Abstract
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.
Highlights
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]
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 robust against quantum fluctuations
We have worked out an explicit example of a three component Bose-Einstein condensate (BEC) with U(2) × U(1) symmetry
Summary
We consider a three-component BEC with U(1) × U(2) symmetry. The Lagrangian for the Gross-Pitaevskii equation is. 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. We first consider the zeroth order, corresponding to the static case, and study the vortex profile functions f0 and g0 The equations for these two functions correspond to those in eq (2.6) without the world-sheet derivatives. The expansion of eq (2.6) in powers of the worldsheet coordinates gives new differential linear equations for the variables f1 and g1 (see appendix A) Since these equations are linear in the first order corrections, f1(r) end g1(r) depend only on the zeroth order profile functions f0 and g0. There appears one type-II translational zero mode corresponding to two broken translational symmetries [32]
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