Self-consistent Analytic Solutions in Twisted $\mathbb{C}P^{N-1}$ Model in the Large-$N$ Limit

We construct self-consistent analytic solutions in the ${\mathbb C}P^{N-1}$ model in the large-$N$ limit, in which more than one Higgs scalar component take values inside a single or multiple soliton on an infinite space or on a ring, or around boundaries of a finite interval.


Introduction
Nonlinear sigma models in two spacetime dimensions share a lot of non-perturbative properties with Yang-Mills theories or QCD in four dimensions, such as asymptotic freedom, dynamical mass gap, dynamical chiral symmetry breaking and instantons, and therefore the former is regarded as a toy model of the latter [1][2][3][4][5][6][7][8][9][10]. In particular, the CP N −1 model [1][2][3] corresponding to the SU(N) gauge theory has been studied extensively, such as the supersymmetric CP N −1 model [11,12] for which the exact Gell-Mann-Low function was obtained [13] and dynamical mass gap was proved by the mirror symmetry [14]. The low-energy dynamics of a non-Abelian vortex string in a U(N) gauge theory in four dimensions can be described by the CP N −1 model defined on a two-dimensional worldsheet [15][16][17] (see Refs. [18][19][20][21] as a review), giving a more precise correspondence between the CP N −1 model in two dimensions and the U(N) gauge theory in four dimensions [22,23]. 1 The CP N −1 model on a compact direction with twisted boundary conditions have been also studied extensively. In this case, an instanton is decomposed into several fractional instantons [26,27]. Recently, bions which are composite states of fractional instanton and anti-instantons have been studied in the CP N −1 model for the application to the resurgence theory [28,29]. The CP N −1 model at finite density has been also studied [30]. Recently, there is a growing interest on the CP N −1 model defined on a finite region such as a ring [31,32], a finite interval [33][34][35][36], and a disk [37,38]. In particular, the case of a finite interval may correspond to a non-Abelian vortex string stretched between two heavy non-Abelian monopoles [39,40] or heavy non-Abelian monopole and antimonopole [41].
Since the CP N −1 model was defined forty years ago, only constant solutions have been studied for long time except for few cases: the model on a finite interval studied recently around whose boundaries a Higgs scalar takes non-zero values [34][35][36]. Recently, in Ref. [42], a class of self-consistent analytic solutions to gap equations has been obtained in the CP N −1 model in the large-N limit in infinite space, by constructing a map from the gap equations in the CP N −1 model to those in the Gross-Neveu (GN) [43] or Nambu-Jona-Lasino [44] model, or equivalently to the Bogoliubov-de Gennes (BdG) equation. The self-consistent analytic solutions that have been found include inhomogeneous Higgs configurations, such as a soliton in which a Higgs scalar field is localized, a lattice of such the soliton and multiple solitons with arbitrary separations, constructed from a real kink [45,46], a real kink crystal [47,48], and multiple kink-anti-kink configurations [49][50][51][52][53], respectively in the GN model. The appearance of the nonzero Higgs field, and consequently Nambu-Goldstone modes, are consistent with the Coleman-Mermin-Wagner theorem [54,55] prohibiting Nambu-Goldstone modes in two spacetime dimensions, because the Higgs field is confined in finite regions in our cases. The integrable structure [56] behind the GN model [57,58] and the map give infinite species of self-consistent analytic solutions. The self-consistent analytic solutions for the case of a finite interval have been also obtained [36] from the corresponding solutions in the GN model [59]. The GN model on a ring was also studied numerically [60] but the map to the CP N −1 model was not applied to this case. However, all the previous works include only solutions in which only one Higgs field component takes a value and so are essentially solutions in the CP 1 model.
In this paper, we construct self-consistent analytic solutions in which more than one Higgs scalar component take nonzero values. This is possible when we impose twisted boundary conditions or equivalently introduce a Wilson line for global flavor symmetry along spatial direction. We first construct a single soliton and multiple solitons on an infinite space. Around solitons, the Higgs phase appears with nonzero Higgs scalar components. We then construct a soliton on a ring. Finally, we construct the Higgs and confining phases in a finite interval. For both cases, the Higgs field diverges around the boundaries. For the former the Higgs fields are nonzero everywhere, while for the latter the Higgs fields are nonzero almost everywhere except for the center of the interval. All of them are genuine solutions in the CP N −1 model with N > 2. These solutions correspond to a non-Abelian vortex string stretched between non-Abelian monopole and anti-monopole whose orientational CP N −1 modes are not aligned.
This paper is organized as follows. In Sec. 2 we define the CP N −1 model and give the gap equations in the large-N limit and explain our method. In Sec. 3 we give several examples of self-consistent analytic solutions to the gap equation. Sec. 4 is devoted to summary and discussion.

Model and method
In the following, we consider the CP N −1 model whose Lagrangian is given by where n i (i = 1, · · · , N) are N complex scalar fields, D 0 µ = ∂ µ − iA 0 µ , and A 0 µ and λ are auxiliary gauge and scalar fields, respectively. We set A 0 µ = 0 in the following. We impose a twisted boundary condition which generates a twist of the flavor degrees of freedom. By picking up the first m components, we consider a boundary condition under which n 1 , · · · , n m are twisted. The twisted boundary condition is equivalent to the presence of a Wilson line for a background SU(m) non-dynamical gauge field. Then, the Lagrangian can be rewritten as where we have defined the m-component vector Σ = (n 1 , n 2 , · · · , n m ) T and the covariant derivative D µ = ∂ µ −iA µ with the background non-dynamical gauge potential A µ which is an m × m matrix and A 0 = 0. The stationary condition for λ and the Σ † together with the Lorenz gauge ∂ µ A µ = 0 yields the gap equations By using the redefinition of the field Σ = exp(i x dxA)Σ, one obtains From now on, we focus on the case of m = 2 for simplicity, though the following argument is straightforwardly applicable to arbitrary m. In the case of m = 2, one can further rewrite the model as where σ =ñ 1 + iñ 2 or (ñ 1 ,ñ 2 ) = (ℜσ, ℑσ) vice versa. Now the gap equations to solve become In order to ensure the real eigenvalues for ω 2 n , we consider λ to be real such that −∂ 2 x +λ to be Hermitian. Eq. (2.13) describes a zero mode solution for the Schrödinger equation (−∂ 2 x + λ)u = Eu. By rewriting the λ as one can solve Eq. (2.13) as We note that the other linearly independent solution is not normalizable. The remaining two equations (2.11) and (2.12) can be solved by the mapping given in the previous work (See Appendix A).

Self-consistent analytic solutions on various spaces
In this section we construct self-consistent analytic solutions on an infinite space, a ring, and a finite interval in each subsection.

Infinite system
We first consider the infinite size system. Some solutions for Eqs. (2.11)-(2.13) are given by [42] Here we note that a localized Higgs soliton Ae iφ cosh −1 mx or a localized Higgs soliton lattice which we describe below are not inhibited from the Coleman-Mermin-Wagner theorem since these solutions do not have a long range order.
The CP 1 modes are given by As an example, if we choose A = ασ x , we obtain For the case of A = βσ y , we obtain Without loss of generality, we choose A = βσ y and ℜσ = m, ℑσ = 0, yielding In the case of A = γσ z , one obtains In Fig. 1, we plot the Higgs lattice solution given by for A = βσ y which corresponds to ∆ = sn(x, ν). Here sn and dn are the Jacobi's elliptic functions and ν is the elliptic parameter. In Fig. 2, we plot a doubly localized Higgs soliton solution corresponding to where ω b = √ m 2 − k 2 , R = (1/2) ln(m + /m − ), and m ± = m ± k. The left and right panels represent the twisted two soliton solution with k = 0.9 (left) and k = 0.999 (right), respectively. The other parameters are set to be δ = 0 and m = 1. By making the separation larger, the relative twist of the two peaks becomes larger.

Ring system
Next we consider the case of the CP N −1 model on a ring with the circumference L. In this case, the twisted boundary condition can be generated by the Aharonov-Bohm(AB)-like effect for the flavor degrees of freedom. We consider the periodic condition for all {n i }. By the singular gauge transformation (Σ T , n 3 , · · · , n N ) = (e i x dyAΣT , n 3 , · · · , n N ),  whereas the background gauge field is completely eliminated from the self-consistent equations. Here β is 2 × 2 Hermitian matrix. If the gauge field is chosen to compensates the twisting of the boundary, one can use the solutions obtained for the non twisting boundary conditions. In the case of the SU(2) generator, we have exp(iασ i x) = cos αx + iσ i sin αx. In the case of A(y) = ασ i , the above twisted boundary condition becomes (cos αL + iσ i sin αL)Σ(x + L) = e iβΣ (x).
This shows the periodic structure on α which is similar to the AB oscillation effect. Because of this periodicity, α has 2π ambiguity. The lowest energy state corresponds to the case of −π ≤ α < π and the solutions with π ≤ |α| corresponds to the higher energy states. This can be easily shown as follows. If we move to the non-twisted boundary problem with the gauge potential, all of those solutions corresponds to the homogeneous solution. Thus the energy difference comes only from the gauge field part given by ∝ A 2 and one can show the solution (a) has the lowest energy.
Apart from the phase winding induced by the boundary twisting, we have another condition forΣΣ (x + L) †Σ (x + L) =Σ(x) †Σ (x), (3.13) since e iασ i L is an unitary matrix. This means that the amplitude of the Higgs field also need to be periodic. In Fig. 3 we plot a homogeneous solution with the flavor rotation which corresponds to σ = me iφ . This solution is not allowed in the infinite system since the gap equation is not satisfied due to the absence of the infrared cutoff played by 1/L in the ring case. The absence of the long-range-ordered solution in the infinite system is consistent with the Coleman-Mermin-Wagner theorem.
This solution obviously satisfies the condition (3.13). The Fig. 3 (a) shows the smallest winding solution. The higher winding states are also shown in (b) and (c) for the same boundary condition. The physical meaning of those solutions can be understood from the schematic figure (d). Because the 2π periodicity of αL, we have infinite branches of the solutions which gives the same twisting of the boundary. For example, the solution (b) is the solution with the second smallest phase winding in which the flavor rotates opposite way compared with the solution (a). The solution (b) can be interpreted as the case of αL = π/2−2π which means that the solution (b) belongs to the neighbor branch to the branch for (a). In the same way, the solution (c) is understood as the case of αL = π/2 + 2π.

Finite interval system
Finally we consider the case of the CP N −1 model on a finite interval which is relevant to describe the CP N −1 modes on a vortex string connecting heavy monopole and (anti-)monopole in the U(N) gauge theory. If we have the flavor twisting between the opposite edges of the string, the solution (3.2), which matches the twisted boundary conditions, becomes a solution. The other important point for this boundary condition is that the Higgs field Σ unavoidably diverges at the edges.
One of self-consistent analytical solutions of this problem can be given by where we considered the twisted boundary condition on Σ such as Here γ is a constant parameter which characterizes the twisting of the boundary condition, ν is the elliptic parameter, and K(ν) is the complete elliptic integral of the first kind. We also have another parameter α which could be set to α = 0 by a field redefinition since it corresponds to the overall U(1) factor of the n 1 + in 2 . Thus, we obtain This vanishes in the infinite size limit (L → ∞, ν → 1). In the same limit, the mass gap function λ becomes constant and thus this solution corresponds to the confining phase in the large size limit. We plot the confining phase solutions in Fig. 4. There is another solution which corresponds to the Higgs phase in the infinite size limit where the Higgs field becomes the plane-wave like solution Σ(x) = const · exp(iσ iγ x) · (1, 0) T : We plot the Higgs phase solutions in Fig. 5. This solution is inhibited in the infinite system since the gap equation is no longer satisfied in the limit. In other words, this solution is possible only in a finite system, to be consistent with the Coleman-Mermin-Wagner theorem.   Figure 5. The winding solution for the Dirichlet boundary condition corresponding to the Higgs phase. The left panel is an untwisted solution constructed in Ref. [36] while the right panel corresponds to the case with a twist of γ = π/3. For the both cases we set ν = 0. The Higgs field does not vanish anywhere in contrast to the case of the confining phase.

Summary and discussion
In the paper, we have constructed self-consistent analytic solutions of the CP N −1 model in the large-N limit with the twisted boundary condition or equivalently with the background SU(m) (m < N) gauge field in the flavor space. The resulting solutions describe the various Higgs configurations with the SU(m) flavor rotation.
In the present analysis, we have assumed that the flavor rotation is uniform since a nonuniform rotation costs more energy. However, nonuniform backgrounds could appear in a specific setup, which remains as a future problem.
In this paper, we have used the mapping from the Gross-Neveu model to the CP N −1 model. It might be possible to generalize this mapping to the case of the chiral Gross-Neveu model, where the complex kink solution [61], complex kink crystal solution [62], and the complex kink with arbitrary separation [63] have been obtained.
Our solutions could also be applicable to the condensed matter physics, for example, to the magnetic order [64][65][66][67].
In the case of a single component at a finite interval with the Dirichlet boundary condition, the Casimir force depending on the size of the system was discussed before [35,36], where the Casimir force gives either attractive or repulsive pressure to the system size. For the case of the twisted boundary conditions studied in this paper, one can further discuss a Casimir force acting on the flavor internal space of CP N −1 . In this case, the force gives either attraction or repulsion between the CP N −1 modes on the boundaries. In the context of a non-Abelian string stretched between a monopole and (anti-)monopole, this force attains ferromagnetic or antiferromagnetic properties, respectively, on the monopole and (anti-)monopole.
The twisted boundary condition in the temporal direction has also been investigated. The large-N volume independence and the absence of the Affleck transition have been shown in the setup [68]. Our formalism and inhomogeneous solutions may possibly be used also in this case.
During completion of this paper, we were informed that the authors of Ref. [35] were writing a paper on similar configurations. In this appendix, we review the mapping between the GN model and the CP N −1 model [42]. We consider the gap equations (2.11)-(2.13) which have to be solved self-consistently. By using ∆ defined in Eq. (2.14), the Klein-Gordon-like equation