$\mathbb{Z}_n$ modified XY and Goldstone models and vortex confinement transition

The modified XY model is a modification of the XY model by addition of a half-periodic term. The modified Goldstone model is a regular and continuum version of the modified XY model. The former admits a vortex molecule, that is, two half-quantized vortices connected by a domain wall, as a regular topological soliton solution to the equation of motion while the latter admits it as a singular configuration. Here we define the ${\mathbb Z}_n$ modified XY and Goldstone models as the $n=2$ case to be the modified XY and Goldstone models, respectively. We exhaust all stable and metastalble vortex solutions for $n=2,3$ and find a vortex confinement transition from an integer vortex to a vortex molecule of $n$ $1/n$-quantized vortices, depending on the ratio between the term of the XY model and the modified term. We find for the case of $n=3$, a rod-shaped molecule is the most stable while a Y-shaped molecule is metastable. We also construct some solutions for the case of $n=4$.The vortex confinement transition can be understood in terms of the ${\mathbb C}/{\mathbb Z}_n$ orbifold geometry.


I. INTRODUCTION
The XY model is a lattice model describing a lot of physical systems such as superconductors and superfluids. Its Hamiltonian is given by H XY = −J i,j cos(ϑ i − ϑ j ), where i, j label the lattice cites and i, j implies a pair of nearest neighbours. In the continuum limit, it becomes just a free U(1) scalar field theory or nonlinear O(2) model. One of the most nontrivial features of this model is to exhibit a topological phase transition called the Berezinskii-Kosterlitz-Thouless (BKT) transition [1][2][3][4] in 2+1 dimensions, which separates bound vortices at low temperature and liberated pairs of vortex and anti-vortex at high temperature. The BKT transition yields quasi-long-range order with algebraically decaying correlations, although long-range order with continuous symmetry is forbidden by the Coleman-Mermin-Wagner (CMW) theorem [5][6][7]. The BKT transition has been confirmed experimentally in various condensed matter systems such as 4 He films [8], thin superconductors [9][10][11][12][13], Josephson-junction arrays [14,15], colloidal crystals [16][17][18][19], and ultracold atomic Bose gases [20]. One of drawbacks of the XY model may be the fact that vortices are singular configurations but not solutions to the equation of motion. To overcome this problem, one can introduce a Higgs (amplitude) degree of freedom together with a potential term along the Higgs direction, then the model becomes the Goldstone or linear O(2) model, allowing vortices as regular solutions to the equation of motion. The vortex core singularity is resolved by the Higgs field while the large distance behavior can be capture by the XY model. The modified XY model is a modification of the XY model by addition of a half-periodic term [21][22][23][24]: where the second term is the half-periodic term. This model admits a vortex molecule, that is, two half-quantized vortices connected by a domain wall, as a singular configuration, and its existence is crucial in the phase diagram, as is so for the XY model. When the coupling J of the modified term is large enough compared with the coupling J, there exists an Ising type phase transition [22][23][24] as a consequence of the presence of domain walls. The modified model in Eq.(1) and its various modifications [25][26][27][28][29][30][31][32][33] are of great importance and interest, because of applicability to various systems such as superfluidity in atomic Bose gases [34], arrays of unconventional Josephson junctions [35], or high temperature superconductivity [36]. The modified Goldstone or modified linear O(2) model is a regular (complemented by the Higgs mode) and continuum version of the modified XY model [37]: where φ = exp r(iϑ) is a complex scalar field (r is the Higgs field), and λ, a and b are positive coupling constants determined from the lattice model. This model admits a vortex molecule of half-quantized vortices connected by a domain wall as a regular topological soliton solution to the equation of motion when b is large enough [37], while for small b the molecule collapses to an integer vortex. The phase diagram is quite rich and there is a two-step phase transition of BKT type and of Ising type [37].
In this paper, as a generalization of the modified XY and Goldstone models, we define the Z n modified XY and Goldstone models: and respectively. The case of n = 2 corresponds to the usual modified XY and Goldstone models.
We study vortex solutions in this model with particular attension to the cases of n = 2, 3. We exhaust stable and metastable vortex solutions for these cases, and find a vortex confinement transition from an integer vortex to a vortex molecule, depending on the ratio between a and b. We find for the case of n = 3 that a rod-shaped molecule is the most stable while a Y-shaped molecule is metastable. We also give some examples of (meta)stable vortices in the case of n = 4. This transition can be understood in terms of the C/Z n orbifold geometry; the model can be written in the form of a nonlinear sigma model with the target space C/Z n with a possible orbifold singularity resolved. If vacua are at large distance from the origin in the target space, a vortex becomes a molecule of 1/n fractional vortices, while if the vacua are close to the origin the vortex becomes an integer vortex.
This paper is organized as follows. In Sec. II, we introduce our model and discuss geometry. In Sec. III, we construct vortex solutions. Section IV is devoted to a summary and discussion.

II. THE MODEL AND GEOMETRY
In this section, we formulate our model and discuss geometric properties. The Lagrangian of the Z n modified Goldstone model is given by The vacua are S 1 defined by |φ| 2 = v 2 . This model is just a nonlinear sigma model with the target space metric In the limit of λ → ∞, the model reduces to an O(2) nonlinear sigma model (or the XY It is sometimes useful to rewrite the Lagrangian by a new field Φ = φ n as Let us discuss the asymptotic behaviour of the target space geometry. Writing φ = re iθ or Φ = Re iΘ (R = r n and Θ = nθ), the geometry behaves two different ways separated by the critical radius r = r c defined by Then, we can see that the metric behaves differently at large and short distances as follows: • For the large distance, r r c (R R c ), the first term in the metric -in Eq. (6) is negligible and the Lagrangian reduces to One observes that Φ is a good coordinate rather than φ. The kinetic term of the Lagrangian in Eq. (9) is just a free scalar field in terms of Φ, but the target space is rather an orbifold: This is because all φω a with a = 0, 1, 2, · · · n − 1 yield the same Φ, where ω n = 1, ω = exp(2πi/n). This metric has an orbifold singularity in the origin, but it is not the case for the whole metric.
• In fact, the short distance behaviour (r r c (R R c )) is dominated by the first term in the metric, and the Lagrangian reduces to The Lagrangian is nothing but the usual Goldstone model in terms of φ. In this case, φ is a good coordinate in which the metric is smooth at the origin φ = 0. Therefore, we have seen that a possible singularity in the orbifold C/Z n is resolved in the full metric, and the whole target space is smooth.

III. VORTICES
Comparing the vacua r = v and the critical radius r = r c around which the geometry behaves differently, we find two different scheme of the structure of vacua and consequently that of vortices. When the vacua exist inside the critical radius r = r c , that is v r c , we do not need the outside geometry, in which case the Lagrangian reduces to the usual Goldstone model of φ admitting the S 1 vacua and integer global vortices. A single vortex configuration is of the form of φ = f (ρ) exp(iϕ) with the polar coordinates (ρ, ϕ).
On the other hand, when r c v, the Lagrangian is well described by Φ in Eq. (7), which is asymptotically reducing the Lagrangian Eq. In summary, we have the following two cases: • r c v, R c v n : integer vortex scheme.
• r c v, R c v n : fractional vortex scheme.
We have numerically obtained the stationary solution with one integer vortex in 2dimensional space by minimizing the energy The solution can be calculated by finding the solution under the boundary condition φ = ve iϕ (on ∂Ω). As a numerical parameters, we have chosen  In the case of n = 4, there appear many metastable solutions and rod-shaped structure such as those shown in Figs. 1 (a) and 2 (a) is no more stable. We show some of them in Fig. 4. The number of metastable solutions seems to be huge and it was impossible to exhaust all of them. We will report the detailed analysis for n ≥ 4 elsewhere.

IV. SUMMARY AND DISCUSSION
As a generalization of the modified XY and Goldstone models, we have defined the Z n modified XY and Goldstone models, having a 2π/n periodic term in addition to the usual XY (2π periodic) term. We have pointed out that the modified Goldstone model can be When two (or n) complex scalar fields coupled by a Josephson term(s) also admits a vortex molecule of half-quantized (1/n quantized) vortices, see Refs. [38][39][40][41][42][43][44][45][46] for two-components and Refs. [47][48][49][50][51] for n components. This describes two-component BECs. In this case, in contrast to the case of the modified model, there is no two-step phase transition [52,53], but it is unclear what is a crucial difference between the two cases althouth the both admit similar solutions.
If we gauge the U (1) symmetry of the Goldstone model, we have an Abelian-Higgs model.
While the former admits a global vortex as we have discussed in this paper, the latter admits a local Abrikosov-Nielsen-Olesen vortex [54,55]. It is an interesting question whether a modified Abelian-Higgs model admits local vortices of the molecule type. There are several questions such as whether there is a critical (BPS) coupling and whether it admits supersymmetric extension, see e.g. Ref. [56]. Whether there is any superconductor described by such models is also an interesting question.