[Paper Review] Canonical approach to the finite density QCD with winding number expansion

Atsushi Nakamura, Shotaro Oka+1 more|arXiv (Cornell University)|Apr 16, 2015

High-Energy Particle Collisions Research3 citations

TL;DR

This paper introduces a winding number expansion method to perform fugacity expansion in lattice QCD, enabling sign-problem-free calculations of finite-density observables. By expanding the Dirac determinant in the hopping parameter and regrouping terms by temporal winding number, the authors compute the canonical partition function from zero-chemical-potential gauge configurations, successfully reproducing chiral condensate and quark number density with a clear phase transition-like behavior at high chemical potential and low temperature.

ABSTRACT

The canonical partition function is related to the grand canonical one through the fugacity expansion and is known to have no sign problem. In this paper we perform the fugacity expansion by a method of the hopping parameter expansion in temporal direction for the lattice QCD: winding number expansion. The canonical partition function is constructed for Nf=2 QCD starting from gauge configurations at zero chemical potential. After derivation of the canonical partition function we calculate hadronic observables like chiral condensate and quark number density and the pressure at the real chemical potential.

Motivation & Objective

• To address the sign problem in finite-density lattice QCD by leveraging the canonical ensemble, which is free of the sign problem.
• To develop a systematic fugacity expansion method using hopping parameter expansion in the temporal direction, specifically through winding number counting.
• To compute hadronic observables such as chiral condensate and quark number density in the grand canonical ensemble at real chemical potential using data from zero-chemical-potential simulations.
• To validate the method by observing a phase transition-like behavior in the chiral condensate at high chemical potential and low temperature.

Proposed method

• The grand canonical partition function is related to the canonical one via a fugacity expansion: $ Z_G = \sum_n Z_C(n) \xi^n $, with $ \xi = e^{\mu/T} $.
• The Dirac determinant is expanded in the hopping parameter $ \kappa $, leading to $ \text{Tr} \log D_W = -\sum_{n=1}^\infty \frac{\kappa^n}{n} \text{Tr}(Q^n) $.
• Each term in the expansion is regrouped by its temporal winding number $ n $, yielding $ \text{Tr} \log D_W = \sum_n w_n \xi^n $, where $ \xi^n = e^{n\mu/T} $.
• The determinant is rewritten as $ \text{Det} D_W = \sum_n z_n(U) \xi^n $, with coefficients $ z_n(U) $ obtained via Fourier transformation of the winding-number-resolved trace.
• The canonical partition function $ Z_C(n) $ is extracted from the Fourier transform of $ Z_G(e^{i\theta}, T, V) $, avoiding numerical sign problems.
• Hadronic observables are computed via $ \langle O \rangle_C(n) = O_C(n)/Z_C(n) $, with $ O_C(n) $ obtained by similar fugacity expansion of the operator's quark contraction.

Experimental results

Research questions

• RQ1 Can the fugacity expansion of the grand canonical partition function be systematically performed using the hopping parameter expansion in the temporal direction?
• RQ2 Can the canonical partition function be accurately reconstructed from zero-chemical-potential gauge configurations using winding number counting?
• RQ3 Does the method reproduce physical observables such as chiral condensate and quark number density at real chemical potential without the sign problem?
• RQ4 Is a phase transition-like behavior in the chiral condensate observable at high chemical potential and low temperature?

Key findings

• The chiral condensate in the canonical ensemble shows a rapid drop at high baryon number, indicating a transition-like behavior.
• In the grand canonical ensemble, the chiral condensate decreases sharply with increasing chemical potential, especially at low temperatures, suggesting chiral restoration.
• The quark number density increases with chemical potential, showing a clear rise at high $ \mu/T $, consistent with a phase transition.
• The method successfully computes observables at real chemical potential using only zero-μ gauge configurations, avoiding the sign problem.
• The results are consistent across different $ \beta $ values, with the transition region shifting to higher $ \mu/T $ at lower temperatures.
• The approach is validated on an $ 8^3 \times 4 $ lattice, showing feasibility for baryon numbers around $ n_B \sim 30 $.