Non-Abelian Sine-Gordon Solitons

We point out that non-Abelian sine-Gordon solitons stably exist in the $U(N)$ chiral Lagrangian. They also exist in a $U(N)$ gauge theory with two $N$ by $N$ complex scalar fields coupled to each other. One non-Abelian sine-Gordon soliton can terminate on one non-Abelian global vortex. They are relevant in chiral Lagrangian of QCD or in color-flavor locked phase of high density QCD, where the anomaly is suppressed at asymptotically high temperature or density, respectively.

Sine-Gordon kinks also explain relations between topological defects or solitons in different dimensions. Since-Gordon kinks inside the world-volume of a topological defect represent some other topological defects in the bulk; Sine-Gordon kinks inside a domain wall are vortices, lumps or baby Skyrmions in the bulk [16,[22][23][24], which explains a relation between sine-Gordon kinks and CP 1 instantons [25,26]. Sine-Gordon kinks inside a domain wall ring are baby Skyrmions [23]. They represent Skyrmions in the bulk if residing in a domain wall within a domain wall [27][28][29] or in a vortex string [30,31], they are Hopfions in the bulk if residing in a toroidal domain wall [32], and are Yang-Mills instantons in the bulk if residing inside a monopole string in Yang-Mills theory in d = 4 + 1 dimensions [33].
There have been many proposal of generalizations of the sine-Gordon model. One of such is a complex sine-Gordon model describing a vortex motion in superfluids [34], the O(4) model [35], conformal field theories [36], and a domain wall junction [37]. There have been non-Abelian generalizations such as the matrix sine-Gordon model [38], the symmetric space sine-Gordon model [39] and so on.
In this paper, we discuss yet another non-Abelian generalization of sine-Gordon kinks.
We point out that the U(N) chiral Lagrangian admits a non-Abelian sine-Gordon kink and that it carries non-Abelian moduli CP N −1 ≃ SU(N)/[SU(N − 1) × U(1)]. Here, the term "non-Abelian" is used in the same way with that of non-Abelian vortices [40][41][42][43] carrying non-Abelian CP N −1 moduli, see Refs. [44][45][46] for a review. As in the same manner with a non-Abelian vortex with non-Abelian moduli which can terminate on a non-Abelian monopole because of the matching of the moduli CP N −1 [47,48], non-Abelian sine-Gordon kink here can terminate on a non-Abelian global vortex [49][50][51][52], see Ref. [4] as a review. We then promote the non-Abelian sine-Gordon solitons to those in non-Abelian U(N) gauge theories with two N by N complex scalar fields coupled to each other by a non-Abelian extension of linear or quadratic Josephson interaction. The Abelian case reduces to phase solitons in two-gap superconductors [8][9][10], while the non-Abelian extension is relevant to a color superconductor of the color-flavor locking phase of dense QCD matter [4,53]. This paper is organized as follows. In Sec. II, after reviewing sine-Gordon kinks in the conventional sine-Gordon model, we discuss non-Abelian sine-Gordon kinks in the U(N) chiral Lagrangian. In Sec. III, sine-Gordon kinks with a modified mass term and their non-Abelian U(N) generalization are discussed. In Sec. IV, these sine-Gordon kinks are promoted to gauge theories. The U(1) gauge theory is nothing but two-gap superconductors or chiral p-wave superconductors corresponding to the conventional or modified mass term, respectively. In Sec. V, we discuss that a sine-Gordon kink can terminate on a non-Abelian global vortex. Sec. VI is devoted to summary and discussion.

A. The sine-Gordon model
The sine-Gordon kink is characterized by the first homotopy group π 1 [U(1)] ≃ Z. The Lagrangian density of conventional sine-Gordon model is with µ = 0, 1, · · · , d − 1 and 0 ≤ θ < 2π. We consider static configurations depending on one spatial direction x. The static energy density is B. Non-Abelian sine-Gordon model as chiral Lagrangian Here we consider the U(N) group: with the first homotopy group is nontrivial: The Lagrangian for a U(N) principal chiral model (chiral Lagrangian) for a U(N)-valued field U(x) is given by This Lagrangian is invariant under the chiral SU(N) L × SU(N) R symmetry The Lagrangian admits the unique vacuum U = 1 N . The chiral symmetry is spontaneously broken to the vector-like symmetry The energy density for static configuration and its Bogomol'nyi completion are given as with the topological charge, defined by The BPS equation is obtained as This equation is invariant under the SU(N) symmetry in Eq. (16).

Let us construct solutions to this equation. The simplest ansatz is given by the following
Abelian solution By substituting this ansatz into Eq. (19), we find that u(x) again satisfies Eq. (9). The tension (energy per unit area) of this configuration is T = NT SG .
Next, we construct non-Abelian solutions. Let us consider the following ansatz: By substituting this ansatz into Eq. (19), we find that u(x) satisfies Eq. (9) From this expression, one can see that the U(1) group element rotates only 2π/N while the rest is compensated by an SU(N) group element T 0 . Namely at x = ∞ (θ = 2π) the U(1) group element becomes exp i 2π Since there exists a redundancy for the action of V , V in fact takes a value in the coset Therefore, the one-kink solution has the moduli In terms of the group elements, the general solution can be rewritten as Let us introduce the orientational vector φ ∈ C N with a constraint which represents homogeneous coordinates of CP N −1 . The generator T and the general solution in Eq. (26) can be rewritten by using the orientational vector as

III. THE MODIFIED SINE-GORDON MODEL AND CHIRAL LAGRANGIAN
A. The modified sine-Gordon model We consider the Lagrangian density of a sine-Gordon model with an unconventional potential, given by with µ = 0, 1. This model admits two vacua θ = 0, π in the defined range 0 ≤ θ ≤ 2π. We concentrate on static configurations. The static energy density is The Bogomol'nyi completion for the energy density is obtained as with the topological charge density The inequality is saturated by the BPS equation A one-kink solution interpolating between θ = 0 at x → −∞ to θ = π at x → +∞ can be given as with the position X in the x-coordinate and the width 1/m. The topological charge for this solution is In terms of u(x) = e iθ(x) , the BPS equation is rewritten as and the topological charge density is rewritten as The one-kink solution is B. Non-Abelian sine-Gordon model as chiral Lagrangian with modified mass The Lagrangian for U(N) principal chiral model with a modified mass is This model admits two vacua U = ±1 N . The energy density for static configuration and its Bogomol'nyi completion are given as with the topological charge, defined by The BPS equation is obtained as As in the same manner, the Abelian kink in Eq. (39) can be embedded into a conner as in Eq. (21) to obtain a non-Abelian kink. Also, it allows the CP N −1 moduli as Eq. (23).

IV. NON-ABELIAN SINE-GORDON SOLITON IN GAUGE THEORIES
A. Abelian gauge theory: two-gap superconductors and chiral p-wave superconductors Let us consider a U(1) gauge theory coupled with two complex scalar fields φ i (x) (i = 1, 2), given by with D µ φ i = (∂ µ − iA µ )φ i . L J is a Josephson term either linear or quadratic: The gauge transformation is defined by while a U(1) global transformation is explicitly broken by γ = 0.
Let us take strong coupling limit (with keeping γ finite): giving constraints With taking a gauge A µ = ∂ µ θ 2 and defining the phase difference θ(x) ≡ θ 1 (x) − θ 2 (x), the covariant derivative terms in Lagrangian in Eq. (45) become while the Josephson terms in Eq. (46) become The gauge theory Lagrangian in Eq.
instead of the first term in the Lagrangian in Eq. (45). Here a = 1, 2, (3) is a spatial index.
The linear Josephson term L J,1 in Eq. (46) is relevant for the Landau-Ginzburg description of two-gap superconductors such as MgB 2 , in which the term proportional to γ is called the (internal) Josephson coupling and θ(x) is called the Leggett mode. The sine-Gordon soliton is called the phase soliton in this context, which was first pointed out theoretically [8] and was found experimentally [9]. It is also relevant for a Josephson junction of two superconductors. On the other hand, the case with the quadratic Josephson interaction L J,2 in Eq. (46) is relevant for chiral p-wave superconductors [11], such as Sr 2 RuO 4 .
A non-relativistic version of the Lagrangian (53) in which overall U(1) is not gauged (e = 0) yields the Gross-Pitaevskii equation for two-component Bose-Einstein condensates of ultracold atmoc gasses such as Rb 87 , in which the term proportional to γ is called a Rabi oscillation term. (In addition, the term g 12 |φ 1 | 2 |φ 2 | 2 is also present but it is not important for the phase solitons.) The sine-Gordon (phase) soliton in this case was studied in Ref. [12].

B. Non-Abelian gauge theory
Let us consider a U(N) gauge theory coupled with two N × N matrix-valued complex scalar fields Φ i (x) (i = 1, 2), whose Lagrangian is given by L J is a non-Abelian Josephson term either linear or quadratic: The U(N) V gauge transformation is defined by is explicitly broken by γ = 0.
Let us take strong coupling limit (with keeping γ finite): giving constraints These constraints can be solved as With taking a gauge A µ = iÛ † ∂ µÛ and defining U(x) ≡Û 2 (x), the covariant derivative terms in Lagrangian in Eq. (54) become and the Josephson terms reduce to γ ≡ m 2 .

The relativistic Lagrangian in Eq. (54) is relevant for a linear model description of chiral
Lagrangian using a hidden local gauge symmetry for which gauge bosons of U(N) gauge symmetry is vector mesons of the hidden local symmetry, see, e. g. Ref. [54].
A non-relativistic version of the Lagrangian has the kinetic and gradient terms instead of the first term in the Lagrangian in Eq. (54) The non-relativistic case with N = 3 with ungauged U(1) is relevant for the Landau-Ginzburg description of the color-flavor locking phase (a color superconductor) for high density QCD [4,53]. In this case, (Φ 1 ) αi = ǫ αβγ ǫ ijk q L jβ q L kγ and (Φ 2 ) αi = ǫ αβγ ǫ ijk q R jβ q R kγ are diquark condensates of left and right handed quarks q L jβ and q R jβ , respectively, where α, β, γ = 1, 2, 3 and i, j, k = 1, 2, 3 are color and flavor indices, respectively.
Here, we have considered the potential for the U(1) symmetry induced from quark mass in chiral Lagrangian in QCD. On the other hand, there is another potential term V ∼ det Φ 1 + det Φ 2 induced from the U(1) A anomaly at quantum level. The non-Abelian sine-Gordon kink should be deformed by this potential accordingly [55]. Therefore, in real QCD, our solutions are relevant in asymptotically high temperature or high density, in which the U(1) A anomaly disappears. Let (r, ϕ, z) be cylindrical coordinates of space. Then, the asymptotic form of a non-Abelian global vortex can be written as In the limit of no mass term (m = 0), the unit winding solution is simply given by θ = ϕ so that the vortex is axisymmetric. The configuration in Eq. (64) can be rewritten as It is obvious that the configuration of the vortex breaks the SU(N) V symmetry of the vacuum to a subgroup SU(N − 1) × U(1) so that there appear moduli CP N −1 , although these moduli are non-normalizable [50,51].
In the presence of the mass term (m = 0), the global vortex configuration is deformed and is no more axisymmetric. In this case, the potential term appears for the field θ(ϕ) in the vortex ansatz in Eq. (64). This is of course the sine-Gordon potential discussed in the previous sections. Only the difference is the argument of θ is θ(ϕ) here and θ(x) before. The final configuration is a non-Abelian vortex attached by a non-Abelian sine-Gordon kink, as schematically drawn in Fig. 1. Both the non-Abelian vortex and non-Abelian sine-Gordon kink have the CP N −1 moduli, and consequently they match at a junction line [60]. This fact implies the instability of sine-Gordon kinks in the U(N) linear sigma model as in the same manner with an axion string [5]. In d = 2 + 1, the sine-Gordon domain line can terminate on a global non-Abelian vortex [4,[49][50][51][52]. The domain line can decay by creating a pair of a non-Abelian vortex and a non-Abelian anti-vortex, as shown in Fig. 2 (a). In d = 3 + 1, the non-Abelian sine-Gordon domain wall can decay by creating a hole bound by a closed non-Abelian vortex string, as illustrated in Fig. 2 (b). This process can occur either thermally or by quantum tunneling. More details will be discussed elsewhere.
However, note that the instability does not exist in the nonlinear model, the U(N) chiral Lagrangian. This is the same situation with an axion string [5].

VI. SUMMARY AND DISCUSSION
We have pointed out that the U(N) chiral Lagrangian admits a non-Abelian sine-Gordon constructing the low-energy effective theory by the moduli approximation [56], which is the CP N −1 model. One then can construct CP N −1 lumps on it that would represent U(N) Skyrmions as was so for N = 2 [27,28].
The interaction between two kinks located at x = X 1,2 with the orientations φ 1,2 can be considered. Like the Abrikosov-type ansatz for vortices, we can give an ansatz for the total configuration as U tot (x) = U 1 (x−X 1 , φ 1 )U 2 (x−X 2 , φ 2 ) for well-separated kinks |X 1 −X 2 | >> m −1 . In particular, an Abelian sine-Gordon kink would be separated into N non-Abelian kinks without cost of energy, which can be expected from the fact that an Abelian kink has energy N multiple of those of non-Abelian kinks. A similar calculation was done for the force between two non-Abelian global vortices [4,51].
In two-gap superconductors, a unit winding vortex can be split into two fractional vortices winding around different components, which are connected by a sine-Gordon kink [10,17,18].
The same happens for coherently coupled multi-component BECs [19][20][21]. In the same way, a local non-Abelian vortex can be split into a set of two global non-Abelian vortices connected by a non-Abelian sine-Gordon domain wall discussed here. In the case of the color-flavor locked phase of dense quark matter, a non-Abelian vortex [4,57] has 1/3 fractional U(1) winding in both Φ 1 and Φ 2 , but it may be decomposed into a global vortex with 1/6 U(1) winding (1/3 U(1) winding in only one of Φ 1 and Φ 2 ). This will be also discussed elsewhere.
Non-Abelian U(N) Sine-Gordon kinks can be extended to the case of arbitrary gauge groups G in the form of G×U (1) Zr with the center Z r of G, since non-Abelian vortices with this type of gauge groups were studied before [58], such as SO(N) and USp(2N) groups [59].
Finally, the sine-Gordon model is integrable. Therefore, we expect the non-Abelian sine-Gordon model presented here is also integrable.