Lesser Green's Function and Chirality Entanglement Entropy via the In-Medium NJL Model

TL;DR

This work links chiral symmetry restoration in hot/dense QCD to quantum information by formulating the von Neumann chirality entropy $S_\chi$ from the left-handed reduced correlator $C_L(k)=P_L G^{<}(k) P_L$ within the in-medium NJL model. Using self-consistent NJL gap equations, it shows that $S_\chi$ rises monotonically with temperature and chemical potential, signaling progressive loss of left-right coherence as $M_q\to0$ and chiral symmetry is restored. Near the chiral transition in the chiral limit, the order-parameter mass $M_q$ scales as $M_q\sim (T_c-T)^{β_{M_q}}$ with $β_{M_q}\approx 0.52$, while the chirality entropy scales as $S_χ\sim S_χ(T_c)+(T_c-T)^{β_{S_χ}}$ with $β_{S_χ}\approx 1$, highlighting a derivative-type critical response. The results establish $S_χ$ as an information-theoretic probe of chiral restoration, offering a complementary perspective to conventional order parameters and suggesting extensions to PNJL and lattice-QCD investigations of entanglement in QCD matter.

Abstract

We study chiral symmetry restoration in hot-and-dense quark matter using the von Neumann chirality entropy within the in-medium Nambu-Jona-Lasinio (NJL) model. Starting from the lesser Green function $G^{<}(k)$, the reduced correlator $C_L=P_LG^{<}P_L$ is constructed, and the associated entropy $S_χ=-\mathrm{Tr}[C_L\ln C_L+(1-C_L)\ln(1-C_L)]$ is introduced to quantify the entanglement between left- and right-handed quark sectors. The dynamical quark mass $M_q(T,μ_q)$ obtained from the gap equation exhibits the expected QCD-like phase structure: A second-order transition in the chiral limit and a smooth crossover for finite $m_q$. The chirality entropy $S_χ$ increases monotonically with temperature and chemical potential, approaching a maximal value as $M_q\to0$. We also explore the critical exponents and scaling behavior of those quantities, yielding $β_{S_χ}\simeq1$. This demonstrates that $S_χ$ serves as an information-theoretic probe for chiral symmetry restoration, linking dynamical mass generation to quantum entanglement in strongly interacting matter.

Figures (4)

• Figure 1: (a) QCD phase diagram of the NJL model, denoted by the effective quark mass $M_q$, as a function of $T$ and $\mu_q$ for $m_q=0$ (a) and $m_q=5.25$ MeV (b).
• Figure 2: Effective occupation probability for $q_L$, $\nu_k$ in Eq. (\ref{['eq:nu_final']}) for $M_q=0.1$ GeV (a) and $0.3$ GeV (b) with $|\bm{k}|=1$ GeV.
• Figure 3: Chirality entanglement entropy $S_\chi$ in Eq. (\ref{['eq:entropy']}), being normalized as $S_\chi/S^{(T,\mu)=0}_\chi$ as a function of $T$ and $\mu_q$ for $m_q=0$ (a) and $m_q=5.25$ MeV (b). Panels (c) and (d) show the corresponding 2-D slices at fixed $\mu_q$.
• Figure 4: (a) Normalized $dS_{\chi}/dT$ (solid) and $|dM_q/dT|$ (dotted) as functions of temperature at $\mu=0$. The pseudo-critical temperatures extracted from the two quantities are $T_{\mathrm{pc}}^{S_{\chi}}=167~\mathrm{MeV}$ and $T_{\mathrm{pc}}^{M_q}=185~\mathrm{MeV}$, respectively, corresponding to a temperature difference of $\Delta T \simeq 18~\mathrm{MeV}$ between the two peaks. (b) Log-log scaling of the dynamical mass $M_q$ and the chirality entropy $S_\chi$ near the chiral critical temperature at $(m_q,\mu_q)=0$, i.e., $-\ln\left[M_q(T)/\mathrm{GeV}\right]$ and $-\ln\left[\Delta S_\chi(T)/\mathrm{GeV}^3\right]$ as functions of $-\ln[\Delta T/\mathrm{GeV}]$, indicating the slopes $\beta_{M_q}\simeq0.5$ and $\beta_{S_\chi}\simeq1$. See the text for more details.