<p>Low-energy dynamics of QCD can be described by pion degrees of freedom in terms of the chiral perturbation theory. A chiral soliton lattice, an array of solitons, is the ground state due to the chiral anomaly in the presence of a magnetic field larger than a certain critical value at finite density. Here, we show in a model-independent and fully analytic manner (at the leading order of the chiral perturbation theory) that the chiral soliton lattice phase transits to a “domain-wall skyrmion phase” when the chemical potential is larger than the critical value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mi mathvariant="normal">c</mi></mrow></msub><mo>=</mo><mn>16</mn><mi>π</mi><msubsup><mrow><mi>f</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>/</mo><mn>3</mn><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>∼</mo><mn>1.03</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></mrow></math> with the pion’s decay constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mi>π</mi></msub></math> and mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>π</mi></msub></math>, which can be regarded as the nuclear saturation density. There spontaneously appear stable two-dimensional skyrmions or lumps on a soliton surface, which can be viewed as three-dimensional skyrmions carrying even baryon numbers from the bulk. They behave as superconducting rings with persistent currents due to a charged pion condensation. This phase is in scope of future heavy-ion collider experiments.</p>

Japan

Minoru Eto

Kentaro Nishimura

Muneto Nitta

Domain-Wall Skyrmion Phase in Dense QCD at Strong Magnetic Fields Using Leading-Order Chiral Perturbation Theory

The ground state of QCD in sufficiently strong magnetic field at finite baryon density is an inhomogeneous state consisting of an array of solitons, called the chiral soliton lattice (CSL). It is, however, replaced in a region with higher density and/or magnetic field by the so-called domain-wall Skyrmion (DWSk) phase where Skyrmions are created on top of the CSL. This was previously proposed within the Bogomol’nyi-Prasad-Sommerfield (BPS) approximation neglecting a gauge field dynamics and taking into account its effect by a flux quantization condition. In this paper, by taking into account dynamics of the gauge field, we show that the phase boundary between the CSL and DWSk phases beyond the BPS approximation is identical to the one obtained in the BPS approximation. We also find that domain-wall Skyrmions are electrically charged with the charge one as a result of the chiral anomaly.

Yuki Amari

Domain-wall Skyrmion phase of QCD in magnetic field: gauge field dynamics

We study topological lumps supported by the second homotopy group π 2(S 2) ⋍ ℤ in a gauged O(3) model without any potential term coupled with a (non)dynamical U(1) gauge field. It is known that gauged-lumps are stable with an easy-plane potential term but are unstable to expand if the model has no potential term. In this paper, we find that these gauged lumps without a potential term can be made stable by putting them in a uniform magnetic field, irrespective of whether the gauge field is dynamical or not. In the case of the non-dynamical gauge field, only either of lumps or anti-lumps stably exists depending on the sign of the background magnetic field, and the other is unstable to shrink to be singular. We also construct coaxial multiple lumps whose size and mass exhibit a behaviour of droplets. In the case of the dynamical gauge field, both the lumps and anti-lumps stably exist with different masses; the lighter (heavier) one corresponds to the (un)stable one in the case of the nondynamical gauge field. We find that a lump behaves as a superconducting ring and traps magnetic field in its inside, with the total magnetic field reduced from the background magnetic field.

Topological solitons stabilized by a background gauge field and soliton-anti-soliton asymmetry

We examine neutrino zeromode solutions on the electroweak Z-string and their effect on the stability of the string in the standard model and its extensions. We propose using topological invariants constructed from the momentum (and real) space topology of Green’s functions, often used for investigating edge modes in condensed matter physics. We analyze the standard model and then examine type-I and type-II extensions of the neutrino sector as well as their hybrid. Based on this analysis, we also comment on proposals in the literature to stabilize the Z-string.

Germany

Yu Hamada

Ryusuke Jinno

China

Masatoshi Yamada

Neutrino zeromodes on electroweak strings in light of topological insulators

The ground state of QCD with two flavors at a finite baryon chemical potential under rapid rotation is a chiral soliton lattice (CSL) of the η meson, consisting of a stack of sine-Gordon solitons carrying a baryon number, due to the anomalous coupling of the η meson to the rotation. In a large parameter region, the ground state becomes a non-Abelian CSL, in which due to the neutral pion condensation each η soliton decays into a pair of non-Abelian sine-Gordon solitons carrying S 2 moduli originated from Nambu-Goldstone (NG) modes localized around it, corresponding to the spontaneously broken vector symmetry SU(2)V. There, the S 2 modes of neighboring solitons are anti-aligned, and these modes should propagate in the transverse direction of the lattice due to the interaction between the S 2 modes of neighboring solitons. In this paper, we calculate excitations including gapless NG modes and excited modes around non-Abelian and Abelian (η) CSLs, and find three gapless NG modes with linear dispersion relations (type-A NG modes): two isospinons (S 2 modes) and a phonon corresponding to the spontaneously broken vector SU(2)V and translational symmetries around the non-Abelian CSL, respectively, and only a phonon for the Abelian CSL because of the recovering SU(2)V. We also find in the deconfined phase that the dispersion relation of the isospinons becomes of the Dirac type, i.e. linear even at large momentum.

Non-Abelian chiral soliton lattice in rotating QCD matter: Nambu-Goldstone and excited modes

Based on the chiral perturbation theory at the leading order, we show a signal of the presence of a new phase in rapidly rotating QCD matter with two flavors, that is a domain-wall Skyrmion phase. Based on the chiral Lagrangian with a Wess-Zumino-Witten (WZW) term responsible for the chiral anomaly and chiral vortical effect, it was shown that the ground state is a chiral soliton lattice (CSL) consisting of a stack of η-solitons in a high density region under rapid rotation. In a large parameter region, a single η-soliton decays into a pair of non-Abelian solitons, each of which carries SU(2)V /U(1) ≃ ℂP 1 ≃ S 2 moduli as a consequence of the spontaneously broken vector symmetry SU(2)V. In such a non-Abelian CSL, we construct the effective world-volume theory of a single non-Abelian soliton to obtain a d = 2 + 1 dimensional ℂP 1 model with a topological term originated from the WZW term. We show that when the chemical potential is larger than a critical value, a topological lump supported by the second homotopy group π 2(S 2) ≃ ℤ has negative energy and is spontaneously created, implying the domain-wall Skyrmion phase. This lump corresponds in the bulk to a Skyrmion supported by the third homotopy group π 3[SU(2)] ≃ ℤ carrying a baryon number. This composite state is called a domain-wall Skyrmion, and is stable even in the absence of the Skyrme term. An analytic formula for the effective nucleon mass in this medium can be written only in terms of the meson’s constants as   <math> <mn>4</mn> <msqrt> <mn>2</mn> </msqrt> <mi>π</mi> <msub> <mi>f</mi> <mi>π</mi> </msub> <msub> <mi>f</mi> <mi>η</mi> </msub> <mo>/</mo> <msub> <mi>m</mi> <mi>π</mi> </msub> <mo>~</mo> <mn>1.21</mn> </math>  $$ 4\sqrt{2}\pi {f}_{\pi }{f}_{\eta }/{m}_{\pi}\sim 1.21 $$  GeV with the decay constants f  π  and f  η  of the pions and η meson, respectively, and the pion mass m  π . This is reasonably heavier than the nucleon mass in the QCD vacuum.

Domain-wall Skyrmion phase in a rapidly rotating QCD matter

An axion cosmic string is known to be a chiral superconductor when the axion couples to an electrically charged fermion. After the QCD phase transition, a QCD axion string is attached by N domain walls. We would like to elucidate the fate of massless fermions on a global string after domain walls attached not only in the axion model but also in general models having string-wall composites. We investigate the Dirac equation under various string-wall composite backgrounds both in the axion(-like) models and in the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetry inspired Abelian-Higgs models. We give an answer to the elementary question of whether massless fermions exist, and if so, where they are localized. The answer depends on fermion/boson masses in the models, and the massless fermion can be localized either on the string, on one of the domain walls, or in one of the vacua. We find analytic solutions for the fermion zero mode function by which we prove the existence of the massless fermion on the string-wall composites. We also show supercurrents flowing along the string-wall composites and anomalous electric currents flowing in from outside.

Yuito Suzuki

Massless fermions and superconductivity of string-wall composites

We study a simple axionlike model with a charged scalar <math><mi>ϕ</mi></math> and a double-charged scalar <math><mi>ζ</mi></math> of global <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> symmetry. A particular feature of our model is that a vacuum manifold is a torus knot. We consider a hierarchical symmetry-breaking scenario where <math><mi>ζ</mi></math> first condenses, giving rise to cosmic <math><mi>ζ</mi></math>-strings, and the subsequent condensation of <math><mi>ϕ</mi></math> leads to domain-wall formation spanning the <math><mi>ζ</mi></math>-strings. We find that the formation of the walls undergoes two different regimes depending on the magnitude of an explicit breaking term of the relative <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> between <math><mi>ζ</mi></math> and <math><mi>ϕ</mi></math>. One is the weakly interacting regime where the walls are accompanied by another cosmic <math><mi>ϕ</mi></math> strings. The other is the strongly interacting regime where no additional strings appear. In both regimes, neither a <math><mi>ζ</mi></math>-string, a <math><mi>ϕ</mi></math>-string nor a wall alone is topological, but the composite of an appropriate number of strings and walls as a whole is topologically stable, characterized by the fundamental homotopy group of the torus knot. We confirm the formation and the structure of the string-wall system by first-principle cosmological two-dimensional simulations. We find stable string-wall composites at equilibrium, where the repulsive force between <math><mi>ζ</mi></math>-strings and the tension of walls balances, and a novel reconnection of the string-wall composites.

Takashi Hiramatsu

Izumi Saito

Yuki Sakakihara

String-wall composites winding around a torus knot vacuum in an axionlike model

QCD matter in strong magnetic field exhibits a rich phase structure. In the presence of an external magnetic field, the chiral Lagrangian for two flavors is accompanied by the Wess-Zumino-Witten (WZW) term containing an anomalous coupling of the neutral pion π 0 to the magnetic field via the chiral anomaly. Due to this term, the ground state is inhomogeneous in the form of either chiral soliton lattice (CSL), an array of solitons in the direction of magnetic field, or domain-wall Skyrmion (DWSk) phase in which Skyrmions supported by π 3[SU(2)] ≃ ℤ appear inside the solitons as topological lumps supported by π 2(S 2) ≃ ℤ in the effective worldvolume theory of the soliton. In this paper, we determine the phase boundary between the CSL and DWSk phases beyond the single-soliton approximation, within the leading order of chiral perturbation theory. To this end, we explore a domain-wall Skyrmion chain in multiple soliton configurations. First, we construct the effective theory of the CSL by the moduli approximation, and obtain the ℂP 1 model or O(3) model, gauged by a background electromagnetic gauge field, with two kinds of topological terms coming from the WZW term: one is the topological lump charge in 2+1 dimensional worldvolume and the other is a topological term counting the soliton number. Topological lumps in the 2+1 dimensional worldvolume theory are superconducting rings and their sizes are constrained by the flux quantization condition. The negative energy condition of the lumps yields the phase boundary between the CSL and DWSk phases. We find that a large region inside the CSL is occupied by the DWSk phase, and that the CSL remains metastable in the DWSk phase in the vicinity of the phase boundary.

Phase diagram of QCD matter with magnetic field: domain-wall Skyrmion chain in chiral soliton lattice

We study various composites of global solitons consisting of domain walls, strings, and monopoles in linear O(N) models with N = 2 and 3. Spontaneous symmetry breaking (SSB) of the O(N) symmetry down to O(N – 1) results in the vacuum manifold S  N−1, together with a perturbed scalar potential in the presence of a small explicit symmetry breaking (ESB) interaction. The O(2) model is equivalent to the axion model admitting topological global (axion) strings attached by N DW domain walls. We point out for the N DW = 2 case that the topological stability of the string with two domain walls is ensured by sequential SSBs (ℤ2)2 → ℤ2 → 1, where the first SSB occurs in the vacuum leading to the topological domain wall as a mother soliton, only inside which the second SSB occurs giving rise to a subsequent kink inside the mother wall. From the bulk viewpoint, this kink is identical to a global string as a daughter soliton. This observation can be naturally ex- tended to the O(3) model, where a global monopole as a daughter soliton appears as a kink in a mother string or as a vortex on a mother domain wall, depending on ESB interactions. In the most generic case, the stability of the composite system consisting of the monopole, string, and domain wall is understood by the SSB (ℤ2)3 → (ℤ2)2 → ℤ2 → 1, in which the first SSB at the vacuum gives rise to the domain wall triggering the second one, so that the daughter string appears as a domain wall inside the mother wall triggering the third SSB, which leads to a granddaughter monopole as a kink inside the daughter vortex. We demonstrate numerical simulations for the dynamical evolution of the composite solitons.