The ground state of low-energy QCD matter in strong magnetic fields is either a chiral soliton lattice (CSL), a periodic array of neutral pion domain walls (chiral solitons) perpendicular to the magnetic field, or domain-wall Skyrmion phase, in which Skyrmions are induced on top of the CSL. Previously found domain-wall Skyrmions are bosons with the baryon number two. In this paper, we show that the minimum domain-wall Skyrmions are fermions with baryon number one; a bosonic domain-wall Skyrmion can be separated without energy cost into two fermionic domain-wall Skyrmions attached on the opposite sides of a chiral soliton. The phase boundary between the CSL and domain-wall Skyrmion phases is unchanged. In the chiral limit, the CSL reduces to a linearly dependent neutral pion on the direction of the magnetic field, while fermionic domain-wall Skyrmions sit in an equal distance of half a period.

Japan

Patrick Copinger

Minoru Eto

Muneto Nitta

Zebin Qiu

Fermionic domain-wall Skyrmions of QCD in a magnetic field

We derive the explicit formula for fractional BPS lumps (or fractional instantons) in the $\textit{ℂP}$ nonlinear sigma model on a two-dimensional torus under various shift-clock twisted boundary conditions. After regularizing the $\textit{ℂP}$ model by an $\textit{N}$-component Abelian-Higgs model, those twisted boundary conditions introduce nontrivial ’t Hooft fluxes $\textit{p}$/$\textit{N}$ for the U(1) gauge field, and the topological charge becomes fractionalized as $\textit{k}$ + $\textit{p}$/$\textit{N}$ ∈ $\textit{ℤ}$ + $\textit{p}$/$\textit{N}$. The moduli space is globally determined as the $\textit{ℂP}$ -fiber bundle on a 2-torus, which is a Kähler manifold of complex dimension $\textit{Nk}$ + $\textit{p}$ as predicted by the index theorem. We present two different parametrizations of the moduli space: one of them immediately identifies the small-lump singularity appearing in the $\textit{ℂP}$ limit, while the other makes the modular invariance manifest.

Yui Hayashi

Tatsuhiro Misumi

Keisuke Ohashi

Yuya Tanizaki

Fractional instantons in 2d $\textit{ℂP}$ $\textit{N}$ − 1 model and 4d Yang-Mills theory with ’t Hooft twists

We investigate the chiral soliton lattice (CSL) in the framework of holographic QCD in magnetic field. Under appropriate boundary conditions for the gauge field and the quark mass deformation, we demonstrate that the ground state in the gravitational dual of QCD is given by the CSL in the background magnetic field and the baryon number density. In the presence of the background magnetic field, we show that the CSL is interpreted as a uniformly distributed D4-branes in the holographic setup, where the chiral soliton is identified with a non-self-dual instanton vortex or a center vortex in the five dimensional bulk gauge theory. While the baryon numbers are given to chiral solitons as well as Skyrmions due to the different terms in the Wess-Zumino-Witten (WZW) term in the chiral perturbation theory, these baryon numbers with different origins are unified in terms of the instanton charge density in five dimensions. With bulk analysis of the WZW term, we find that the pion decay constant becomes dependent on the magnetic field. For the massless pion case, we obtain an analytical form that is in qualitative agreement with lattice QCD results for strong magnetic fields.

Markus Amano

Shin Sasaki

Holographic QCD matter: chiral soliton lattices in strong magnetic field

We investigate a baryonic structure in low-energy QCD via a model-independent way using the chiral perturbation theory at the leading order, in the presence of the baryon chemical potential $\textit{μ}$ , the isospin chemical potential $\textit{μ}$ , and the electromagnetic coupling. For such a scenario in the chiral limit, it has been known that the neutral pion winds like in the chiral soliton lattice, confined within an Abrikosov-Nielsen-Olesen (ANO) vortex of the charged pions. This structure undergoes a drastic transformation when the pion mass is introduced, i.e., both charged and neutral pions condense in the bulk, allowing two distinct types of vortices: the charged pions constitute a local ANO-like vortex, while the neutral pion configures a global vortex which is further attached to a domain wall also known as the chiral soliton. Remarkably, the ANO vortex forms a topological linking with the closed global vortex line, when $\textit{μ}$ exceeds its critical value as a function of $\textit{μ}$ . The linking number has the physical meaning of the baryon number in view of the Wess-Zumino-Witten term. In this sense, the linked configuration realizes a stable Skyrmion-type solution, but innovatively without the Skyrme term. We therefore propose a novel phase of dense baryonic matter comprised of such vortices, which shall play a role in the low-energy QCD phase diagram.

Germany

Yu Hamada

Baryons as linked vortices in QCD matter with isospin asymmetry

Dislocations, as topological defects in crystal lattices, are fundamental to understanding plasticity in materials. Similar periodic structures also arise in continuum field theories, such as chiral soliton lattices (CSLs), which appear in condensed matter systems like chiral magnets and in high-energy contexts such as quantum chromodynamics in strong magnetic field or under rapid rotation. This work investigates whether dislocations can dynamically form within such emergent CSLs. The chiral sine-Gordon model, reduced from the aforementioned examples by certain truncations, is useful to determine the ground state but it cannot describe time evolution, lacks dynamical formation or leads to singular dislocations, because its equations of motion do not contain a topological term. We propose a field-theoretical model including the topological term coupled to external fields resolving these issues by modifying the topological term so it affects the dynamics. Using numerical simulations, we study the real-time formation of CSLs in two and three spatial dimensions. In 2D, edge dislocations emerge spontaneously, guiding soliton growth and later annihilating to leave a stable CSL. In 3D, both edge and screw dislocations form; the latter exhibits helical structure influenced by the external field. We find stable double helical screw dislocations looking like a double helix staircase or DNA. We then demonstrate the formation of helical dislocations and analyze how the external field strength affects CSL density and formation speed. Our results provide a novel theoretical framework for understanding dislocations in solitonic structures, connecting high-energy field theory with materials science phenomena.

Kentaro Nishimura

Dislocations and crystallization dynamics of chiral soliton lattices

<p>In this paper, we study chiral non-Abelian domain walls in a phase of unconventional vacua of the Ginzburg-Landau model for dense QCD by considering a wider range of parameters space not directly deduced from QCD. The phase is characterized by asymmetric vacuum expectation values (VEVs), for example with the left scalar field, corresponding to the left quark-quark condensate, having a nonvanishing VEV and the right field having a vanishing one. The domain wall soliton interpolates between this vacuum and another where the left and right scalar fields switch roles. We study this formal possibility, but not any mechanism to generate these vacua nonperturbatively at finite density or finite temperature. Using a strong-coupling, or sigma-model limit, we are able to reduce the full dynamical complex matrix valued equations of motion to the sine-Gordon, a generalization of the sine-Gordon, and a generalization of the double sine-Gordon equations. In this limit, we prove nonexistence of domain walls in one of the vacua studied here, and we find full numerical computations to converge to the sigma-model limit for many cases, with some exceptions that we discuss.</p>

China

Sven Bjarke Gudnason

Chiral non-Abelian domain walls in the Ginzburg-Landau theory

Nuclear matter with a strong magnetic field is prevalent inside neutron stars and heavy-ion collisions. In a sufficiently large magnetic field, the ground state is either a chiral soliton lattice (CSL), an array of solitons of the neutral pion field, or a domain-wall Skyrmion phase in which Skyrmions emerge inside the chiral solitons. In the region of large chemical potential and a magnetic field lower than its critical value for CSL, a Skyrmion crystal is expected to take up the ground state based on the chiral perturbation theory at the next leading order. We determine the phase boundary between such a Skyrmion crystal and the QCD vacuum. We examine the previous conjecture that a Skyrmion in magnetic field could be in a form of a neutral pion domain wall bounded by a superconducting ring of charged pions with the radius determined by the quantization condition of the penetrating magnetic flux. We also validate that a Skyrmion would shrink to null without the Skyrme term, although Derrick’s scaling law is modified by a background magnetic field, and the stability at the leading order is not ruled out in theory.

Yuki Amari

Phase boundary of nuclear matter in magnetic field

We investigate a relation between non-invertible symmetries and selection rules of topological solitons such as axionic domain walls and magnetic strings in the (3 + 1)-dimensional axion electrodynamics with a massive axion or a massive photon. In the low-energy limit of the phases where either the axion or the photon is massive, we identify non-invertible 0- or 1-form symmetry generators as axionic domain walls or magnetic strings, respectively. By non-invertible transformations on magnetic monopoles or axionic strings, we give constraints on possible configurations of topological solitons in the presence of the monopoles or axionic strings. Our results are consistent with a solution to the axionic domain wall problem by the magnetic monopole. Further, we give a new constraint on a linked configuration of the magnetic and axionic strings.

Yoshimasa Hidaka

Ryo Yokokura

Selection rules of topological solitons from non-invertible symmetries in axion electrodynamics

We study the capture of a magnetic Skyrmion into a domain wall (DW) structure in chiral magnets and find that in the respective ground states of the DW and the Skyrmion, they repel each other. This means that an isolated magnetic Skyrmion cannot <italic>a priori</italic> enter the DW and become a DW-Skyrmion. However, rotating the DW's phase away from the stable phase may cause the successful capture and hence creation of a stable bound state of a magnetic Skyrmion and a DW: the DW-Skyrmion. At certain distances, the DW may also destroy the magnetic Skyrmion by inducing shrinkage of the latter. This happens as the isolated magnetic Skyrmion has negative DMI energy, whereas the DW-Skyrmion has positive DMI energy; in a finite range from the DW, the magnetic Skyrmion can be pushed to have vanishing DMI energy, for which it collapses to a point. <italic>En passant</italic>, we find the possibility of pair creation of a Skyrmion anti-DW-Skyrmion pair and a creation of more than one DW-Skyrmions by a Kibble-like mechanism.

Creation of domain-wall Skyrmions in chiral magnets

<p>Knots emerge in various fields of mathematics and physics today. We show that knots indeed appear as stable solitons in a realistic extension of the standard model of particle physics that provides the QCD axion and right-handed neutrinos. This result suggests that, during the early Universe, a “knot dominated era” may have existed, where knots were a dominant component of the Universe, and this scenario can be tested through gravitational wave observations. Furthermore, we propose that the end of this era involves the collapse of the knots via quantum tunneling, leading to the generation of matter-antimatter asymmetry in the Universe.</p>