Thermodynamics of magnetized BPS baryonic layers and the effects of the Isospin chemical potential

TL;DR

This work constructs and analyzes magnetized BPS baryonic layers within a gauged non-linear sigma model coupled to Maxwell theory, providing closed-form relations between baryon number, topological charge, magnetic flux, and thermodynamic quantities. A Hamilton-Jacobi-based BPS bound yields first-order equations that relate the SU(2) profile to the magnetic flux, enabling analytic expressions for energy, pressure, and susceptibility in a finite-volume setting. The authors also extend the formalism to nonzero Isospin chemical potential, showing that a modified BPS bound remains and that the grand canonical partition function can be expressed in terms of Riemann zeta functions under suitable approximations. The results offer a rare analytic handle on strongly interacting, magnetized baryonic matter at finite density, with potential applications to nuclear pasta and neutron-star physics, while acknowledging limitations from neglected liquid/electron components. Overall, the paper provides a coherent framework to study nonperturbative magnetized QCD-like systems with precise thermodynamic predictions and connects to number-theoretic structures via the partition function.

Abstract

Through the Hamilton-Jacobi equation of classical mechanics, BPS magnetized Baryonic layers (possessing both baryonic charge and magnetic flux) have been constructed in the gauged non-linear sigma model (G-NLSM) minimally coupled to Maxwell theory, which is one of the most relevant effective theories for Quantum Chromodynamics (QCD) in the strongly interacting low-energy limit which also takes into account the electromagnetic interactions. Since the topological charge that naturally appears on the right hand side of the BPS bound is a non-linear function of the baryonic charge, the thermodynamics of these magnetized Baryonic layers is highly non-trivial. In this work, using tools from the theory of Casimir effect, we derive analytical relationship between baryonic charge, topological charge, magnetic flux and relevant thermodynamical quantities (such as pressure, specific heat and magnetic susceptibility) of these layers. The critical Baryonic chemical potential is identified. Quite interestingly, the grand canonical partition function can be related with the Riemann zeta function. On the technical side, it is quite a remarkable result to derive explicit expressions for all these thermodynamics quantities of a strongly interacting magnetized system at finite Baryon density. The effects of the Isospin chemical potential can be included as well: in particular, we will be able to construct explicitly the BPS bound and the corresponding BPS configurations also in the case in which the Isospin chemical potential is non-zero. The physical interpretations of our analytical results will be discussed.

Figures (29)

• Figure 1: Topological charge as a function of the baryon charge
• Figure 2: Analytical approximation numerical integration for equation \ref{['FVero']}, where $\Phi=3.5$, $K=2$, $p=1$, $L_r=1$. The graph \ref{['Fig:FApproxSmall']} represents the approximation for small $I_0$, which becomes quite accurate when $I_0\rightarrow 0$. The graph \ref{['Fig:FApproxBig']} is the approximation valid for $I_0>0$ big enough. The constant parameters have been fixed to $p=1$, $K=2$, $L_r =1$, $L=1$.
• Figure 3: In this graph, it is compared the analytical expression of $v(2\pi)$ given in \ref{['ApproxSol']} and the evaluation obtained from a numerical solution of $F(\Phi,I_0) = 2\pi L_r \sqrt{K}$, where $F$ is expressed in \ref{['FVero']}. The first graph represents the approximation for small values of $I_0$ and the second one for big values of $I_0$. The constant parameters have been fixed to $K=2$, $L_r =1$, $L=1$.
• Figure 4: In this graph, it is shown the values of $B/p^2$ in terms of $I_0$, where $K=2$, $L_r =1$, $L=1$. The dashed line represents the maximal value obtained for $I_0\rightarrow\infty$.
• Figure 5: Values of $v(2\pi)(n)$ as a solution of the equation $B/p^2=n$, where $K=2$, $L_r =1$, $L=1$. For big values of $n$, $v(2\pi)(n)$ enters in a linear regime.