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QCD phase diagram in a magnetic field with baryon and isospin chemical potentials

Yu Hamada, Muneto Nitta, Zebin Qiu

TL;DR

The study targets the phase diagram of low-energy QCD in a magnetic field with finite baryon and isospin chemical potentials using leading-order chiral perturbation theory, incorporating the Goldstone-Wilczek current through the gauged WZW term. This framework reveals an array of topological phases, including the CSL, uniform charged pion condensation, an Abrikosov vortex lattice (AVL), and a novel baryonic vortex lattice (BVL), plus an AVL–CSL intersection carrying baryon number; the latter arises at relatively modest fields $B^c\sim10^{17}$ G, while CSL itself requires $B_{\text{CSL}}\sim10^{19}$ G. A key result is that the baryon-number densities in CSL, BVL, and AVL–CSL intersections are equal to $eB/2\pi$, reflecting anomaly-induced topology and linking to neutron-star phenomenology where $\mu_B\sim1$–$2$ GeV and $B\sim10^{16}$–$10^{18}$ G. The boundaries are analytic up to one numerical constant, and the findings provide a robust, model-independent map of exotic phases potentially realizable in dense hadronic matter under strong magnetic fields.

Abstract

Based on the chiral perturbation theory at the leading order, we present the phase diagram of low-energy QCD in a magnetic field at finite baryon and isospin chemical potentials. The phase diagram consists of the QCD vacuum, the chiral soliton lattice, the uniform charged pion condensation, an Abrikosov vortex lattice of the charged pions, a baryonic vortex lattice composed of neutral and charged pion vortices with their topological linking number being the baryon number, and a hybrid phase of chiral soliton and vortex lattices, with their intersections carrying the baryon number. While the chiral soliton lattice demands ultra-strong magnetic field $\sim 10^{19}$ G,the intersection phase appears at $\sim10^{17}$ G, which is more realistic in neutron stars.

QCD phase diagram in a magnetic field with baryon and isospin chemical potentials

TL;DR

The study targets the phase diagram of low-energy QCD in a magnetic field with finite baryon and isospin chemical potentials using leading-order chiral perturbation theory, incorporating the Goldstone-Wilczek current through the gauged WZW term. This framework reveals an array of topological phases, including the CSL, uniform charged pion condensation, an Abrikosov vortex lattice (AVL), and a novel baryonic vortex lattice (BVL), plus an AVL–CSL intersection carrying baryon number; the latter arises at relatively modest fields G, while CSL itself requires G. A key result is that the baryon-number densities in CSL, BVL, and AVL–CSL intersections are equal to , reflecting anomaly-induced topology and linking to neutron-star phenomenology where – GeV and – G. The boundaries are analytic up to one numerical constant, and the findings provide a robust, model-independent map of exotic phases potentially realizable in dense hadronic matter under strong magnetic fields.

Abstract

Based on the chiral perturbation theory at the leading order, we present the phase diagram of low-energy QCD in a magnetic field at finite baryon and isospin chemical potentials. The phase diagram consists of the QCD vacuum, the chiral soliton lattice, the uniform charged pion condensation, an Abrikosov vortex lattice of the charged pions, a baryonic vortex lattice composed of neutral and charged pion vortices with their topological linking number being the baryon number, and a hybrid phase of chiral soliton and vortex lattices, with their intersections carrying the baryon number. While the chiral soliton lattice demands ultra-strong magnetic field G,the intersection phase appears at G, which is more realistic in neutron stars.
Paper Structure (9 sections, 31 equations, 4 figures, 1 table)

This paper contains 9 sections, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Leading Order Chiral Perturbation
  3. Vortex and Chiral Soliton Lattices
  4. Phase Diagram in $(B,\mu_B,\mu_I)$
  5. Summary
  6. Appendix A: Characteristic Lengths
  7. Appendix B: Vortex tension
  8. Appendix C: Fit for $c_v$
  9. Appendix D: Further details on the phase diagram

Figures (4)

  • Figure 1: Phase diagram in one slice of $(B,\mu_B,\mu_I)$ space at the typical $\mu_B=1\text{ GeV}$. $B_\text{CSL}$ (green), $\tilde{B}_\text{CSL}$ (orange) and $\mu_I^c$ (red) depend on the value of $\mu_B$. $B_{c1}$ (grey) almost overlaps the horizontal axis because it is far below the scale of this figure. The intersection of the red, blue and orange curves is located at $\mu_I^c = 307 \,{\rm MeV}$ and $B^c = 4.28 \times 10^{17}\text{ G } \ll B_{\rm CSL}=1.02\times 10^{19}\text{ G}$.
  • Figure 2: ANO vortex tension by numerical calculations.
  • Figure 3: The fitting for $c_v$. Baryonic vortices constitute the BVL while ANO vortices comprise the AVL.
  • Figure 4: Phase diagrams at $\mu_B\rightarrow\infty$ (left) and $\mu_B=0$ (right).