phase of quark determinant in lattice QCD with finite chemical potential

TL;DR

This paper analyzes the phase of the quark determinant in lattice QCD at finite chemical potential, demonstrating bounds and convergence properties, and applying phase reweighting to explore high-density regions beyond previous methods.

Contribution

It introduces an analytic approximation for the phase bounds, confirms these with numerical studies, and applies phase reweighting to investigate high-density QCD regions.

Findings

The phase's absolute value has an upper bound that grows with volume and decreases with lattice temporal extent.

The winding number expansion converges well beyond /T=1.

Phase reweighting reveals first-order like behavior at /T0.8.

Abstract

We investigate the phase of the quark determinant with finite chemical potential in lattice QCD using both analytic and numerical methods. Applying the winding number expansion and the hopping parameter expansion to the logarithm of the determinant, we show that the absolute value of the phase has an upper bound that grows with the spatial volume but decreases exponentially with an increase in the temporal extent of the lattice. This analytic but approximate result is confirmed with a numerical study in four-flavor QCD in which the phase is calculated exactly. Since the phase is well controlled on lattices with larger time extents, we try the phase reweighting method in a region beyond $\mu/T=1$ where the Taylor expansion method cannot be applied. Working in four-flavor QCD, we find a first-order like behavior on a $6^3\times 4$ lattice at $\mu /T\approx 0.8$ which was previously observed by Kentucky group with the canonical method. We also show that the winding number expansion has a nice convergence property beyond $\mu/T=1$. We expect that this expansion is useful to study the high density region of the QCD phase diagram at low temperatures.