Tensor renormalization group study of cold and dense QCD in the strong coupling limit

TL;DR

The study demonstrates that tensor renormalization group techniques can robustly address the cold and dense region of (3+1)d QCD in the strong coupling limit, enabling precise determination of the chiral and nuclear critical endpoints at $N_\\tau=8$ and confirming a first-order transition in the thermodynamic limit at zero temperature. By constructing a Grassmann tensor network representation and employing ATRG with substantial parallelization, the authors compute the chiral condensate and quark number density as functions of the chemical potential, finding $m_c^{\\chi}(N_\\tau=8)=2.0545(34)$ and $m_c^{n}(N_\\tau=8)=2.075(23)$ with $\\mu_c^{\\chi}=\\mu_c^{\\n}$, and observing a sharp first-order transition on a $1024^4$ lattice at $m=2.07$ and $\\mu\\approx 1.5915$. These results are consistent with mean-field expectations and offer an independent nonperturbative check in a regime inaccessible to standard Monte Carlo methods due to the sign problem. The work paves the way for future extensions to finite coupling, gauge fields, and Wilson quarks, potentially illuminating high-density QCD phases such as color–flavor locked states.

Abstract

We study the phase structure of the (3+1)-dimensional cold and dense QCD with the Kogut--Susskind quark in the strong coupling limit using the tensor renormalization group method. The chiral and nuclear transitions are investigated by calculating the chiral condensate and the quark number density as a function of the chemical potential. For a fixed temporal extent $N_τ=8$, we determine the critical quark masses $m_c^χ$ and $m_c^{n}$ for the chiral condensate and the quark number density, respectively, at which the first-order phase transition terminates with the vanishing discontinuity in thermodynamic quantities. We find that both quantities at the same quark mass exhibit a discontinuity at the same chemical potential, and the resulting critical quark masses are consistent with each other. We also compare our results for the critical quark masses with those obtained from the Monte Carlo simulation in the dual formulation and from the mean-field analysis. We further confirm the first-order phase transition at finite quark mass on a $1024^4$ lattice, which is essentially in the thermodynamic limit at zero temperature, as expected from the mean-field analysis.

Figures (9)

• Figure 1: Relative error of thermodynamic potential in terms of $D_{3d}$ at $m=2.00$, for $\mu=1.56$ and 1.58, on a $32^3\times 8$ lattice.
• Figure 2: Relative error of thermodynamic potential in terms of $D$ at $m=2.00$, for $\mu=1.56$ and 1.58, on a $32^3\times 8$ lattice.
• Figure 3: $\mu$ dependence of $f(m,\mu)$ on a $32^3\times 8$ lattice for $m\in[1.90,2.025]$. The inset figure shows the $\mu$ dependence of $f(m=1.9,\mu)$ for various volumes.
• Figure 4: $\mu$ dependence of $\langle n \rangle(m,\mu)$ at $N_\tau=8$ with $m\in[1.90,2.025]$. Triangles denote $\langle n \rangle (m,\mu\rightarrow (\mu_++\mu_-)/2\pm 0)$ (see main text for details). The data at $m=2.10$ is also plotted for comparison.
• Figure 5: Fit of $\Delta \langle n \rangle(m)$ as a function of $m$. Solid curve denotes the fit result.