[Paper Review] Phase Quenched Lattice QCD at Finite Density and Temperature

D. K. Sinclair, J. B. Kogut|ArXiv.org|Sep 14, 2007

High-Energy Particle Collisions Research9 references3 citations

TL;DR

This study investigates 3-flavor lattice QCD at finite density and temperature using the phase-quenched approximation to circumvent the sign problem. Simulations on $8^3\times4$, $12^3\times4$, and $16^3\times4$ lattices near the critical quark mass find no evidence for a critical endpoint at small isospin chemical potential $\mu_I$, with the critical mass $m_c(\mu_I)$ decreasing as $\mu_I$ increases, suggesting a softened transition due to reduced symmetry.

ABSTRACT

We simulate 3-flavour lattice QCD at finite quark-number chemical potential mu in the phase-quenched approximation, close to the finite temperature transition. Working close to the critical quark mass, we find no evidence for the expected critical endpoint at small mu. We are performing further simulations aimed at calculating the equation-of-state of this theory outside of the superfluid domain, where its phase structure is expected to mimic the full theory.

Motivation & Objective

• To investigate the phase structure of 3-flavor QCD at finite chemical potential and temperature using the phase-quenched approximation to avoid the sign problem.
• To determine whether a critical endpoint exists in the $(T,\mu_I)$ plane for small $\mu_I$ near the critical quark mass $m_c(0)$.
• To calculate the equation of state for phase-quenched QCD outside the superfluid phase, where it is expected to mirror the full theory.
• To assess the validity of using the chiral condensate as a magnetic order parameter in Binder cumulant and finite-size scaling analyses.

Proposed method

• Simulate 3-flavor QCD on $8^3\times4$, $12^3\times4$, and $16^3\times4$ lattices at finite $\mu_I$ using exact RHMC with rational approximations enabled by a speculative lower bound on the Dirac operator spectrum.
• Use the 4th-order Binder cumulant of the chiral condensate $B_4(\bar{\psi}\psi)$ to detect critical points, with values crossing the Ising universality class value $B_4=1.604(1)$ as a signature.
• Apply Ferrenberg-Swendsen reweighting to continue simulations to the critical $\beta$ value that minimizes $B_4(\bar{\psi}\psi)$, using 300,000 trajectories per $\beta$ with five noisy estimators per trajectory.
• Numerically integrate the derivative of $\ln Z$ with respect to $\beta$ and $\mu_I$ to reconstruct the partition function and pressure, using the plaquette action and isospin density $j_0^3$.
• Perform finite-size scaling by plotting $L^{-\gamma/\nu}\chi_{\bar{\psi}\psi}$ as a function of $m$ to locate critical points where curves intersect.
• Use the running coupling $\beta(a)$ at $\mu_I=0$ to convert lattice units to physical units and compute thermodynamic observables like energy density $\epsilon$.

Experimental results

Research questions

• RQ1 Does a critical endpoint exist in the phase-quenched 3-flavor QCD theory at small $\mu_I$ near the critical quark mass $m_c(0)$?
• RQ2 How does the critical mass $m_c(\mu_I)$ evolve with increasing $\mu_I$, and does it decrease as observed in this study?
• RQ3 Is the chiral condensate a valid proxy for the magnetic order parameter in the Binder cumulant analysis?
• RQ4 Does the phase structure of phase-quenched QCD match that of full QCD outside the superfluid phase?
• RQ5 What is the equation of state of phase-quenched QCD for $\mu_I < m_\pi$ and across the plasma phase at high $\beta$?

Key findings

• No evidence for a critical endpoint is found at small $\mu_I$; the Binder cumulant $B_4(\bar{\psi}\psi)$ increases with $\mu_I$ on $12^3\times4$ and $16^3\times4$ lattices, contrary to expectations for a critical point.
• $m_c(\mu_I)$ decreases with increasing $\mu_I$: $m_c(0)=0.0265(3)$, $m_c(0.2)=0.0259(5)$, and $m_c(0.3)=0.0256(4)$, indicating a softened transition.
• The Binder cumulant curves for $8^3\times4$ and $12^3\times4$ lattices cross near the Ising value $B_4=1.604(1)$, confirming the critical point belongs to the 3D Ising universality class.
• Finite-size scaling of the chiral susceptibility $\chi_{\bar{\psi}\psi}$ shows intersection points between $m=0.025$ and $m=0.03$, consistent with the $B_4$-based critical point estimate.
• The isospin density $j_0^3$ saturates at $\mu_I \approx 2$ on high-$\beta$ lattices, indicating the onset of Pauli blocking and the breakdown of the Fermi gas picture.
• The equation of state for phase-quenched QCD is being computed along lines of constant $\beta$ to compare with full QCD, with simulations covering $\mu_I < m_\pi$ at low $\beta$ and full $\mu_I$ range at high $\beta$.