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

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 π0 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 π3[SU(2)] ≃ ℤ appear inside the solitons as topological lumps supported by π2(S2) ≃ 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 ℂP1 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.


Introduction
The determination of matter phases stands as a pivotal challenge in modern physics.Quantum Chromodynamics (QCD) serves as the foundational theory of strong interactions, encapsulating descriptions of quarks, gluons, and hadrons (both baryons and mesons) as bound states of the aforementioned entities.Remarkably, the lattice QCD offers a comprehensive description of these bound states.The QCD phase diagram, especially under extreme conditions like high baryon density, pronounced magnetic fields, and rapid rotation, garners significant attention [1].Such conditions are not merely of theoretical interest but pertain to real-world scenarios like the interior of neutron stars and phenomena observed in heavy-ion collisions.While lattice QCD is adept at addressing scenarios with zero baryon density, its extension to finite baryon density is hampered by the infamous sign problem.Contrastingly, in situations where chiral symmetry undergoes spontaneous breaking, the emergence of massless Nambu-Goldstone (NG) bosons or pions, which are dominant at low energy, is observed.This low-energy dynamics is aptly described by either the chiral Lagrangian or the chiral perturbation theory (ChPT) centered on the pionic degree of freedom.Importantly, this description is predominantly dictated by symmetries and only modulated by certain constants, including the pion's decay constant f π and quark masses m π [2,3].
As an extreme condition, QCD in strong magnetic fields has received quite intense attention because of the interior of neutron stars and heavy-ion collisions.In the presence of an external magnetic field, the chiral Lagrangian 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 [4,5] in terms of the Goldstone-Wilczek current [6,7].It was determined to reproduce the so-called chiral separation effect [1,4,[8][9][10] in terms of the neutral pion π 0 .Then, at a finite baryon chemical potential µ B under a sufficiently strong magnetic field B, if the inequality < l a t e x i t s h a 1 _ b a s e 6 4 = " A 8 / k D 5 X S G R Y j h W 3 l 0 x 6 b e / p d C q I = " > A A A C h n i c h V G 7 S g N B F D 2 u r x h f U R v B J h o i F h I m 4 g u r o I W W v v K A J I T d d a K L + 2 J 3 E t Q l t e A P W F g p i I j Y 6 g f Y + A M W f o J Y K t h Y e L N Z E Q 3 q X X b m 3 D P 3 3 D k z o 9 i 6 5 g r G H l u k 1 r b 2 j s 5 Q V 7 i 7 p 7 e v P z I w m H G t i q P y t G r p l p N T Z J f r m s n T Q h M 6 z 9 k O l w 1 F 5 1 l l d 6 m + n q 1 y x 9 U s c 1 P s 2 7 x o y N u m V t Z U W R B V i o w W j E r J K w i + J x z D W 6 z V C p P R / G e 6 z D O 1 1 y H D p y y M J B p u 4 I j z i H E K a v 8 5 R Q 5 i 0 F a r i V C E T u 0 v j N m X 5 g D U p r / d 0 f b V K u + j 0 O 6 S M I s 4 e 2 C V 7 Y f f s i j 2 x 9 1 9 7 e X 6 P u p d 9 m p W G l t u l / q P h j b d / V Q b N A j t f q j 8 9 C 5 Q x 7 3 v V y L v t M / V T q A 1 9 9 e D 4 Z W N h P e 6 N s z P Phase diagram of QCD matter with magnetic fields.The blue curve denotes the boundary of QCD vacuum by Son and Stephanov in eq.(1.1).The green dotted curve denotes the instability curve of the CSL by the charged pion condensation in eq.(4.17), given below, that asymptotically reduces to eq. (1.2) for large B. The red curve is our new finding of the phase boundary between the CSL and DWSk phases in eq.(4.16), given below, that behaves asymptotically as eq.(4.20), given below.The blue, red and green dotted curves meet at the tricritical point in eq.(1.3).The CSL configuration is a metastable state in the region between the red and green dotted curves.
holds, the ground state of QCD with two flavors (up and down quarks) becomes inhomogenous in the form of a chiral soliton lattice (CSL) consisting of a stack of domain walls or solitons carrying a baryon number [5,11,12]. 1owever, Brauner and Yamamoto found that such a CSL state is unstable against a charged pion condensation in a region of higher density and/or stronger magnetic field [12].The asymptotic expression of the instability curve at large B is : the green dotted curve (at large B) in fig. 1 (1.2) above which the CSL is unstable, where "CPC" denotes the charged pion condensation.The full expression of the boundary is given in eq.(4.17), below, and is denoted by the green dotted curve in fig. 1.In Ref. [28], an Abrikosov's vortex lattice was proposed as a consequence of the charged pion condensation (see also a recent paper [29]).This instability curve ends at the tricritical point: In our previous paper [30], we proposed that there is the domain-wall Skyrmion (DWSk) phase in the region inside the CSL, with µ c in eq. ( 1.3), in which Skyrmions are created on top of the solitons in the ground state.To show this, the effective world-volume theory on a single soliton was constructed as an O(3) sigma model or the CP 1 model with topological terms induced from the WZW term.Then, there appear topological lumps (or baby Skyrmions) supported by π 2 (CP 1 ) ≃ Z on the world volume, corresponding to 3+1 dimensional Skyrmions supported by π 3 [SU(2)] ≃ Z in the bulk point of view.Such a composite state of a domain wall and Skyrmions are called domain-wall Skyrmions.2However, we used a single-soliton approximation, considering the domain-wall Skyrmion on a single soliton [30].In other words, we assumed that the solitons are well separated, and this assumption can be justified only at the phase boundary between the QCD vacuum and the CSL phase in eq.(1.1), namely at the tricritical point in eq.(1.3).On the other hand, the instability curve of the CSL due to a charged pion condensation also ends at the same point in eq.(1.3) [12].Therefore, a natural question that arises is the compatibility between the DWSk phase and instability curve.
In this paper, we determine the phase boundary between the CSL and DWSk phases beyond the single-soliton approximation, in which the boundary was the straight vertical line in eq.(1.4) ending on the tricritical point represented by the white dot in fig. 1.To this end, we explore domain-wall Skyrmion chains in multiple soliton configurations.A similar domain-wall skyrmion chain has been also studied in chiral magnets [44].As is well known, the CSL configuration is analytically given by the elliptic function.We construct the effective theory of the CSL by the moduli approximation [51][52][53], in which we promote the CP 1 moduli of the CSL to fields depending on the worldvolume (x 0 , x 1 , x 2 ) and integrate over one period of the lattice in the codimensional direction x 3 .We obtain the CP 1 model or O(3) model with two kinds of topological terms coming from the WZW term: one is topological lump charge responsible for π 2 (S 2 ) ≃ Z in 2+1 dimensional worldvolume and the other is a topological term counting the soliton number.We then construct lumps in the 2+1 dimensional worldvolume theory.Since the electromagnetic U(1) gauge symmetry is spontaneously broken around a ring surrounding the lump, the lump can be regarded as a superconducting ring.Then its size modulus is fixed by the flux quantization condition of the superconducting ring, enhancing its stability.The lumps in the soliton worldvolume correspond to Skyrmions in the bulk, Skyrmions periodically sit on each soliton in the CSL, and thus the configuration is a domain-wall Skyrmion chain.The condition that a lump has negative energy yields the phase boundary between the CSL and DWSk phases denoted by the red curve in fig. 1.In the strong magnetic field limit, the phase boundary asymptotically behaves as The important is that the boundary curve has the lower critial chemical potential µ B and lower critical magnetic field B than those of the instability curve (the green dotted curve in fig. 1) of the CSL in eq.(1.2).Therefore, the CSL state remains metastable in the region between the red and green dotted curves.This paper is organized as follows.In sec. 2 we present a CSL in the strong magnetic field.In sec. 3 we construct the effective worldvolume theory of a one period of the CSL, by the moduli approximation.In sec. 4 we construct topological lumps in the soliton's worldvolume theory and determine the phase boundary between the CSL and DWSk phases.Sec. 5 is devoted to a summary and discussion.

Chiral soliton lattice in strong magnetic field
We focus on the phase where chiral symmetry is spontaneously broken.The effective field theory of pions, known as ChPT, can describe the low-energy dynamics.The pion fields are represented by a 2 × 2 unitary matrix, where τ a (with a = 1, 2, 3) are the Pauli matrices, normalized as tr(τ a τ b ) = 2δ ab .The field Σ transforms under SU(2) L × SU(2) R chiral symmetry as where both L and R are 2×2 unitary matrices.Then, the effective Lagrangian at the where f π and m π are pion's decay constant and mass, respectively, and D µ is a covariant derivative defined by where Q is a matrix of the electric charge of quarks.The U(1) EM transformation is given by The external U(1) B gauge field A B µ can couple to Σ via the Goldstone-Wilczek current [6,7].The conserved and gauge-invariant baryon current in the external magnetic field is detailed in refs.[5,6]: with A B µ = (µ B , 0), and introducing the notations

The effective Lagrangian that couples to A B
µ is expressed as which is recognized as the WZW term [4,5].Thus, the total Lagrangian is An important observation is warranted at this point.In order to formulate an effective Lagrangian, we adopt a modification of the standard power counting scheme of ChPT as presented in ref. [20]: In this power-counting scheme, eq.(2.8) is of order O(p 2 ) and is consistent with eq. ( 2.

3).
It is significant to note that µ B only manifests in the WZW term of eq.(2.8), which allows us to attribute a negative power counting to µ B .The effective field theory up to O(p 2 ) must incorporate both terms in eq.(2.7).However, previous studies on the CSLs have not taken into account the first term in eq.(2.7).We emphasize that the inclusion of an O(p 4 ) term, such as the Skyrme term and the chiral anomaly term (which includes π 0 E • B), is not essential for our results.As a result, our analysis maintains its model-independece.Moreover, it should be noted that at the leading order, the gauge field is nondynamical due to its kinetic term being of order O(p 4 ).
We note that our effective theory admits a parallel stack of the sine-Gordon soliton expanding perpendicular to the external magnetic field, which is called the chiral soliton lattice.This state is stable under a sufficiently large magnetic field, as shown in [5].If we consider the case of no charged pions Σ 0 = e iτ 3 χ 3 , the effective Lagrangian reduces to The ordinary QCD vacuum corresponds to χ 3 = 0.However, the third term in eq.(2.11) modifies the ground state of QCD at finite µ B and B. The anticipated time-independent neutral pion background χ 3 is obtained by minimization of the energy functional.The static Hamiltonian depending only on the z coordinate is given by (2.12) Without loss of generality, we will orient the uniform external magnetic field along the z-axis; B = (0, 0, B).We note that eq.(2.12) have the first derivative term proportional to ∂ z χ 3 .Then, the configuration of the ground state will have a nontrivial z-dependence.In order to determine the static configuration of χ 3 , let us solve the EOM of eq.(2.11).The equation of motion for such a one-dimensional configuration χ 3 (z) then reads which can be analytically solved by the elliptic functions: with a real constant κ (0 ≤ κ ≤ 1) called the elliptic modulus.This solution is a lattice state of the π 0 (= f π χ 3 ) meson with a period ℓ.The period ℓ satisfies the following equations: with the complete elliptic integral of the first kind K(κ).Substituting eq.(2.14) into eq.(2.12) and integrating it between one period ℓ, we get the tension of a single soliton inside the CSL, that is the energy density per unit area integrated over one period with the complete elliptic integral of the second kind E(κ).Minimizing the energy density per unit length E/ℓ with respect to k gives me the following condition: (2.18) Since the left-hand side of Eq. (2.18) is bounded from below as E(κ)/κ ≥ 1 (0 ≤ κ ≤ 1), the CSL solution exists if and only if the following condition is satisfied [5,12]: denoted by the blue curve in fig. 1. Inserting the minimization condition (2.18) into Eq.(2.17), we evaluate the energy density at the optimized k as: which is lower than that of the QCD vacuum.Therefore, the CSL is energetically more stable than the QCD vacuum.
3 Effective worldvoume theory of soliton in chiral soliton lattice The preceding section concentrated exclusively on the π 0 meson.General solutions that encompass charged pions can be derived from Σ 0 through an SU(2) V transformation, where g represents an SU(2) matrix.It is clear that Σ 0 is invariant under SU(2) V when g = e iτ 3 θ .Consequently, each soliton possesses moduli originated from the spontaneous symmetry breaking SU(2) V → U(1) 3 in the vicinity of the soliton: Such a sine-Gordon soliton carrying non-Abelian CP 1 moduli is called a non-Abelian sine-Gordon soliton [35,54] (see also refs.[36,55,56]).However, unlike these references, our solitons are nontopological because they are not protected by topology and are rather stabilized by the WZW term.
For subsequent discussions, we characterize the CP 1 moduli using the homogeneous coordinates ϕ ∈ C 2 of CP 1 , which fulfill the relations [35] In the context of ϕ, eq.(3.1) can be recast as u ≡ e iχ CSL 3 = exp 2 i am m π z κ , κ + πi . (3.5) Given that the moduli space is S 2 , we can also employ the real three-component vector n defined as to describe this space.The π 0 CSL in eq.(2.14) corresponds to n 3 = 1.
Let us construct the low-energy effective field theory of the CSL based on the moduli approximation [51][52][53].In the following, we will promote the moduli parameter ϕ as the fields on the 2 + 1-dimensional soliton's world volume.We first calculate the effective action coming from eq. (2.3).Substituting eq.(3.4) into eq.( 2.3), we get where x α (α = 0, 1, 2) are world-volume coordinates.Integrating over z, the effective action stemming from eq. (3.7) can be calculated as with the Kähler class C(κ) defined by and the tension E of a signle soliton in eq.(2.17), where we have used the integrals, and the contribution of the gauge field can be summarized into the covariant derivative: The first term in eq.(3.8) is the kinetic term of ϕ, being equivalent to the gauged CP 1 model.The second term in eq.(3.8) is the minus tension of each soliton, that is the energy density of the CSL in one period ℓ.
We next calculate the effective action coming from the WZW term in eq.(2.8).The contribution of the first term in eq.(2.7) to the WZW term in eq.(2.8) can be expressed as µ B B. Here, we refer to the Skyrmoin charge density as which can be factorized as where the CP 1 lump topological charge density is defined as Integration of q over x and y gives the quantized lump charge k, being associated with π 2 (CP 1 ) : Integrating B from 0 to ℓ, we get where we have used (3.17 We thus have seen that one lump on one soliton corresponds to two Skyrmions (baryons) in the bulk.This one-to-two correspondence is in contrast to the domain-wall Skyrmions in QCD under rapid rotation [17], in which case one lump on a soliton corresponds to one Skyrmion.
The contribution of the second term in eq.(2.7) to the WZW term in eq.(2.8) can be divided into two terms as follows: We consider the uniform external magnetic field along the z-axis, B = (0, 0, B).Then, the first term in eq.(3.19) becomes In terms of the projection operator P ≡ ϕϕ † satisfying P 2 = P , R k and L k can be expressed as Since ϕ does not depend on z, the second and third terms in L 3 and R 3 vanish.Therefore, eq.(3.20) becomes and integrating over z, we get where we have used the boundary condition of χ CSL 3 for a single soliton, χ CSL 3 (ℓ)−χ CSL 3 (0) = 2π.We next calculate the second term in eq.(3.19).Substituting eq.(3.4) into ∂ i Σ∂ k Σ † and ∂ k Σ † ∂ i Σ, these two quantities can be calculated as Hence, we can represent in terms of u and ϕ as follows: Inserting this expression into the second term in eq (3.19), it becomes Integrating over z, we get where we have used the integral eq.(3.17).Summing up eqs.(3.24) and (3.29), the effective Lagrangian from the second term in eq.(2.7) can be calculated as Finally, we arrive at the effective Lagrangian of the non-Abelian sine-Gordon soliton under the magnetic field: This is a background-gauged CP 1 model or O(3) model with the topological terms.Note that in the single-soliton limit (κ = 1), the Kähler class in eq.(3.9) reduces to C(κ = 1) = 16f 2 π /3m 2 π recovering our previously result [30].

Domain-wall Skyrmion chain and domain-wall Skyrmion phase
We examine Skyrmions within the domain-wall effective theory described by eq.(3.31).Initially, we neglect the gauge coupling by setting D µ → ∂ µ , considering the effects of the WZW term from eqs. (2.7) and (2.8).Subsequently, we incorporate the effects of the gauge coupling.The resulting static Hamiltonian is given by Since the constants in eq.(3.31)only give the condition of whether the domain wall appears or not, it is sufficient to consider B > B c , and thus have been omitted in eq.(4.1).Then, the total energy E DW = d 2 x H DW is bounded from below as which is called the Bogomol'nyi bound, where we have used The inequality in eq. ( 4.2) is saturated only when the fields satisfy the (anti-)Bogomol'nyi-Prasad-Sommerfield (BPS) equation [57] where the upper (lower) sign corresponds to the (anti-)BPS equation.It is interesting to observe that the second term in eq. ( 4.1) splits energies between BPS lumps k > 0 and anti-BPS lumps k < 03 .The BPS solutions to this equation characterized by the winding number k (> 0) is given by [57] where w ≡ x + iy, and the set of complex parameters {a are the moduli parameters.We examine the gauge coupling between n and A α .With the electromagnetic gauge symmetry U (1) EM generated by τ 3 , the transformation of n 1 + in 2 is given by while n 3 remains neutral.The covariant derivative is Let C be a closed curve where n 3 = 0 and D its interior.Due to the spontaneous breaking of U (1) EM symmetry around |n 1 + in 2 | = 1, curve C functions as a superconducting loop, carrying a persistent current.Expressing n 1 + in 2 = e iψ on C, the gauge field configuration along C is determined by the gradient energy minimization, leading to |D α (n 1 + in 2 )| 2 = 0 and ∂ α ψ = eA α .Consequently, the flux and area quantization on D becomes where k is the lump number on D and S D is the area of D. The isosurface of baryon number density B = 1/(10π 2 ) (orange), and the sine-Gordon soliton π/2 < θ < 3π/2 (blue).From the left to right panels, the elliptic modulus κ decreases (Bµ B increases) and the periodicity of the CSL decreases.The size modulus is fixed by the quantization condition, the physical width of the right configuration is smaller than that of the left configuration since smaller κ corresponds to larger B. In the rescaled coordinates, the shape changes from round (left) to crushed (right), and they look like periodic macarons (left) to a pancake tower (right).
For a single lump with k = 1, represented by f = b 0 w , the size and phase moduli are |b 0 | and arg b 0 , respectively.The relationship for n 3 is and the region size D defined by n 3 = 0 is |w| = |b 0 |.The flux quantization requires the size modulus to be For axially symmetric k-lumps, and the flux quantization requires the size modulus to be Since Skyrmions sit in the same positions (x 1 , x 2 ) at each soliton, the total configuration in 3+1 dimensions is domain-wall Skyrmion chains.In fig.2, we plot our solutions of domain-wall Skyrmion chains for various elliptic modulus κ.The orange regions denote the isosurface of the baryon number density B = 1/(10π 2 ), and the blue regions denote the soliton π/2 < θ < 3π/2.One can confirm that one lump on one soliton is composed of two Skyrmions as can be expected from eq. (3.18).From the left to right panels, the elliptic modulus κ decreases corresponding to the situation that Bµ B increases, and the periodicity of the CSL decreases.From left to right, these look like from periodic macarons to a pancake tower.We note that a similar domain-wall skyrmion chain has been studied in chiral magnets in 2 + 1 simensions [44].Now let us discuss a constraint from the second term of WZW term following ref.[30].The integration of the last term in eq. ( 4.2) can be rewritten as where we have used the explicit solution in eq.(4.5) in the last expression.We thus reach the energy of domain-wall Skyrmions, given by For a single lump (k = 1), we have E DWSk = 2πC(κ) with the cancellation between the second and third terms due to the flux quantization in eq.(4.10) which is always positive.For higher winding k ≥ 2, we have a further constraint to minimize the domain-wall Skyrmion energy in eq.(4.14), which can become negative for sufficiently large µ B from its second term.Finally, let us discuss the DWSk phase in which Skyrmions are created spontaneously.From the above consideration, the phase boundary between the CSL and DWSk phases is determined to be When µ B ≥ µ c , the lumps have negative energy and are spontaneously created, implying the DWSk phase.Note that in the single-soliton limit κ = 1, µ c in eq.(4.16) reduces to a constant 16πf 2 π 3mπ (thus a vertical line) reproducing the previous result in eq.(1.4) [30].Since the elliptic modulus κ is determined from B and µ B in general, eq.(4.16) gives a nontrivial curve beyond the one-soliton approximation, represented by the red curve in fig. 1.This is our main result.It is interesting to observe that µ c can be interpreted as the effective nucleon mass in this medium (inside solitons with the chemical potential µ B and magnetic field B), which is 16πf 2 π 3mπ ∼ 1.03 GeV at the tricritical point (the white dot in fig.1), and it becomes lighter as the magnetic field is stronger.
Let us compare this boundary with the instability curve of the CSL configuration via the charged pion condensation in eq.(1.2) by Brauner and Yamamoto [12].The full expression for the instability curve is determined by eliminating the elliptic modulus κ from the following two equations [12]  and is denoted by the green dotted curve in fig. 1.One can observe that this curve is entirely above the phase boundary between the CSL and DWSk phases in eq.(4.16), denoted by the red curve in fig. 1.The CSL configuration remains locally stable (metastable) in the region between the red and green dotted curves.
Here, let us investigate the large B behaviours of these two curves.To this end, we expand the equations around κ = 0. Expanding eqs.(4.16) and (2.18) around κ = 0, we obtain respectively.Eliminating κ from these two equations, we obtain On the other hand, expanding the instability curve of eq.(4.17) in the same way, we find that it asymptotically behaves as in eq.(1.2) Clearly, this is above the phase boundary between the CSL and DWSk phases in eq.(4.20).

Summary and discussion
In this paper, in the phase diagram of QCD with finite baryon density and magnetic field, we have determined the phase boundary between the CSL and DWSk phases beyond the singlesoliton approximation at the leading order O(p 2 ) of ChPT.The key point to go beyond the single-soliton approximation is considering domain-wall Skyrmion chains in multiple soliton configurations.We have constructed the low-energy effective theory of one period of the CSL by the moduli approximation.We have obtained in eq.(3.31) the backgroundgauged CP 1 model or O(3) model with topological terms originated from the WZW term, topological lump charge in 2+1 dimensional worldvolume and the topological term for the soliton number.A single topological lump in 2+1 dimensional worldvolume theory is a superconducting ring.Due to the flux quantization condition in eq.(4.8), the size modulus is fixed.We have determined the phase boundary between the CSL and DWSk phases in eq.(4.16) denoted by the red curve in fig. 1 from the negative energy condition of the lumps.
We have found that a large region in the CSL phase is occupied by the DWSk phase, and that the CSL configuration is metastable in the region between the red curve and green dotted curve given by eq.(4.17), beyond which the CSL is unstable.The blue, red and green dotted curves meet at the tricritical point in eq.(1.3).
We have worked out at the leading order O(p 2 ) of the ChPT for which we have not needed higher derivative terms such as the Skyrme term.At this order, the magnetic field is a background field.At the next leading order O(p4 ), one needs higher derivative terms as well as the kinetic term of the electromagnetic gauge field.The stability beyond the leading order remains a future problem.
The phase transition between the CSL and DWSk phases in eq. ( 4.16) denoted by the red curve in fig. 1 would be the so-called second order of nucleation type in the classification by de Gennes [59] In such a case, the configuration of one side of the boundary often remains metastable on the other side, which is in fact our case.Similarly, the phase boundary between the QCD vacuum and CSL denoted by the blue curve in fig. 1 was recently shown to be of the second order [21], and it should be of the nucleation type.Quantum nucleation, explored in the case of the transition from the vacuum to CSL [22,23], should be applied to the transition from the CSL to DWSk phase.Investigating this transition is one of important future directions.
In this paper, we have obtained the periodic structure of Skyrmions: a domain-wall Skyrmion chain.Another approach is to construct an effective theory of a soliton lattice.The effective theory of each soliton is a CP 1 model or O(3) model.As was studied for a non-Abelian vortex lattice in ref. [60], we can construct a lattice effective theory as follows.A neighboring pair of solitons interacts as H int = −J ⟨i,i+1⟩ n i n i+1 with the CP 1 moduli n i of the i-th soliton.In our case, the lattice behaves as a ferromagnet with J > 0, and thus the moduli tend to be aligned.When one constructs a lump on a soliton, the system prefers to place the same lumps on its neighboring solitons.Then, that theory admits an array of lumps along the lattice direction, which is nothing but our Skyrmion chain.One can also take a continuum limit (large Bµ B ) resulting in a 3+1 dimensional anisotropic CP 1 model , in which we need a careful treatment for the terms from the WZW term.Then, the continuum theory should admit a lump string along the z-direction, which should have negative energy.
Let us discuss a possible relation between our configuration of the domain-wall Skyrmion chain and an Abrikosov vortex lattice in the charged pion condensation proposed in Ref. [28].It was shown in Refs.[36,56] that when (ungauged) Skyrmions are periodically arranged with a twisted boundary condition, they reduce to global vortices in the small periodicity limit. 4See the rightmost panel of Fig. 2. In our case, these vortices should carry baryon numbers [34,73,74].This may offer a possible crossover between our configuration of the Skyrmion chain and an Abrikosov vortex lattice [28].However, there is a significant difference.If we turn on a dynamical electromagnetic gauge field at the next leading order O(p 4 ), they would reduce to superconducting strings since charged pions are condensed in the vortex cores.Thus, our Skyrmion chains at the next leading order O(p 4 ) are superconducting strings in the short period limit (that is the continuum limit of the Heisenberg spin chain at large Bµ B as mentioned above).
Before concluding this paper, we wish to comment on a domain-wall Skyrmion phase analogous to the one found in QCD matter under rapid rotation [17].Recent years have seen a surge in interest regarding rotating QCD matter [13][14][15][16][75][76][77][78][79][80][81][82][83][84][85], primarily due to the observation of an exceptionally large vorticity of the order of 10 22 /s in quark-gluon plasmas produced in non-central heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) [86,87].In ChPT, the anomalous term for the η ′ meson was derived in [13,14] by matching it with the chiral vortical effect (CVE) [10,[88][89][90][91][92] in the context of mesons.Analogous to the effect of a magnetic field, this term suggests a CSL composed of the η ′ meson during rapid rotation [13][14][15].For two flavor scenarios, the phenomenon manifests as an η-CSL made up of the η meson.In a significant parameter region, a single η-soliton energetically decays into a couple of non-Abelian solitons, leading to neutral pion condensation in its vicinity.A lone non-Abelian soliton breaks the vector symmetry SU(2) V down to its U(1) subgroup. Ths results in NG modes described by SU(2) V /U(1) ≃ CP 1 ≃ S 2 , which localize near the soliton as discussed in [16].Therefore, mirroring the π 0 soliton in a magnetic setting, each non-Abelian soliton carries CP 1 moduli and is termed a non-Abelian sine-Gordon soliton [35,36,[54][55][56].Relying on the single-soliton approximation, we posited the DWSk phase for rapid rotations in [17].Consequently, our present study on a domain-wall Skyrmion chain based on multiple solitons can be extended to rotational scenarios.
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Figure 2 .
Figure 2. The minimal (k = 1) Skyrmion chain in the CSL with f (w) = 2/eB/w.The z coordinate is rescaled by m π as m π z, and the x and y coordinates are rescaled by the lump size 2/eB as 2/eB(x, y).The isosurface of baryon number density B = 1/(10π2 ) (orange), and the sine-Gordon soliton π/2 < θ < 3π/2 (blue).From the left to right panels, the elliptic modulus κ decreases (Bµ B increases) and the periodicity of the CSL decreases.The size modulus is fixed by the quantization condition, the physical width of the right configuration is smaller than that of the left configuration since smaller κ corresponds to larger B. In the rescaled coordinates, the shape changes from round (left) to crushed (right), and they look like periodic macarons (left) to a pancake tower (right). )