non-Abelian Vortices Research Articles

Physics of non-Abelian vortices in Bose-Einstein condensates

A wide order-parameter manifolds of Bose-Einstein condensates (BECs) with spin degrees of freedom enables various kinds of topological defects which never appear in conventional scalar BECs. In this paper, we focus on the characteristic example of them; non-Abelian vortices and their dynamics in the cyclic phase of a spin-2 BEC. Non-Abelian vortices shows the unique effect in their collision dynamics, i.e. unlike Abelian vortices, they do neither reconnect themselves nor pass through each other but create a rung vortex between them.

Majorana meets Coxeter: Non-Abelian Majorana fermions and non-Abelian statistics

We discuss statistics of vortices having zero-energy non-Abelian Majorana fermions inside them. Considering the system of multiple non-Abelian vortices, we derive a non-Abelian statistics that differs from the previously derived non-Abelian statistics. The new non-Abelian statistics presented here is given by a tensor product of two different groups, namely the non-Abelian statistics obeyed by the Abelian Majorana fermions and the Coxeter group. The Coxeter group is a symmetric group related to the symmetry of polytopes in a high-dimensional space. As the simplest example, we consider the case in which a vortex contains three Majorana fermions that are mixed with each other under the SO(3) transformations. We concretely present the representation of the Coxeter group in our case and its geometrical expressions in the high-dimensional Hilbert space constructed from non-Abelian Majorana fermions.

Topological interactions of non-Abelian vortices with quasiparticles in high density QCD

Non-Abelian vortices are topologically stable objects in the color-flavor locked (CFL) phase of dense QCD. We derive a dual Lagrangian starting with the Ginzburg-Landau effective Lagrangian for the CFL phase, and obtain topological interactions of non-Abelian vortices with quasiparticles such as $U(1)_B$ Nambu-Goldstone bosons (phonons) and massive gluons. We find that the phonons couple to the translational zero modes of the vortices while the gluons couple to their orientational zero modes in the internal space.

Non-Abelian Multiple Vortices in Supersymmetric Field Theory

In this paper, we consider a system of non-Abelian multiple vortex equations governing coupled SU(N) and U(1) gauge and Higgs fields which may be embedded in a supersymmetric field theory framework. When the underlying domain is doubly periodic, we prove the existence and uniqueness of an n-vortex solution under a necessary and sufficient condition explicitly relating the domain size to the vortex number n and the Higgs boson masses. When the underlying domain is the full plane, we use a constructive approach to establish the existence and uniqueness of an n-vortex solution.

Geometry and energy of non-Abelian vortices

We study pure Yang–Mills theory on Σ × S2, where Σ is a compact Riemann surface, and invariance is assumed under rotations of S2. It is well known that the self-duality equations in this setup reduce to vortex equations on Σ. If the Yang–Mills gauge group is SU(2), the Bogomolny vortex equations of the Abelian Higgs model are obtained. For larger gauge groups, one generally finds vortex equations involving several matrix-valued Higgs fields. Here we focus on Yang–Mills theory with gauge group \documentclass[12pt]{minimal}\begin{document}$\mathrm{SU}(N)/\mathbb {Z}_N$\end{document} SU (N)/ZN and a special reduction which yields only one non-Abelian Higgs field. One of the new features of this reduction is the fact that while the instanton number of the theory in four dimensions is generally fractional with denominator N, we still obtain an integral vortex number in the reduced theory. We clarify the relation between these two topological charges at a bundle geometric level. Another striking feature is the emergence of nontrivial lower and upper bounds for the energy of the reduced theory on Σ. These bounds are proportional to the area of Σ. We give special solutions of the theory on Σ by embedding solutions of the Abelian Higgs model into the non-Abelian theory, and we relate our work to the language of quiver bundles, which has recently proved fruitful in the study of dimensional reduction of Yang–Mills theory.

Proposed Aharonov-Casher interference measurement of non-Abelian vortices in chiralp-wave superconductors

We propose a two-path vortex interferometry experiment based on the Aharonov-Casher effect for detecting the non-Abelian nature of vortices in a chiral p-wave superconductor. The effect is based on observing vortex interference patterns upon enclosing a finite charge of externally controllable magnitude within the interference path. We predict that when the interfering vortices enclose an odd number of identical vortices in their path, the interference pattern disappears only for non-Abelian vortices. When pairing involves two distinct spin species, we derive the mutual statistics between half quantum and full quantum vortices and show that, remarkably, our predictions still hold for the situation of a full quantum vortex enclosing a half quantum vortex in its path. We discuss the experimentally relevant conditions under which these effects can be observed.

Abelian and non-Abelian anyons in integer quantum anomalous Hall effect and topological phase transitions via superconducting proximity effect

We study the quantum anomalous Hall effect described by a class of two-component Haldane models on square lattices. We show that the latter can be transformed into a pseudospin triplet p+ip-wave paired superfluid. In the long wave length limit, the ground state wave function is described by Halperin's (1,1,-1) state of neutral fermions analogous to the double layer quantum Hall effect. The vortex excitations are charge e/2 abelian anyons which carry a neutral Dirac fermion zero mode. The superconducting proximity effect induces `tunneling' between `layers' which leads to topological phase transitions whereby the Dirac fermion zero mode fractionalizes and Majorana fermions emerge in the edge states. The charge e/2 vortex excitation carrying a Majorana zero mode is a non-abelian anyon. The proximity effect can also drive a conventional insulator into a quantum anomalous Hall effect state with a Majorana edge mode and the non-abelian vortex excitations.

Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from L2 metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by Ψ0, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.

Sharp existence and uniqueness theorems for non-Abelian multiple vortex solutions

Vortices in non-Abelian gauge field theory play essential roles in the mechanism of color confinement and are governed by systems of nonlinear elliptic equations of complicated structure. In this paper, we present a series of sharp existence and uniqueness theorems for multiple vortex solutions of the non-Abelian BPS equations over R 2 and on a doubly periodic domain. Our methods are based on calculus of variations which may be used to analyze more extended problems. The necessary and sufficient conditions for the existence of a unique solution in the doubly periodic situation are expressed in terms of physical parameters involved explicitly.

Topological interactions between non-Abelian vortices and the surrounding environment

Topological interactions between non-Abelian vortices and the surrounding environment

Large-Nsolution of the heterotic weighted nonlinear sigma-model

We study a heterotic two-dimensional N=(0,2) gauged non-linear sigma-model whose target space is a weighted complex projective space. We consider the case with N positively and N^~=N_F - N negatively charged fields. This model is believed to give a description of the low-energy physics of a non-Abelian semi-local vortex in a four-dimensional N=2 supersymmetric U(N) gauge theory with N_F > N matter hypermultiplets. The supersymmetry in the latter theory is broken down to N=1 by a mass term for the adjoint fields. We solve the model in the large-N approximation and explore a two-dimensional subset of the mass parameter space for which a discrete Z_{N-N^~} symmetry is preserved. Supersymmetry is generically broken, but it is preserved for special values of the masses where a new branch opens up and the model becomes super-conformal.

Collision Dynamics of Non-Abelian Vortices in Spinor Bose-Einstein Condensates

Bose-Einstein condensates with spin degrees of freedom admit various topological phases and excitations. Non-Abelian vortices offer a remarkable example which is expected to be realized in the cyclic phase of the spin-2 spinor Bose-Einstein condensates. We demonstrate that the non-Abelian property of non-Abelian vortices shows the unique effect in their collision dynamics, i.e., unlike Abelian vortices, they do neither reconnect themselves nor pass through each other but create a rung vortex between them.

NEW RESULTS ON NON-ABELIAN VORTICES — FURTHER INSIGHTS INTO MONOPOLE, VORTEX AND CONFINEMENT

We discuss some of the latest results concerning the non-Abelian vortices. The first concerns the construction of non-Abelian BPS vortices based on general gauge groups of the form G = G′×U(1). In particular detailed results about the vortex moduli space have been obtained for G′ = SO(N) or U Sp(2N). The second result is about the "fractional vortices", i.e., vortices of the minimum winding but having substructures in the tension (or flux) density in the transverse plane. Thirdly, we discuss briefly the monopole-vortex complex.

On the [formula omitted]-metric of vortex moduli spaces

We derive general expressions for the Kähler form of the L 2 -metric in terms of standard 2-forms on vortex moduli spaces. In the case of abelian vortices in gauged linear sigma-models, this allows us to compute explicitly the Kähler class of the L 2 -metric. As an application we compute the total volume of the moduli space of abelian semi-local vortices. In the strong coupling limit, this then leads to conjectural formulae for the volume of the space of holomorphic maps from a compact Riemann surface to projective space. Finally we show that the localization results of Samols in the abelian Higgs model extend to more general models. These include linear non-abelian vortices and vortices in gauged toric sigma-models.

Group theory of non-abelian vortices

We investigate the structure of the moduli space of multiple BPS non-Abelian vortices in U(N) gauge theory with N fundamental Higgs fields, focusing our attention on the action of the exact global (color-flavor diagonal) SU(N) symmetry on it. The moduli space of a single non-Abelian vortex, \( \mathbb{C}{P^{N - 1}} \), is spanned by a vector in the fundamental representation of the global SU(N) symmetry. The moduli space of winding-number k vortices is instead spanned by vectors in the direct-product representation: they decompose into the sum of irreducible representations each of which is associated with a Young tableau made of k boxes, in a way somewhat similar to the standard group composition rule of SU(N) multiplets. The Kähler potential is exactly determined in each moduli subspace, corresponding to an irreducible SU(N) orbit of the highest-weight configuration.

Majorana Fermions in Non-Abelian QCD Vortices

We study the internal structure of a non-Abelian vortex in color superconductivity with respect to quark degrees of freedom. Stable non-Abelian vortices appear in the Color-FlavorLocked phase whose symmetry SU(3)C+L+R is further broken to SU(2)C+L+R ⊗ U(1)C+L+R in the vortex cores. Fermionic description of vortices is made possible by the Bogoliubov-de Gennes (B-dG) equation. Quark spectra in the vortex are obtained from the B-dG equation by treating the diquark gap having the vortex configuration as a background field. We find that there are massless modes (zero modes) well-localized around a vortex, in the triplet and singlet states of the unbroken symmetry SU(2)C+L+R. §1. Vortices in superconductors have internal structure At various scales in nature, there appear vortices, for instance, Cosmic strings in space, typhoons on Earth, water flow in a sink, and quantized vortices in superconductors and liquid Helium. The vortices that we discuss here appear at the shortest scales, the scale of strong interaction, in particular in the color-flavor locked (CFL) phase of color superconductivity in high density quark matter. All the vortices are essentially two-dimensional objects and are characterized by non-zero winding numbers of order parameters or velocity fields. However, vortices in superconductors are unique because they have non-trivial internal structure originating from their microscopic degrees of freedom. Although theoretical framework of this problem was established long time ago in the context of inhomogeneous superconductors, 1) it has been attracting people’s attention with renewed interests. One of the recent developments is the possibilities of “unconventional” superconductors where the order parameters have non-trivial symmetries. 2) It is argued that one can determine the symmetry-breaking pattern of the order parameters by looking at the vortex internal structure. In some cases called “topological” superconductors, there exists a zero-energy fermionic mode (“Majorana fermions”). On the other hand, we know several examples where massless modes are present around defects or nontrivial configuration. For instance, domain-wall fermions used in lattice QCD, fermion zero modes of the Dirac operator in the presence of instantons, and the edge modes in quantum Hall effects. Hence, it is quite a natural interest to investigate the internal structure of vortices. We have a novel type of vortices with non-Abelian symmetry, which is realized in the CFL phase of high-density quark matter. Thus, we study the internal structure of such non-Abelian vortices. 3)

Topological Excitations in Spinor Bose-Einstein Condensates

A rich variety of order parameter manifolds of multicomponent Bose-Einstein condensates (BECs) admit various kinds of topological excitations, such as fractional vortices, monopoles, skyrmions, and knots. In this paper, we discuss two topological excitations in spinor BECs: non-Abelian vortices and knots. Unlike conventional vortices, non-Abelian vortices neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. We discuss the collision dynamics of non-Abelian vortices in the cyclic phase of a spin-2 BEC. In the latter part, we show that a knot, which is a unique topological object characterized by a linking number or a Hopf invariant [$\pi_3 (S^2)=Z$], can be created using a conventional quadrupole magnetic field in a cold atomic system.

Moduli space metric for well-separated non-Abelian vortices

The moduli space metric and its Kahler potential for well-separated non-Abelian vortices are obtained in U(N) gauge theories with N Higgs fields in the fundamental representation.

Non-Abelian Yang-Mills–Higgs vortices

In this Letter we present new, genuinely non-Abelian vortex solutions in SU(2) Yang-Mills--Higgs theory with only one {\it isovector} scalar field. These non-Abelian solutions branch off their Abelian counterparts (Abrikosov-Nielsen-Olesen vortices) for precise values of the Higgs potential coupling constant $\lambda$. For all values of $\lambda$, their energies lie below those of the Abelian energy profiles, the latter being logarithmically divergent as $\lambda\to\infty$. The non-Abelian branches plateau in the limit $\lambda\to\infty$ and their number increases with $\lambda$, this number becoming infinite. For each vorticity, the gaps between the plateauing energy levels become constant. In this limit the non-Abelian vortices are non-interacting and are described by the {\it self-dual} vortices of the O(3) sigma model. In the absence of a topological lower bound, we expect these non-Abelian vortices to be {\it sphalerons}.

Low-energyU(1)×USp(2M)gauge theory from simple high-energy gauge group

We give an explicit example of the embedding of a near BPS low-energy (U(1) x USp(2M))/Z_2 gauge theory into a high-energy theory with a simple gauge group and adjoint matter content. This system possesses degenerate monopoles arising from the high-energy symmetry breaking as well as non-Abelian vortices due to the symmetry breaking at low energies. These solitons of different codimensions are related by the exact homotopy sequences.