<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

<p>There exist metastable domain walls of the flavor-singlet meson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>η</mi></mrow></math> for the U(1) axial symmetry in two-flavor color superconductivity (2SC) in quantum chromodynamics (QCD) at large baryon density. We show that, due to the coupling of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi></math> to confined SU(2) gluons in the 2SC phase, the effective theory on the domain wall is described by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mo stretchy="false">)</mo><mrow><mo>−</mo><mn>1</mn></mrow></msub></math> Chern-Simons theory, which is dual to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo><mn>2</mn></msub></math> Chern-Simons theory. This theory has a spin-1 droplet excitation that does not carry a baryon number, which we identify as a vector meson. We also discuss the effective theories and baryonic droplet excitations on the domain walls of the flavor-singlet mesons in the superfluid phases of QCD at large isospin density and two-color QCD at large baryon density.</p>

Naoki Yamamoto

Ryo Yokokura

Quantum Hall liquids in high-density QCD

We study one-flavor SU(2) and SU(3) lattice QCD in (1 + 1) dimensions at zero temperature and finite density using matrix product states and the density matrix renormalization group. We compute physical observables such as the equation of state, chiral condensate, and quark distribution function as functions of the baryon number density. As a physical implication, we discuss the inhomogeneous phase at nonzero baryon density, where the chiral condensate is inhomogeneous, and baryons form a crystal. We also discuss how the dynamical degrees of freedom change from hadrons to quarks through the formation of quark Fermi seas.

Tomoya Hayata

Yoshimasa Hidaka

Dense QCD2 with matrix product states

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

Continuous phase transitions can be classified into ones characterized by local-order parameters and others that need additional topological constraints. The critical dynamics near the former transitions have been extensively studied, but the latter is less understood. We fill this gap in knowledge by studying the transition dynamics to a parity-breaking topological ground state called the chiral soliton lattice in quantum chromodynamics at finite temperature, baryon chemical potential, and an external magnetic field. We find a slowing down of the soliton’s translational motion as the critical magnetic field approaches, while the local relaxation rate to the solitonic state remains finite. The time required to converge to the stationary state strongly depends on the symmetry of the initial configuration, determining whether translational motion occurs during the dynamic process.

United States

China

Noriyuki Sogabe

Novel transition dynamics of topological solitons

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

The chiral soliton lattice (CSL) is a lattice structure composed of domain walls aligned in parallel at equal intervals, which is energetically stable in the presence of a background magnetic field and a finite (baryon) chemical potential due to the topological term originated from the chiral anomaly. We study its formation from the vacuum state, with describing the CSL as a layer of domain-wall disks surrounded by the vortex or string loop, based on the Nambu-Goto-type effective theory. We show that the domain wall nucleates via quantum tunneling when the magnetic field is strong enough. We evaluate its nucleation rate and determine the critical magnetic field strength with which the nucleation rate is no longer exponentially suppressed. We apply this analysis to the neutral pion in the two-flavor QCD as well as the axionlike particles (ALPs) with a finite (baryon) chemical potential under an external magnetic field. In the former case, even though the CSL state is more energetically stable than the vacuum state and the nucleation rate becomes larger for a sufficiently strong magnetic field, it cannot be large enough so that the nucleation of the domain walls is not exponentially suppressed and promoted, without suffering from the tachyonic instability of the charged pion fluctuations. In the latter case, we confirm that the effective interaction of the ALPs generically includes the topological term required for the CSL state to be energetically favored. We show that the ALP CSL formation is promoted if the magnetic field strength and the chemical potential of the system is slightly larger than the scale of the axion decay constant.

Tetsutaro Higaki

Kohei Kamada

Formation of a chiral soliton lattice

A chiral soliton lattice (CSL), proposed as the ground state of rotating baryonic matter at a finite density, is shown to be unstable in a large parameter region for two flavors owing to pion condensations, leading to two types of non-Abelian (NA) CSL phases (dimer and deconfining phases). We determine the phase diagram where the dimer phase meets the other phases and QCD vacuum at three tricritical points. The critical angular velocity of NA-CSLs is lower than that of η-CSL. Each NA soliton carries an isospin, and an antiferro-isospin chain is formed leading to gapless isospinons. The anomalous coupling to the magnetic field makes the NA-CSL (η-CSL) ferrimagnetic (ferromagnetic).

Phases of rotating baryonic matter: non-Abelian chiral soliton lattices, antiferro-isospin chains, and ferri/ferromagnetic magnetization

We study the ground states of low-density hadronic matter and high-density color-flavor locked color superconducting phase in three-flavor QCD at finite baryon chemical potential under rotation. We find that, in both cases under sufficiently fast rotation, the combination of the rotation-induced topological term for the η′ meson and the QCD anomaly leads to an inhomogeneous condensate of the η′ meson, known as the chiral soliton lattice (CSL). We find that, when baryon chemical potential is much larger than isospin chemical potential, the critical angular velocity for the realization of the η′ CSL is much smaller than that for the π 0 CSL found previously. We also argue that the η′ CSL states in flavor-symmetric QCD at low density and high density should be continuously connected, extending the quark-hadron continuity conjecture in the presence of the rotation.