Domain-wall Skyrmion phase in a rapidly rotating QCD matter

Based on the chiral perturbation theory at the leading order, we show 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 $\eta$-solitons in a high density region under rapid rotation. In a large parameter region, a single $\eta$-soliton decays into a pair of non-Abelian solitons, each of which carries ${\rm SU}(2)_{\rm V}/{\rm U}(1) \simeq {\mathbb C}P^1 \simeq S^2$ moduli as a consequence of the spontaneously broken vector symmetry ${\rm SU}(2)_{\rm V}$. 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 ${\mathbb C}P^1$ 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 $\pi_2(S^2) \simeq {\mathbb Z}$ 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 $\pi_3[ {\rm SU}(2)] \simeq {\mathbb Z}$ 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 is obtained as $4\sqrt{2}\pi f_{\pi}f_\eta/m_{\pi} \sim 1.21$ GeV with the decay constants $f_{\pi}$ and $f_\eta$ of the pions and $\eta$ meson, respectively, and the pion mass $m_{\pi}$, which is surprisingly close to the nucleon mass in the QCD vacuum.


Introduction
Quantum Chromodynamics (QCD) is the fundamental theory of the strong interaction described by quarks and gluons.QCD at extreme conditions such as high baryon density, strong magnetic field, and rapid rotation has been paid much attention since it is relevant for neutron star interiors and heavy-ion collisions [1].Lattice QCD cannot be extended to finite baryon density because of the notorious sign problem.Instead, at least at low energy, the chiral Lagrangian or the chiral perturbation theory (ChPT) offers a powerful tool since the theory is thoroughly determined by symmetry up to some constants, the pion's decay constant, quark masses, and so on [2,3].When the chiral symmetry mixing different species of quarks (up-quarks, down-quarks and so on) is spontaneously broken, there appear Nambu-Goldstone(NG) bosons or pions.Thus, the low-energy dynamics can be described by the aforementioned ChPT.
One of the most important extreme conditions for QCD is strong magnetic fields 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 (GW) current [6,7].It was determined to reproduce the so-called chiral separation effect (CSE) [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, the ground state of QCD with two flavors (up and down quarks) was found to be a chiral soliton lattice (CSL) consisting of a stack of domain walls or solitons carrying a baryon number [5,11,12].However, such a CSL state was found to be unstable against a charged pion condensation in a region of higher density and/or stronger magnetic field [12].In such a region, there appears a new phase, the domain-wall Skyrmion phase in which Skyrmions are created on top of the solitons in the ground state [13,14]. 1 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, topological lumps (or baby Skyrmions) supported by π 2 (CP 1 ) ≃ Z on the world volume, corresponding to 3D Skyrmions supported by π 3 [SU (2)] ≃ Z in the bulk point of view, appear in the ground state for a sufficiently large chemical potential.The composite states of a domain wall and Skyrmions are called domain-wall Skyrmions.Such domain-wall Skyrmions were previously proposed and studied in field theory [17][18][19][20][21][22]. 2 Domain-wall Skyrmions of a 2+1 dimensional version were also proposed in field theory [24][25][26] and have been recently observed experimentally [27,28] in chiral magnets [29][30][31][32][33] (see also Refs.[34][35][36]).
Another important extreme condition for QCD that we focus on in this paper is a rapid rotation.Quark-gluon plasmas produced in non-central heavy-ion collision experiments at the Relativistic Heavy Ion Collider (RHIC) reach the largest vorticity observed thus far, of the order of 10 22 /s [37,38].This has triggered significant attention to rotating QCD matter in recent years [39][40][41][42][43][44][45][46][47][48][49][50][51][52][53].In particular, similar but different type of CSL appears in QCD at finite density under rapid rotation instead of a strong magnetic field. 3The anomalous term for the η ′ meson was obtained [50,51] by matching with the chiral vortical effect (CVE) [10,[64][65][66][67][68] in terms of mesons.Although the full WZW term is not known unlike the case of the magnetic field, this CVE term is sufficient to yield a CSL phase made of the η ′ meson as the ground state in a certain parameter region [50][51][52], instead of that of the neutral pion π 0 in the case of the magnetic field. 4In the two-flavor case, there appears a CSL phase made of the η meson.However, it was shown in ref. [53] that in a large parameter region, a single η-soliton energetically decays into a pair of non-Abelian solitons, around which the neutral pion condensation occurs.A single non-Abelian soliton spontaneously breaks the vector symmetry SU(2) V into its U(1) subgroup, resulting in NG modes SU(2) V /U(1) ≃ CP 1 ≃ S 2 localized in the vicinity of the soliton.Thus, as the case of a π 0 soliton in the magnetic field, each non-Abelian soliton carries CP 1 moduli and are called non-Abelian sine-Gordon solitons [21,69] (see also refs.[22,70,71]).In a lattice of non-Abelian solitons, the CP 1 modes at two neighboring solitons repel each other, and thus they are antialigned.The lattice behaves as a Heisenberg anti-ferromagnet, in which we call one soliton an up-soliton and its neighbors down-solitons, and then up and down solitons appear alternately.Such a non-Abelian CSL can be classified into the two cases, the deconfined and dimer phases.In the deconfined phase, an up-soliton and down-soliton repel each other, and they are separated with the equal distance.In the dimmer phase, they attract each other at large distances and repel at short distances, and thus constitute a molecule.The lattice can be regarded as a lattice of molecules.On the other hand, in the confined phase, the up and down-solitons attract and are completely overlapped to become η solitons.This is the previously known η-CSL [51], in which the vector symmetry SU(2) V is unbroken, and no soliton carries CP 1 modes.
In this paper, we establish the presence of a new phase in rapidly rotating QCD matter, namely a domain-wall Skyrmion phase inside the non-Abelian CSL, similar to the case of a strong magnetic field.Within non-Abelian CSLs, either in the deconfined or dimer phase, the vector symmetry SU(2) V is spontaneously broken into its U(1) subgroup, thus being accompanied by NG modes SU(2) V /U(1) ≃ CP 1 ≃ S 2 as mentioned above.For our purpose we concentrate on a single non-Abelian soliton in the deconfined phase.We construct the effective theory on a single non-Abelian soliton and obtain a CP 1 model with a topological term originated from the WZW term.It admits topological lumps (baby Skyrmions) ensured by the second homotopy group π 2 (S 2 ) ≃ Z [72].We find that when the chemical potential is larger than a critical value, the topological lumps have negative energy due to the WZW term.The lumps on the non-Abelian soliton are Skyrmions supported by the third homotopy group π 3 (S 3 ) ≃ Z in the bulk point of view [21], and they carry baryon numbers.This implies the domain-wall Skyrmion phase in which lumps are spontaneously created in the ground state, where the both chiral solitons and Skyrmions carry baryon numbers.The lump energy is obtained as 4 √ 2πf π f η /m π which can be interpreted as the effective nucleon mass in this medium (inside a soliton at finite density under rapid rotation) and is evaluated as ∼ 1.21 GeV.This value is close enough the nucleon mass 938 MeV in the QCD vacuum.
This paper is organized as follows.In sec.2, we review non-Abelian CSLs.In sec.3, we construct the effective worldvolume theory of a single non-Abelian soliton in the deconfined phase of non-Abelian CSL.In sec.4, we constuct domain-wall Skyrmions and find the presence of the domain-wall Skyrmion phase.Sec. 5 is devoted to a summary and discussion.

Non-Abelian chiral soliton lattices under rotation
We focus on the phase in which the U(2) L ×U(2) R chiral symmetry is spontaneously broken down.The low-energy dynamics can thus be described by an effective field theory of the pions -ChPT.A 2 × 2 unitary matrix represents the pion fields where τ a (a = 1, 2, 3) are the Pauli matrices with the normalization tr(τ a τ b ) = 2δ ab .This field Σ transforms under the SU(2) L × SU(2) R chiral symmetry as Σ → LΣR † , where L and R are 2 × 2 unitary matrices, while χ 0 transforms under the axial U(1) A symmetry as χ 0 → χ 0 + 2θ 0 .Then, the effective Lagrangian at the leading order is (µ = 0, • • • , 3) where f π and f η are the decay constants of the pions and the U(1) A singlet (η) meson , respectively, m is the quark mass, and A and B are parameters that cannot be determined by symmetry alone.The first and second terms are the kinetic terms of the χ a and χ 0 , respectively.The third term is the mass term of the mesons, stemming from the explicit chiral symmetry breaking due to the finite quark masses.Then, the pion mass m π is related to the quark mass by the Gell-Mann-Oakes-Renner relation Bm = f 2 π m 2 π /4.The fourth term represents the QCD anomaly: U(1) A → Z 4 .The parameter A gives an additional mass term for χ 0 meson given by δm 2 χ 0 = A/f 2 η .Here, g µν is a spacetime metric representing the rotating coordinates, and g µν is its inverse: where Ω stands for the rotation velocity.
The external U(1) B gauge field A B µ can couple to Σ through the GW current [6, 7] where we have introduced the standard notation Then, the effective Lagrangian coupling to A B µ can be written as In order to introduce finite baryon chemical potential µ B , we choose the U(1) B gauge field as A B µ = (µ B , 0).This is a WZW term but the full expression for rotation L WZW is not known thus far, in contrast to the case of the magnetic field in which case the full expression is known [4,5].In the external electromagnetic field, the gauge-invariant and conserved baryon current can be derived by the "trial and error" U(1) em gauging [7].The coupling of U(1) em gauged GW current to A B µ is calculated as [5,6] ) The second term with Σ = e iχ 3 τ 3 becomes eB • ∇χ 3 /(4π 2 ).In fact, the above equation can be derived by reproducing the chiral separation effect (CSE) [1,4,[8][9][10] in terms of χ 3 meson.This procedure is justified by the fact that the chiral anomaly coefficient determines the transport coefficient of the CSE [66,73].Therefore, the anomaly matching of the CSE can derive the part of LWZW .Unfortunately, the GW current is already invariant under general coordinate transformations, so the method applied to the electromagnetic field cannot be used.Hence, the method to derive the full expression is not known.However, when applying this method to a rotating system, it is evident that at least the following terms exist.QCD at finite baryon chemical potential under global rotation is known to exhibit the anomalous current in the direction of the rotation, which is the so-called CVE [8,10,64,67,68] : where q is a quark field.We note that the chiral anomaly determines the transport coefficient of CVE.Therefore, due to the exactness of the chiral anomaly coefficient, it must be reproduced in terms of the χ 0 meson in the ChPT.The anomaly matching for the CVE gives us the anomalous coupling of the χ 0 meson to the rotation [50,51] : Hereafter, we interpret that L CVE is a part of L WZW .
To derive an effective Lagrangian, we adopt a modification to the conventional power counting scheme in ChPT [56]: In this power counting, eq.(2.6) is of order O(p 2 ), consistent with eq.(2.2).The sole appearance of µ B in the WZW term in eq.(2.6) permits the assignment of a negative power counting to µ B .The effective field theory up to O(p 2 ) encompasses the terms in eq.(2.5); however, eq.(2.5) has been overlooked in prior studies of the CSLs under rotation.For discussions related to the magnetic field, we refer to our previous work [13,14].In the QCD vacuum, the effects of the QCD anomaly are generally not suppressed.Hence, we note that it is not feasible to incorporate the QCD anomaly's effects into ChPT (Of course, in the large-N c expansion, the effects of the QCD anomaly are of the order of 1/N c , allowing them to be treated perturbatively [74,75]).We underscore that an O(p 4 ) term, such as a Skyrme term, is unnecessary for our findings.
Our effective theory when ignoring the charged pions χ 1,2 and assuming one-dimensional dependence in the x 3 coordinate reduces to where the prime denotes a differentiation with respect to x 3 and we have introduced the following quantities, and dimensionless variables as follows: The third and fourth terms are the potential terms of the χ 0 and χ 3 : For later convenience, we introduce new fields defined as (2.17) In terms of χ + and χ − , the Hamiltonian is reduced as Eq. (2.13) gives the equation of motions as follows: Let us first consider the case of ϵ = 0 and β = 0.The potential term at β = 0 is sketched as fig. 1.The configuration connecting (0, 0) and (2π, 0) is well-known as a single sine-Gordon soliton, which has the transnational moduli, ζ 0 .On the other hand, the configuration connecting (0, 0) and (π, π) is given by which also has the translational moduli.We note that this soliton spontaneously breaks the SU(2) V symmetry to a U(1) subgroup: g = e iατ 3 . (2.25) Therefore, this soliton has not only translational moduli R but also SU(2)/U(1) ∼ = CP 1 ∼ = S 2 moduli.We call this soliton an up soliton.The configuration connecting (0, 0) and (π, −π) is given by the SU(2) V transformation: which is referred to as a down soliton.The up soliton and down soliton are connected by the CP 1 moduli.Next, let us consider the effects of non-zero ϵ and β.The term depending on ϵ is Since χ + (χ − ) has a peak at the center of the up (down) soliton, the energy density of eq.(2.27) becomes lower when the up soliton and down soliton are separated.Therefore, there is a repulsive (attractive) force between the up and down solitons due to the finite positive (negative) ϵ.As β increases from 0, the χ 3 dependence of the potential decreases.Therefore, the up and down soliton overlap to become the ordinary sine-Gordon soliton.The effect of finite β is the attractive interaction between the up and down solitons.
From the preceding discussion, we identify three distinct cases concerning the arrangement of the up and down solitons: the deconfined phase, the dimer phase, and the confined phase.
1. Deconfined Phase: If the repulsive force significantly exceeds the attractive force, the up and down solitons fully separate.This state is termed the deconfined phase.
For this condition, the relationship between the distance d between the up and down solitons and the distance ℓ between the same type of soliton is given by d = ℓ 2 .
2. Confined Phase: Conversely, when the attractive force strongly prevails over the repulsive force, the up and down solitons overlap entirely, denoted as d = 0.This case is referred to as the confined phase.
3. Dimer Phase: When the attractive and repulsive forces counteract each other equally, a molecular state of the up and down solitons forms, satisfying the condition When ϵ < 0 the inter-soliton force is attractive, so that the CSL is in the confined phase.Namely, the CSL is of the Abelian type with χ 3 = 0 and χ 0 is the same as that given in eq.(2.21).Then the Hamiltonian reads The tension (the mass per unit area) is given by integrating H over z as This becomes zero at S = 4/π, and therefore the Abelian CSL becomes the ground state for S ≥ 4/π.This is shown in fig. 2. When ϵ > 0, the inter-soliton force is repulsive and therefore the ground state is the non-Abelian CSL.There are no analytic solutions to eqs.(2.19) and (2.20), so that we numerically solve them and complete the phase diagram for β = 0 as shown in fig. 2. We emphasize that the critical velocity decreases for ϵ > 0.
3 Non-Abelian sine-Gordon soliton and its effective world-volume theory In this section, we construct the effective field theory of the U(2) non-Abelian sine-Gordon soliton under rotation by using the moduli (Manton) approximation [76][77][78].On the phase transition line between the vacuum and the deconfined phases, the single soliton enters the system alternately, and their distance is infinite.Then, we focus only on the up soliton: Of course, one can choose the down soliton, but since they are infinitely apart, it is sufficient to choose one.Considering a sufficiently small ϵ, we can approximate χ + ≃ 4 tan −1 e ζ .So far, we have neglected the charged pions.Its general solution containing the charged pions can be obtained from U 0 by an SU(2) V transformation, where g is an SU(2) matrix.Since g in eq.(3.2) is redundant with respect to a U(1) subgroup generated by τ 3 , it takes a value in a coset space, SU(2 Together with the translational modulus Z, the single sine-Gordon soliton has the moduli Such a soliton with non-Abelian moduli is called a non-Abelian sine-Gordon soliton [21,69].
Let us parameterize the CP 1 moduli by the homogeneous coordinates ϕ ∈ C 2 of CP 1 , satisfying [21] In terms of ϕ, eq.(3.2) is represented as where we define u ≡ e iχ + .Since CP 1 ≃ S 2 , the moduli space is also parameterized by the three-component real vector n with the unit length condition, |n| = 1.ϕ and n a are related by the following formula: The condition ϕ † ϕ = 1 is solved by using the inhomogeneous coordinate f ∈ C as follow: Then, the up soliton corresponds to n 3 = 1 (f = 0) and the down solitons to Now, we are prepared to formulate the low-energy effective theory for a single soliton using the moduli approximation [76][77][78].Consider a single sine-Gordon soliton perpendicular to the x 3 -coordinate.In what follows, the moduli parameter ϕ will be treated as fields on the soliton's 2 + 1-dimensional worldvolume.However, we will not do the same for the translational modulus Z as its transverse motion is not pertinent to our investigation.By substituting eq.(3.5) into L, we get Here, q is the baby Skyrmion (lump) charge density defined by The integration of which over the two-dimensional space defines the topological lump number In the last expression in eq.(3.11), P is a projection operator P 2 = P [79,80].The integrations of the above Lagrangian (3.8), (3.9) and (3.10) over the codimension x 3 give us the total effective world-volume theory of the non-Abelian sine-Gordon soliton: where α, β = 0, 1, 2 and we have used the integration formulas The equations in the second line represent the tension of the domain wall.The third and fourth lines in eq.(3.13) denote the CP 1 theory in the rotating coordinates and lump charge, respectively.The constant terms in eq.(3.13) are irrelevant in our study; thus we ignore them hereafter.The Lagrangian can be rewritten in terms of the inhomogeneous coordinate f as (3.17) In order to determine the ground state, let us calculate the momentum conjugate for ∂ 0 f and ∂ 0 f * : Then, the Hamiltonian can be calculated as follows: The last term proportional to Ω 2 is at higher order O(p 4 ) which we will omit in the following.

Domain-wall Skyrmion phase
In this section, we construct topological lumps in the domain-wall world-volume theory, and show that they correspond to Skyrmions in the bulk, implying the domain-wall Skyrmion phase.For this purpose, let us introduce the complex coordinate: Using these coordinates, we have When the Bogomol'nyi-Prasad-Semmerfield (BPS) equation for k > 0 [72] or the anti-BPS equation for k < 0 holds, the energy (4.2) saturates the minimum (the Bogomol'nyi bound) of the following inequality: with moduli a . Now, let us show the relation between the topological lumps in the domain-wall worldvolume theory and Skyrmions in the bulk.To this end, the baryon (Skyrmion) number B in the bulk taking a value in π 3 [SU(2)] ≃ Z can be calculated in terms of the Maurer-Cartan one-form with the lump number k defined in eq.(3.12).Therefore, we have found that k topological lumps on the non-Abelian sine-Gordon soliton carry a baryon number k and represent k Skyrmions in the bulk.This one-to-one correspondence between lumps on the soliton and Skyrmions in the bulk has a sharp contrast to the domain-wall Skyrmion in the strong magnetic field, in which case one lump in the domain wall corresponds to two Skyrmions in the bulk.Fig. 3 shows a three-dimensional configuration of a domain-wall Skyrmion.This can be compared with the case of that in strong magnetic field, in which a single lump has two peaks corresponding to two Skyrmions in the bulk.
Finally, let us evaluate the energy of the lump.Substituting eq.(4.6) into eq.(4.2), we obtain the energy of BPS k-lump configurations as5 Thus, when the chemical potential is larger than the critical value, the energy of lumps are negative, E DW ≤ 0 so that lumps are spontaneously created.This is the domain-wall Skyrmion phase for rotation.In the evaluation in eq.(4.9), we have used

Summary and discussion
We have shown the presence of a new phase of rapidly rotating QCD matter in high density region, that is a domain-wall Skyrmion phase.It was previously known based on the chiral Lagrangian with the CVE term [50] that the ground state is a CSL consisting of a stack of η-solitons for two flavors (η ′ -solitons for three flavors) in a high density region under rapid rotation [51].In a large parameter region, a single η-soliton decays into a pair of non-Abelian sine-Gordon solitons [21,69], each of which carries SU(2) V /U(1) ≃ CP 1 ≃ S 2 moduli as a consequence of the spontaneous breaking of the vector symmetry SU(2) V in the vicinity of each soliton [53].In such a non-Abelian CSL, we have shown that the effective world-volume theory of a single non-Abelian soliton is a d = 2 + 1 dimensional CP 1 model [O(3) model] with a topological term originated from the WZW term, eq.(3.13).We have shown that when the chemical potential is larger than a critical value in eq.(4.9), a lump has negative energy to be spontaneously created on the soliton world-volume, implying the domain-wall Skyrmion phase.This lump on the soliton world-volume corresponds to a due to the above topological term (see footnote 5).Then, the lumps would have constraint as the case of those for the strong magnetic field [13,14].While we have considered two flavors in this paper, more realistic case is three flavors including the strange quark.In this case, the chiral symmetry SU(3) L × SU(3) L is spontaneously broken to SU(3) V as well as the axial U(1) A symmetry, and the order parameter manifold is [SU(3) L × SU(3) R × U(1) A ]/[SU(3) V × Z 3 ] ≃ U(3).Then, a single U(3) non-Abelian soliton spontaneously breaks SU(3) V to its subgroup SU(2) × U(1), and thus there appear SU(3)/[SU(2) × U(1)] ≃ CP 2 NG modes in the vicinity of the soliton.Then, CP 2 lumps on the soliton corresponds to SU(3) Skyrmions in the bulk [53].Now, we make comments on the domain-wall Skyrmions in quark matter at large µ B .The ground state in the two-flavors case without rotation is the two-flavor superconducting (2SC) phase [82] in which the chiral symmetry is unbroken.Instead, for the three flavors, the ground state without rotation is the color-flavor locked (CFL) phase [83,84] (see ref. [82] as a review), in which the chiral symmetry SU(3) L × SU(3) R is spontaneously broken as well as U(1) A .As the case of three-flavor nuclear matter, an η ′ -CSL phase appears under the rapid rotation [51], in which a continuity to the three-flavor CSL in nuclear matter was also studied.The instanton effect is suppressed due to the Debye screening [85,86] in the large µ B region, implying small β so that non-Abelian CSL is favored.Then, domainwall Skyrmions can be constructed as CP 2 lumps on a U(3) non-Abelian soliton.It was discussed in ref. [87] that Skyrmions in the CFL phase can be regarded as quarks instead of baryons and are called qualitons.Thus, quarks condensed outside the non-Abelian solitons which is in the CFL phase may not be condensed inside the non-Abelian solitons, similar to Andreev bound states in superconductors.
Apart from application to QCD, there are also interesting points for physics of topological solitons.For instance, more general non-BPS solutions of the CP N −1 model can be constructed by the Din and Zakrzewski's projection method [79,80,88].One of questions is what these correspond to in the bulk.The other is fractional CP N −1 lumps in a twisted boundary condition [89,90].What is the meaning of fractional baryons in the bulk perspective?
t e x i t s h a 1 _ b a s e 6 4 = " t 3 I f g l r e j / p 1 1 n X M X q b e 4 b T S I 8 w i J c J / j E H H S V i m L U 4 Z O 7 D 6 N J V p t R q x N 6 6 C m H 6 p N O s W i 7 p F y B G P s g d 2 w F 3 b P b t k T e / + 1 V i O s E X i p 0 2 w 0 t d w t 9 h + l V t / + V V V o l i h / q v 7 0 L L G L + d C r I O 9 u y A S 3 M J v 6 2 s H J y + p C b q w x z i 7 Z M / m / Y I / s j m 5 g 1 1 7 N q y z P n S N O H 6 B 9 f + 6 f o D C V 1 m b T M 9 l p d X E p + o p O D G M U E / T e c 1 j E C j L I 0 7 l 7 O M Y p z m L P S l J J K U P N V C U W a Q b x J R T 1 A / 5 9 i 9 Y = < / l a t e x i t > + < l a t e x i t s h a 1 _ b a s e 6 4 = " q 2

Figure 2 .
Figure 2. The schemetic picture of the three phases (left) and the phase diagram for β = 0 (right).(Left panel) The deconfined, dimer and confined phases in which up and down solitons repel, form a molecule, and are completely overlapped, respectively.(Right panel) In this parameter choice, the dimer phase does not appear, and the whole non-Abelian CSL implies the deconfined phase.The gree line correspond to the noninteractive case in which the up and down solitons do not interact each other and form lattices independently.

. 5 )
Due to the second term, anti-BPS lumps have more energy than BPS lumps.Let us consider BPS k-lump solution[72]