Thermodynamic properties of non-Hermitian Nambu--Jona-Lasinio models

TL;DR

This work asks how non-Hermitian bilinear extensions affect the thermodynamics of the Nambu–Jona-Lasinio model at finite temperature and density. By incorporating a pseudoscalar term (anti-PT-symmetric) and a PT-symmetric pseudovector term with a background field, the authors derive modified gap equations and grand potentials, map the resulting phase diagrams, and analyze quark-number, entropy, pressure, energy density, and the interaction measure. They find real dynamical masses in both extensions but with distinct phase-structure shifts: the pseudoscalar case tends to raise the transition temperature and induce a fermion excess, while the pseudovector case yields mass changes that depend on the background orientation and can favor antifermionic dominance, sometimes producing negative I. These results imply that non-Hermitian contributions can imprint observable signatures in strongly interacting fermion systems and may offer insights into non-Hermitian baryon asymmetry, while the formalism extends to other four-fermion theories and remains robust under different regulators.

Abstract

We investigate the impact of non-Hermiticity on the thermodynamic properties of interacting fermions by examining bilinear extensions to the $3+1$ dimensional $SU(2)$-symmetric Nambu--Jona-Lasinio (NJL) model of quantum chromodynamics at finite temperature and chemical potential. The system is modified through the anti-$PT$-symmetric pseudoscalar bilinear $\barψγ_5 ψ$ and the $PT$-symmetric pseudovector bilinear $iB_ν\,\barψγ_5γ^νψ$, introduced with a coupling $g$. Beyond the possibility of dynamical fermion mass generation at finite temperature and chemical potential, our findings establish model-dependent changes in the position of the chiral phase transition and the critical end-point. These are tunable with respect to $g$ in the former case, and both $g$ and $|B|/B_0$ in the latter case, for both lightlike and spacelike fields. Moreover, the behavior of the quark number, entropy, pressure, and energy densities signal a potential fermion or antifermion excess compared to the standard NJL model, due to the pseudoscalar and pseudovector extension respectively. In both cases regions with negative interaction measure $I = ε-3p$ are found. Future indications of such behaviors in strongly interacting fermion systems, for example in the context of neutron star physics, may point toward the presence of non-Hermitian contributions. These trends provide a first indication of curious potential mechanisms for producing non-Hermitian baryon asymmetry. In addition, the formalism described in this study is expected to apply more generally to other Hamiltonians with four-fermion interactions and thus the effects of the non-Hermitian bilinears are likely to be generic.

Figures (15)

• Figure 1: Behavior of the effective fermion mass $m$ within the NJL model in MeV at the chemical potential $\mu=0$ and $\mu=0.2 \Lambda$ as a function of the temperature $T$ in MeV.
• Figure 2: Qualitative behavior of the thermodynamic potential $\Omega_{\text{NJL}}$ at vanishing temperature $T$ as a function of the effective mass $m$ for various chemical potentials $\mu$. The different cases illustrate the possible existence of extrema at vanishing and finite mass.
• Figure 3: Behavior of the effective fermion mass $m$ within the NJL model in MeV at vanishing temperature $T$ as a function of the chemical potential $\mu$ in MeV. The stable physical mass solution associated with the global minimum of $\Omega_{\text{NJL}}$ is shown as a solid black line, undergoing a first-order phase transition at $\mu^c$.
• Figure 4: Effective fermion mass $m_{\text{NJL}}$ as a function of the temperature $T$ and the chemical potential $\mu$ in MeV. The chiral phase transition is denoted in red, with a red dot indicating the CEP. At low temperatures the mass undergoes a discontinuous first-order chiral phase transition, while the transition is of second order at small chemical potentials.
• Figure 5: (a) Quark number density at vanishing temperature as a function of the chemical potential $\mu$ in MeV, illustrating that, while the stable, metastable, and unstable mass solutions of the gap equation are a function of the chemical potential with multiple branches, they form a single-valued function of the quark number density. (b) Chiral phase diagram of the NJL model in the temperature-quark number density plane. The phase transition of the stable physical fermion mass is denoted as solid black line. Red lines denote the spinoidals associated with $\mu_+$ (dashed) and $\mu_-$ (solid) marking the transition from the metastable (shaded) to the unstable mass regions.