Chiral soliton lattice in inhomogeneous magnetic fields

Abstract

It has been known that in sufficiently strong uniform magnetic fields, the ground state of quantum chromodynamics (QCD) supports a spatially modulated condensate of neutral pions, dubbed chiral soliton lattice (CSL). In this paper, we investigate whether a similar ordered ground state might exist when the external magnetic field is nonuniform, as appropriate for potential phenomenological applications, including heavy-ion collisions and neutron stars. To that end, we use the low-energy effective field theory of QCD, restricted to the neutral pions as the sole low-energy degrees of freedom in strong magnetic fields. In the limit of vanishing pion mass, we achieve a complete characterization of magnetic fields supporting a CSL-like ground state. Moreover, for a simple but infinite family of magnetic fields, we find the corresponding CSL state analytically. Going away from the massless limit requires full numerical minimization of the energy functional. Here we provide some sample numerical results, focusing on the qualitative differences as compared to the situations with a uniform magnetic field or a vanishing pion mass. The main conclusion remains unchanged: while bending the magnetic field typically reduces the energy gain due to the neutral pion condensation, a CSL-type ground state is still possible. As a byproduct of our work, we map the location of the CSL phase in the phase diagram of QCD in a uniform magnetic field and finite volume.