Quantum Exact Non-Abelian Vortices in Non-relativistic Theories

Non-Abelian vortices arise when a non-Abelian global symmetry is exact in the ground state but spontaneously broken in the vicinity of their cores. In this case, there appear (non-Abelian) Nambu-Goldstone (NG) modes confined and propagating along the vortex. In relativistic theories, the Coleman-Mermin-Wagner theorem forbids the existence of a spontaneous symmetry breaking, or a long-range order, in 1+1 dimensions: quantum corrections restore the symmetry along the vortex and the NG modes acquire a mass gap. We show that in non-relativistic theories NG modes with quadratic dispersion relation confined on a vortex can remain gapless at quantum level. We provide a concrete and experimentally realizable example of a three-component Bose-Einstein condensate with U(1) x U(2) symmetry. We first show, at the classical level, the existence of S^3 = S^1 |x S^2 (S^1 fibered over S^2) NG modes associated to the breaking U(2) ->U(1) on vortices, where S^1 and S^2 correspond to type I and II NG modes, respectively. We then show, by using a Bethe ansatz technique, that the U(1) symmetry is restored, while the SU(2) symmery remains broken non-pertubatively at quantum level. Accordingly, the U(1) NG mode turns into a c=1 conformal field theory, the Tomonaga-Luttinger liquid, while the S^2 NG mode remains gapless, describing a ferromagnetic liquid. This allows the vortex to be genuinely non-Abelian at quantum level.


I. INTRODUCTION
Vortices play a prominent role in various areas of physics such as superfluidity, superconductivity, magnetic materials, nematic liquids, fractional quantum Hall effect, topological quantum computation, quantum field theory and cosmology. In particular in high energy physics, much attention has been recently paid to non-Abelian vortices. Such objects arise when a non-Abelian symmetry is spontaneously broken in the vicinity of the vortex. The gapless Nambu-Goldstone (NG) modes associated to this symmetry breaking are then confined on the vortex, where they can propagate. Non-Abelian vortices were first discovered in supersymmetric gauge theories [1][2][3][4][5] (see Refs. [5][6][7][8] for a review), where their presence elegantly explains a long standing problem about quantum physics in different dimensions, i.e. similarities in the non-perturbative dynamics of sigma models in 1+1 dimensions and Yang-Mills theory in 3+1 dimensions ( e.g. existence of dynamical mass gap, asymptotic freedom, instantons). Non-Abelian vortices also appear in high density quark matter (in this context also known as color flux tubes) [9][10][11][12][13] (see Ref. [14] for a review). In this case, a Coleman-Weinberg quantum potential arising on the vortex world-sheet leads to the presence of monopole mesons confined by color flux tubes [15,16]. This phenomenon explains a quark-hadron duality between low and high density QCD, i.e. quarks are confined when monopoles are condensed at low density while monopoles are confined when quarks are condensed at high density [16]. The examples mentioned above correspond to relativistic systems. In such cases NG modes acquire mass gaps through quantum corrections [17,18] consistently with the Coleman-Mermin-Wagner theorem [19,20] which forbids the existence of a spontaneous symmetry breaking or a long-range order in 1+1 dimensions. In this sense, non-Abelian vortices studied thus far in relativistic theories are Abelianized at quantum level [17,21].
On the other hand, the situation can be different in non-relativistic theories. There are two kinds of NG modes, type-I and II, with linear and quadratic dispersions respectively [22]. This distinction is deeply connected with the geometry of the space parametrized by the NG modes [23,24]; one type-II NG mode corresponds to two broken generators, while one type-I NG mode corresponds to one broken generator. In the relativistic case only type-I NG are allowed.
In this paper, we show that type-II NG modes confined in the core of a vortex remain gapless at quantum level, providing a quantum exact non-Abelian vortex. We consider a concrete example of a three-component Bose-Einstein condensate (BEC) with U (2) × U (1) symmetry, in which there is partial phase separation between one (immiscible) component and the other two (miscible) components. Our setup can be realized in experiments of ultracold atomic gases; (Almost) U (2) symmetric two-component BECs were already realized by ultracold 87 Rb [25] and 23 Na [26] atoms. If one mixes additional atom with repulsive interaction with these atoms, one can experimentally realize our set-up. In this system, we find that there appear an S 1 type-I NG mode and an S 2 CP 1 type-II NG mode in a vortex core. By noting that this vortex acts as a 1+1 dimensional trap for the two miscible components, we describe the modes living in the vortex core as a 1+1 dimensional twocomponent Bose gas with U (2) symmetry. By studying the exact solution of this system through a Bethe ansatz technique, we show that at quantum level the U (1) symmetry recovers and the U (1) sector can be described by a conformal field theory (CFT) with the conformal charge c = 1 or a Tomonaga-Luttinger liquid [27,28]. On the other hand, the SU (2) symmetry breaking holds at quantum level and the S 2 NG mode remains gapless, describing a ferromagnetic liquid [29].
In the cylindrical coordinates (r, θ, z), a vortex along the z-axis is given by with the boundary conditions f 0 → v and g 0 → 0 at r → ∞ and f 0 → 0 and g 0 → 0 at r → 0, see Fig. 1. Here, η is a two-component complex constant that parameterizes S 3 , and identify the NG modes associated with the ground state symmetry breaking U (2) → U (1) that takes place in the vicinity of the vortex core: We construct a low-energy effective theory of the vortex through the moduli approximation [30,31] introducing a world-sheet coordinate (t, z) dependence of the vortex configura- Notice that we have included a world-sheet coordinate dependence on the profile functions too. This is equivalent to the inclusion of "Higgs" excitations H and h in the calculation: f (r, θ, z, t) ≡ f 0 (r) + H(r, θ, z, t), g(r, θ, z, t) ≡ g 0 (r) + h(r, θ, z, t).
This modification to the standard approach in deriving effective theories in the relativistic case is crucial. The field H is a massive bulk mode, while the field h is a lighter nonpropagating mode that contributes to the effective action for the zero-modes η.
By substituting Eq. (3) in the Lagrangian we get: from whose variation one obtains the equations of motion for f and g: In the derivation of the effective action for the moduli η we will keep only terms up to second derivatives in the world-sheet coordinates. This allows us to consider an expansion of the equations above in powers of these coordinates. An important observation is that, thanks to the (static) equation of motions, the corrections to the effective Lagrangian in Eq. (5) enter quadratically at least. This allows us to calculate all terms in the effective action, up the the second order in the world-sheet coordinates, by just calculating corrections to the profile functions up to the first order. As can be seen by inspecting Eq. (6), first order corrections will only involve time derivatives of the moduli η. We proceed using the following expansion: profile functions f 0 and g 0 . The equations for these two functions correspond to those in Eq. (6) without the world-sheet derivatives. A numerical example can be seen in Fig. 1.
We can see how the vortex built in the immiscible component acts as a one dimensional trap for the two miscible components. This is the first crucial observation of our paper.
The asymptotic behavior of the profile functions can be found analytically by studying the As common in systems with only global symmetries, the profile function f 0 for the condensate supporting the winding decays polynomially. On the other hand the expression above shows the profile function for the trapped components g 0 decays exponentially. This is our second important observation. The exponential decay allows us to search for normalizable zeromodes propagating on the vortex line, so that the 1+1 effective action is finite on an infinite volume.
We proceed now to the calculation of the effective action with the derivation of the higher order corrections f 1 and g 1 . The expansion of Eq. (6) in powers of the world-sheet coordinates gives new differential linear equations for the variables f 1 and g 1 (see Appendix A). Since these equations are linear in the first order corrections, f 1 (r) end g 1 (r) depend only on the zeroth order profile functions f 0 and g 0 . Their numerical evaluation can be found in The effective action is obtained after integrating over the transverse direction r, this can be easily done numerically with the knowledge of the profile functions f 0,1 and g 0,1 . The whole calculation has assumed only a derivative expansion in powers of the world-sheet derivatives, keeping terms up to the second order: Since the function g 0 is exponentially decaying, the coefficient α is finite. Moreover, notice that the first order correction i g 0 η †η is exponentially small at large distances, which implies that also the functions f 1 and g 1 decay exponentially. As a consequence, the coefficient β is finite and of the same order of α. This means that the effective action is finite and the zero modes are truly confined on the vortex. The coefficients α and β have been calculated numerically in Appendix A for the special choice of parameters shown in Fig. 1.
In order to see the dispersions of the NG modes, we recall that the geometry of the NG To make this structure explicit, we take a parameterization with n † n = 1, where ϕ and n parametrize S 1 and S 2 . Then, the effective Lagrangian can be rewritten as which shows that ϕ and n are type-I and II NG modes with linear and quadratic dispersions, respectively. Here, remarkably n describes one type-II NG mode corresponding to the two broken generators of S 2 . In addition, there appears one type-II translational zero mode corresponding to two broken translational symmetries [32].

III. QUANTUM EXACT GAPLESS MODES
In the effective theory approach, we have obtained gapless modes at classical level. In order to discuss whether these modes remain gapless quantum mechanically, we should take into account massive modes and quartic interaction.
where ρ = σ Ψ † σ Ψ σ is the density, and L is a system size along the z-direction. The effective parameters λ and µ can be related to the original parameters through a renormalization group, but here we do not need explicit relations. Since there is no spin-dependent coupling, the Hamiltonian has U (2) symmetry in which we can use the Bethe ansatz technique [28,35].
The resultant Bethe ansatz equations are [36,37] where Θ(x) = 2 tan −1 (x), and N and M represent the total number of the particles and particle number for the component "2", respectively. k j and l α are called quasimomenta and isospin rapidities originating from SU (2) symmetry in the system. I j and J α take integer or half-integer values, depending on whether N − M is odd or even. Then, the total energy and momentum are given by By using I j and J α , the ground state can be characterized as [36] {I j } = {−(N − 1)/2, · · · , (N − 1)/2}, M = 0.
In the thermodynamic limit, the above Bethe ansatz equations are then reduced to where ρ(k) is the density of quasimomenta, and cut-off Λ is determined by the density of the system N/L = Λ −Λ ρ(k). This equation corresponds to that in Lieb-Liniger gas [38,39]. Thus, as far as S 1 sector is concerned, the low-energy excitation is linear gapless characterized by the c = 1 CFT while there is no BEC. In addition, M = 0 means that the ground state is ferromagnetic, which is consistent with a general theorem that a ground state of spinful bosons without spin-dependent interaction is always polarized [40] .
We next consider the nature of the isospin excitation originating from S 2 manifold. Although analytic results can be obtained for weak and strong coupling limits, we need to numerically determine it for general couplings. To this end, we consider a dressed energy formalism in the Bethe ansatz [28]. In our model, we can obtain where µ is the chemical potential and we introduced the dressed energy c (k) and s (l), each of which represents S 1 and S 2 excitations. The derivation of the above equations is given in Appendix B. In the low-energy isospin excitation, an isospin rapidity l is related to the real momentum of the isospin excitation p as follows: By solving a couple of the above equations, we can confirm that the isospin excitation is gapless quadratic one [36,37]. This indicates that a type-II NG mode, called a magnon, emerges for the isospin sector due to a spontaneous symmetry breaking of SU (2).

IV. SUMMARY AND DISCUSSION
We have shown that a non-Abelian vortex remains non-Abelian at the quantum level when trapped NG modes are of type-II, e.g. the low-energy behavior of the NG modes is We can also consider the case where the U (2) symmetry is explicitly broken to U (1) 2 , but the trapped components are still miscible. This can be obtained with an additional term of the type −L int = λ 12 |Ψ 1 | 2 |Ψ 2 | 2 + 1 2 λ 11 |Ψ 1 | 4 + 1 2 λ 22 |Ψ 2 | 4 , where λ 12 is smaller than intracomponent couplings in λ 11 and λ 22 (the U (2) symmetric case corresponds to λ 11 = λ 22 = λ 12 = λ).
At the classical level, this symmetry is spontaneously broken in the vortex core and two type-I NG modes are trapped in the vortex. At the quantum level, the U (1) 2 symmetry is recovered and there are two Tomonaga-Luttinger liquids as far as the so-called Tomonaga-Luttinger parameter in the isospin sector is greater than 1 (See Appendix B).
It is straightforward to extend our model to an (N +1)-component BEC with U (N )×U (1) symmetry. There appear NG modes S N −1 S 1 CP N −1 , where S 1 and CP N −1 are type-I and II NG modes at the classical level. At the quantum level, the U (1) symmetry is recovered with the Tomonaga-Luttinger liquid, while the SU (N ) symmetry remains broken and CP N −1 modes remain as type-II NG modes, as ensured by the Eisenberg-Lieb theorem [40].
Finally, fate of type-II NG modes localized on domain walls [41,42] and skyrmion lines [43] in the presence of quantum effects is one interesting future problem to explore.

Appendix A: Derivation of the effective action
Assuming the following derivative expansion for the profile functions as explained in the we obtain the following equations: Since the non-homogeneous term in the effective action is exponentially small, the functions f 1 and g 1 are also exponentially small. A numerical solution for these quantities is shown in Fig. 2. Using Eq. (10), we can evaluate numerically the coefficients α and K. Using the values of the parameters as indicated in the figures captions, we have, for example: where we have taken = m = 1.

Appendix B: Derivation of Bethe ansatz equations
Here we sketch a derivation of the Bethe ansatz equations in two-component bosons, which can be solved by the so-called nested Bethe ansatz method. Our derivation is based on one in Ref. [35], which discusses the Fermionic Hubbard model. We first consider N particle wavefunction as where {P } denotes possible permutations. From the continuity of the wavefunction and its first derivative at equal positions as x 1 = x 2 , we have We next introduce and |σ 1 , · · · , σ b , σ a , · · · , σ N , = Π (a,b) |σ 1 , · · · , σ a , σ b , · · · , σ N , with the identify operator I. Then, Eq. (B3) can be expressed as We next consider the periodic boundary condition: By substituting the above into Eq. (B4), we have which satisfies we have By using Pauli matrices, we have We next introduce the so-called transfer matrix By considering a "vacuum" state as |vac = |1, · · · , 1 , we obtain C(l)|vac = 0, (B18) Since Y (a,b) and T (a) satisfy the celebrated Yang-Baxter relation it is straightforward to show the following commutation relations: By substituting the above into Eq. (B13) and using the commutation relations (B21), (B22), and (B23), we obtain the following Bethe ansatz equations: Finally, by taking the logarithm of the above equations, we obtain Eqs. (14) and (15).
We next introduce Then, the ground state energy can be rewritten as where ε = ( c (k), s (l)) T is the dressed energy. This satisfies which is nothing but Eqs. (19) and (20).
and the similar relations for θ fields. Here, u µ is the velocity and K µ is the so-called Luttinger parameter, and they are given by As can be seen from the above bosonization Hamiltonian, if there is no coupling between up and down particles (κ = 0), the Hamiltonian is quadratic, which is called Tomonaga-Luttinger Hamiltonian.
In the presence of κ, we also have cos φ σ , and therefore physics becomes non-trivial. To see what happens in the presence of this coupling, we can consider the renormalization group analysis, which is one of the most powerful approaches in one dimensional world.
We note that the above are essentially equivalent to renormalization group equations in the so-called BKT transition. We can easily check that cos φ σ is locked since κ flows to strong coupling if K σ < 1 while cos φ σ is irrelevant if K σ > 1.
In a weakly-interacting bosonic system, we naturally expect that K σ > 1, and therefore the fixed point is the Tomonaga-Luttinger liquid phase. Namely, even if we consider the