non-Abelian Vortices Research Articles

Flowing between string vacua for the critical non-Abelian vortex with a deformation of N=2 Liouville theory

It has been shown that non-Abelian solitonic vortex string supported in four-dimensional (4D) N=2 supersymmetric QCD (SQCD) with the U(2) gauge group and Nf=4 quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states was found and interpreted as a spectrum of hadrons in 4D N=2 SQCD. In particular, the lowest string state appears to be a massless Bogomol’nyi-Prasad-Sommerfield (BPS) baryon associated with the deformation of the complex structure modulus b of the conifold. It was recently shown that the Coulomb branch of the associated string sigma model which opens up at strong coupling can be described by N=2 Liouville theory. Building on these results we switch on quark masses in 4D N=2 SQCD and study the interpolation of the initial U(2) SQCD with Nf=4 quarks to the final SQCD with the U(4) gauge group and Nf=8 quarks. To find the true string vacuum which arises due to the mass deformation we solve the effective supergravity equations of motion associated with the deformed world sheet Liouville theory. We show that the massless BPS baryon b survives the deformation and that finding of the spectrum of low-lying massive hadrons in the final SQCD is linked to the Calogero problem. Published by the American Physical Society 2024

Proximity effects of vortices in neutron 3P2 superfluids in neutron stars: Vortex core transitions and covalent bonding of vortex molecules

Neutron $^{3}P_{2}$ superfluids consisting of neutron pairs with the total angular momentum $J=2$, spin-triplet, and $P$ wave are believed to be realized in neutron star cores. Within the Ginzburg-Landau theory it was previously found that a singly quantized vortex is split into two half-quantized non-Abelian vortices connected by one (or three) soliton(s) forming a vortex molecule with the soliton bond(s), in the absence (presence) of a magnetic field parallel to them. In this paper, we investigate proximity effects of two vortex molecules by exhausting all possible two vortex molecule states consisting of four half-quantized vortices and determine the phase diagram spanned by the magnetic field and rotation speed. As the rotation speed is increased, the distance between the two vortex molecules becomes shorter. In the magnetic field below the critical value, we find that as the rotation speed is increased, the two separated vortex molecules transit to a dimerized vortex molecule, where the two vortex molecules are bridged by two solitons that we call ``covalent bonds'' in analogy with chemical molecules. We also find that the orders of the constituent half-quantized vortex cores transit from a ferromagnetic order to a cyclic order as the vortex molecules come closer. On the other hand, no dimerization occurs in the magnetic field above the critical value. Instead, we find a transition for the polarization direction of the vortex molecules from a configuration parallel to the separation to one perpendicular to the separation as they come closer. We also show some examples of three, four, and many vortex molecule states.

Topologically protected vortex knots and links

In 1869, Lord Kelvin found that the way vortices are knotted and linked in an ideal fluid remains unchanged in evolution, and consequently hypothesized atoms to be knotted vortices in a ubiquitous ether, different knotting types corresponding to different types of atoms. Even though Kelvin’s atomic theory turned out incorrect, it inspired several important developments, such as the mathematical theory of knots and the investigation of knotted structures that naturally arise in physics. However, in previous studies, knotted and linked structures have been found to untie via local cut-and-paste events referred to as reconnections. Here, in contrast, we construct knots and links of non-Abelian vortices that are topologically protected in the sense that they cannot be dissolved employing local reconnections and strand crossings. Importantly, the topologically protected links are supported by a variety of physical systems such as dilute Bose-Einstein condensates and liquid crystals. We also propose a classification scheme for topological vortex links, in which two structures are considered equivalent if they differ from each other by a sequence of topologically allowed reconnections and strand crossings, in addition to the typical continuous transformations. Interestingly, this scheme produces a remarkably simple classification.

NS three-form flux deformation for the critical non-Abelian vortex string

It has been shown that the non-Abelian solitonic vortex string supported in four-dimensional (4D) $\mathcal{N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge group and ${N}_{f}=4$ quark flavors becomes a critical superstring. This string propagates in the ten-dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau noncompact threefold, namely, the conifold. The spectrum of low-lying closed string states in the associated type-IIA string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N}=2$ SQCD. In particular, the lowest string state appears to be a massless BPS baryon associated with the deformation of the complex structure modulus $b$ of the conifold. In the previous work the deformation of the ten-dimensional background with nonzero Neveu-Schwarz 3-form flux was considered and interpreted as a switching on a particular choice of quark masses in 4D SQCD. This deformation was studied to leading order at small 3-form flux. In this paper we study the back reaction of the nonzero 3-form flux on the metric and the dilaton introducing the ansatz with several warp factors and solving the gravity equations of motion. We show that 3-form flux produces a potential for the conifold complex structure modulus $b$, which leads to the runaway vacuum. At the runaway vacuum warp factors disappear, while the conifold degenerates. In 4D SQCD we relate this to the flow to the U(1) gauge theory upon switching on quark masses and decoupling of two flavors.

Fermionic vacuum polarization induced by a non-Abelian vortex

In this paper, we analyze the fermionic condensate (FC) and the vacuum expectation value (VEV) of the energy–momentum tensor associated with an iso-spin-[Formula: see text] charged massive fermionic field induced by the presence of a [Formula: see text] vortex, taking into account the effect of the conical geometry produced by this object. We consider the vortex as an idealized topological defect, i.e. very thin, straight and carrying a magnetic flux running along its core. Besides the direct coupling of the fermionic field with the iso-vector gauge field, we also admit the coupling with the scalar sector of the non-Abelian vortex system, expressed as a vector in the three-dimensional iso-space. Due to this interaction, the FC is expressed as the sum of two contributions associated with the two different effective masses for the [Formula: see text] fermionic components of the iso-spin operator, [Formula: see text]. The VEV of the energy tensor also presents a similar structure. The vacuum energy density is equal to the radial and axial stresses. As to the azimuthal one, it is expressed in terms of the radial derivative of energy density. Regarding the magnetic flux, both the FC and the VEV of the energy–momentum tensor can be positive or negative. Another interesting consequence of the interaction with the bosonic sector, the FC and VEV of the energy–momentum tensor, presents different intensity for different values of the ratio between the scalar coupling constant and the mass of the fermionic field. This is a new feature presented by the system.

Non-Abelian half-quantum vortices in 3P2 topological superfluids

${}^{3}P_{2}$ superfluids realized in neutron stars are the largest topological quantum matters in our Universe. We establish the existence and stability of non-Abelian half-quantum vortices (HQVs) in ${}^{3}P_{2}$ superfluids with strong magnetic fields. Using a self-consistent microscopic approach, we find that a singly quantized vortex is energetically destabilized into a pair of two non-Abelian HQVs owing to the strongly spin-orbit-coupled pairing. We find a topologically protected Majorana fermion on each HQV, thereby providing two-fold non-Abelian anyons characterized by both Majorana fermions and a non-Abelian first homotopy group.

Relations among topological solitons

We clarify relations among topological solitons in various dimensions: a domain wall, non-Abelian vortex, magnetic monopole and Yang-Mills instanton, together with a (non-Abelian) sine-Gordon soliton, baby Skyrmion (lump) and Skyrmion. We construct a composite configuration consisting of a domain wall, vortex, magnetic monopole and Yang-Mills instanton (wall-vortex-monopole-instanton) using the effective theory technique or moduli approximation. Removing some solitons from such a composite, we obtain all possible composite solitons in the form of solitons within a soliton, including all the previously known configurations, yielding relations among topological solitons.

Core structures of vortices in Ginzburg-Landau theory for neutron P23 superfluids

We investigate vortex solutions in the Ginzburg-Landau theory for neutron $^3P_2$ superfluids relevant for neutron star cores in which neutron pairs possess the total angular momentum $J=2$ with spin-triplet and $P$ wave, in the presence of the magnetic field parallel to the angular momentum of vortices. The ground state is known to be in the uniaxial nematic (UN) phase in the absence of magnetic field, while it is in the $D_2$ ($D_4$) biaxial nematic (BN) phase in the presence of the magnetic field below (above) the critical value. We find that a singly quantized vortex always splits into two half-quantized non-Abelian vortices connected by soliton(s) as a vortex molecule with any strength of the magnetic field. In the UN phase, two half-quantized vortices with ferromagnetic cores are connected by a linear soliton with the $D_4$ BN order. In the $D_2$ ($D_4$) BN phase, two half-quantized vortices with cyclic cores are connected by three linear solitons with the $D_4$ ($D_2$) BN order. The energy of the vortex molecule monotonically increases and the distance between the two half-quantized vortices decreases with the magnetic field increases, except for a discontinuously increasing jump of the distance at the critical magnetic field. We also construct an isolated half-quantized non-Abelian vortex in the $D_4$ BN phase.

Prime number factorization using a spinor Bose–Einstein condensate-inspired topological quantum computer

Inspired by non-abelian vortex anyons in spinor Bose–Einstein condensates, we consider the quantum double \(\mathcal {D}(\mathbb {Q}_8)\) anyon model as a platform to carry out a particular instance of Shor’s factorization algorithm. We suggest that the \(\mathcal {D}(\mathbb {Q}_8)\) anyon model could be realized by a specific low-temperature phase of a spin-2 Bose–Einstein condensate. We provide the excitation spectrum for this model, as well as the fusion rules, braid group representations, and a circuit architecture that facilitates the computation. All necessary quantum gates, less one, can be compiled exactly for this hybrid topological quantum computer. The required non-topological gate could be implemented using measurement based protocols. To analyze the effect of decoherence on the non-topological gate, a noise model based on stochastic unitary rotations is considered. The computational potential of this quantum double anyon model is similar to that of the Majorana fermion-based Ising anyon model, thus offering a complementary future platform for topological quantum computation.

Critical non-Abelian vortices and holography for little string theory

It has been shown that non-Abelian vortex string supported in four dimensional (4D) ${\mathcal N}=2$ supersymmetric QCD (SQCD) with the U(2) gauge group and $N_f = 4$ quark flavors becomes a critical superstring. This string propagates in the ten dimensional space formed by a product of the flat 4D space and an internal space given by a Calabi-Yau non-compact threefold, namely, the conifold. The spectrum of closed string states of the associated string theory was obtained using the equivalence between the critical string on the conifold and the non-critical string on the semi-infinite cigar described by SL($2, \mathbb{R}$)/U(1) Wess-Zumino-Novikov-Witten model. This spectrum was identified with the spectrum of hadrons in 4D ${\mathcal N}=2$ SQCD. In order to describe effective interactions of these 4D hadrons in this paper we study correlation functions of normalizable vertex operators localized near the tip of the SL($2, \mathbb{R}$)/U(1) cigar. We also compare our solitonic string approach to the gauge-string duality to the AdS/CFT-type holography for little string theories (LSTs). The latter relates off mass-shell correlation functions on the field theory side to correlation functions of non-normalizable vertex operators on the cigar. We show that in most channels holographic approach works in our theory because normalizable and non-normalizable vertex operators with the same conformal dimension are related due to the reflection from the tip of the cigar. However, we find that holography does not work for lightest hadrons with given baryonic charge.

Chiral non-Abelian vortices and their confinement in three flavor dense QCD

We find chiral non-Abelian vortices having windings only in one of the diquark condensations of left-handed and right-handed quarks in the color-flavor locked phase of dense QCD. They are the minimum vortices carrying half color magnetic fluxes of those of non-Abelian semi-superfluid vortices (color magnetic flux tubes) and 1/6 quantized superfluid circulations of Abelian superfluid vortices. These vortices carry ${\mathbb C}P^2$ orientational moduli in the internal space corresponding to their fluxes. The ${\mathbb C}P^2$ moduli of two chiral non-Abelian vortices with chiralities opposite to each other are energetically favored to be aligned while those of a vortex and anti-vortex to be orthogonal, and then these vortices attract each other. They are attached by chiral domain walls in the presence of the mass and axial anomaly terms explicitly breaking axial and chiral symmetries. We numerically show that two chiral non-Abelian vortices with chiralities opposite to each other are connected by a chiral domain wall, consisting a mesonic bound state which is nothing but a non-Abelian semi-superfluid vortex. We also show that Abelian and non-Abelian axial vortices attached by chiral domain walls are all unstable to decay into a set of chiral non-Abelian vortices. Furthermore, we find that chiral non-Abelian vortices exhibit unique features: one is the so-called topological obstruction implying that unbroken symmetry generators in the bulk are not defined globally around the vortices, and the other is color non-singlet Aharonov-Bohm (AB) phases implying that quarks encircling these vortices can detect the colors of magnetic fluxes of them at infinite distances.

Standard model gauge fields localized on non-Abelian vortices in six dimensions

Abstract A brane-world SU(5) grand unified theory model with global non-Abelian vortices is constructed in six-dimensional spacetime. We find a solution with a vortex associated to SU(3) separated from another vortex associated to SU(2). This 3–2 split configuration achieves a geometric Higgs mechanism for SU(5) → SU(3) × SU(2) × U(1) symmetry breaking. A simple deformation potential induces a domain wall between non-Abelian vortices, leading to a linear confining potential. The confinement stabilizes the vortex separation moduli, and ensures the vorticities of the SU(3) and SU(2) groups are identical. This dictates the equality of the numbers of fermion zero modes in the fundamental representation of SU(3) (quarks) and SU(2) (leptons), leading to quark/lepton generations. The standard model massless gauge fields are localized on the non-Abelian vortices thanks to a field-dependent gauge kinetic function. We perform fluctuation analysis with an appropriate gauge fixing and obtain a four-dimensional effective Lagrangian of unbroken and broken gauge fields at quadratic order. We find that SU(3) × SU(2) × U(1) gauge fields are localized on the vortices and exactly massless. Complications in analyzing the spectra of gauge fields with the nontrivial gauge kinetic function are neatly worked out by a vector-analysis-like method.

The moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory

We determine the dimension of the moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory in 2 + 1 dimensions for gauge groups with G′ being an arbitrary semi-simple group and n 0 the greatest common divisor of the abelian charges of the G′ invariants. The calculation is carried out using a Callias-type index theorem, the moduli matrix approach and a D-brane setup in type IIB string theory. We prove that the index theorem gives the number of zeromodes or moduli of the non-abelian vortices, extend the moduli matrix approach to the Yang–Mills–Chern–Simons–Higgs theory and finally derive the effective Lagrangian of Collie and Tong using string theory.

Topological confinement of vortices in two-flavor dense QCD

We find a novel confinement mechanism in the two-flavor dense quark matter proposed recently, that consists of the 2SC condensates and the P-wave diquark condensates of d-quarks. This quark matter exhibiting color superconductivity as well as superfluidity is classified into two phases; confined and deconfined phases of vortices. We establish that the criterion of the confinement is color neutrality of Aharonov-Bohm (AB) phases: vortices exhibiting color non-singlet AB phases are confined by the so-called AB defects to form color-singlet bound states. In the deconfined phase, the most stable vortices are non-Abelian Alice strings, which are superfluid vortices with fractional circulation and non-Abelian color magnetic fluxes therein, exhibiting color non-singlet AB phases. On the other hand, in the confined phase, these non-Abelian vortices are confined to either a baryonic or mesonic bound state in which constituent vortices are connected by AB defects. The baryonic bound state consists of three non-Abelian Alice strings with different color magnetic fluxes with the total flux canceled out connected by a domain wall junction, while the mesonic bound state consists of two non-Abelian Alice strings with the same color magnetic fluxes connected by a single domain wall. Interestingly, the latter contains a color magnetic flux in its core, but this can exist because of color neutrality of its AB phase.

Flux compactification for the critical non-Abelian vortex and quark masses

It has been shown that non-Abelian solitonic vortex strings supported in four-dimensional (4D) ${\mathcal N}=2\;$ supersymmetric QCD (SQCD) with the U($N=2$) gauge group and $N_f=4$ quark flavors behave as critical superstrings. In addition to four translational moduli non-Abelian strings under consideration carry six orientational and size moduli. Together they form a ten-dimensional space required for a superstring to be critical. The target space of the string sigma model is a product of the flat four-dimensional space $\mathbb{R}^4$ and a Calabi-Yau non-compact threefold $Y_6$, namely, the conifold. The spectrum of low lying closed string states in the associated type IIA string theory was found and interpreted as a spectrum of hadrons in 4D ${\mathcal N}=2\;$ SQCD. In particular, the lowest string state appears to be a massless BPS baryon associated with the deformation of the complex structure modulus $b$ of the conifold. Here we address a problem of switching on quark masses in 4D SQCD, which classically breaks the world sheet conformal invariance in the string sigma model. To avoid this problem we follow a standard string theory approach and use a flux "compactification" to lift the complex structure modulus of the conifold. Namely, we find a solution of supergravity equations of motion with non-zero NS 3-form flux. It produces a potential for the baryon $b$, which leads to the run-away vacuum. Using field theory arguments we interpret 3-form flux in terms of a particular choice of quark masses in 4D SQCD. At the run-away vacuum the conifold degenerates to lower dimensions. We interpret this as a flow from a non-Abelian string to an Abelian one.

Some remarks about emission channels of non-Abelian vortices

As is well established, several gauge theories admit vortices whose mean life time is very large. In some cases, this stability is a consequence of the topology of the symmetry group of the underlying theory. The main focus of the present work is, given a putative vortex, to determine if it is non abelian or not by analysis of its physical effects. The example considered here is the simplest one namely, a $SU(2)$ gauge model whose internal orientational space is described by $S^2$. Axion and gravitational emission are mainly considered. It is found that the non abelian property is basically reflected in a deviation of gravitational loop factor $\gamma_l$ found in \cite{vachaspati}-\cite{burden}. The axion emission instead, is not very sensitive to non abelianity, at least for this simple model. Another important discrepancy is that no point of the vortex reaches the speed of light when orientational modes are excited. In addition, the total power corresponding to each of these channels is compared, thus adapting the results of \cite{davis}-\cite{peloso} to the non abelian context. The excitations considered here are simple generalizations of rotating or spike string ansatz known in the literature \cite{ruso1}-\cite{kruczenski}. It is suggested however, that for certain type of semi-local strings whose internal moduli space is non compact, deviations due to non abelianity may be more pronounced.

Dynamics of global and local vortices with orientational moduli

The dynamics of both global and local vortices with non-Abelian orientational moduli is investigated in detail. Head-on collisions of these vortices are numerically simulated for parallel, anti-parallel and orthogonal internal orientations where we find interesting dynamics of the orientational moduli. A detailed study of the inter-vortex force is provided and a phase diagram separating Abelian and non-Abelian vortex types is constructed. Some results on scatterings with non-zero impact parameter and multi-vortex collisions are included.

Non-Abelian gauged fracton matter field theory: Sigma models, superfluids, and vortices

By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that between insulator v.s. superfluid/superconductivity (U(1) symmetry disordered phase described by a topological gauge theory or a disordered Sigma model v.s. U(1) global/gauge symmetry-breaking ordered phase described by a Sigma model with a U(1) target space underlying Goldstone modes): We can regard our tensor gauge theories as disordered phases, and we transient to their new ordered phases by deriving new Sigma models in continuum field theories. While one low energy theory is captured by degrees of freedom of rotor or scalar modes, another side of low energy theory has vortices and superfluids -- we explore non-abelian vortices (two types of vortices mutually interacting non-commutatively beyond an ordinary group structure) and their Cauchy-Riemann relation.

String baryon in four-dimensional N=2 supersymmetric QCD from the 2D-4D correspondence

We study non-Abelian vortex strings in four-dimensional (4D) $\mathcal{N} = 2$ supersymmetric QCD with U$(N=2)$ gauge group and $N_f=4$ flavors of quark hypermultiplets. It has been recently shown that these vortices behave as critical superstrings. The spectrum of closed string states in the associated string theory was found and interpreted as a spectrum of hadrons in 4D $\mathcal{N} = 2$ supersymmetric QCD. In particular, the lowest string state appears to be a massless BPS "baryon." Here we show the occurrence of this stringy baryon using a purely field-theoretic method. To this end we study the conformal world-sheet theory on the non-Abelian string -- the so called weighted $\mathcal{N} = (2,2)$ supersymmetric $\mathbb{CP}$ model. Its target space is given by the six-dimensional non-compact Calabi-Yau space $Y_6$, the conifold. We use mirror description of the model to study the BPS kink spectrum and its transformations on curves (walls) of marginal stability. Then we use the 2D-4D correspondence to show that the deformation of the complex structure of the conifold is associated with the emergence of a non-perturbative Higgs branch in 4D theory which opens up at strong coupling. The modulus parameter on this Higgs branch is the vacuum expectation value of the massless BPS "baryon" previously found in string theory.

Coexistence phase of S01 and P23 superfluids in neutron stars

In neutron star matter, there exist $^{1}S_{0}$ superfluids in lower density in the crust while $^{3}P_{2}$ superfluids are believed to exist at higher density deep inside the core. In the latter, depending on the temperature and magnetic field, either the uniaxial nematic phase, the D$_{2}$-biaxial nematic phase, or the D$_{4}$-biaxial nematic phase appears. In this paper, we discuss a mixture of the $^{1}S_{0}$ and $^{3}P_{2}$ superfluids and find their coexistence. Adopting the loop expansion and the weak-coupling approximation for the interaction between two neutrons, we obtain the Ginzburg-Landau (GL) free energy in which both of the $^{1}S_{0}$ and $^{3}P_{2}$ condensates are taken into account by including the coupling terms between them. We analyze the GL free energy and obtain the phase diagram for the temperature and magnetic field. We find that the $^{1}S_{0}$ superfluid excludes the $^{3}P_{2}$ superfluid completely in the absence of magnetic field, they can coexist for weak magnetic fields, and the $^{1}S_{0}$ superfluid is expelled by the $^{3}P_{2}$ superfluid at strong magnetic fields, thereby proving the robustness of $^{3}P_{2}$ superfluid against the magnetic field. We further show that the D$_{4}$-BN phase covers the whole region of the $^{3}P_{2}$ superfluidity as a result of the coupling term, in contrast to the case of a pure $^{3}P_{2}$ superfluid studied before in which the D$_{4}$-BN phase is realized only under strong magnetic fields. Thus, the D$_{4}$-BN phase is topologically the most interesting phase, e.g., admitting half-quantized non-Abelian vortices relevant not only in magnetars but also in ordinary neutron stars.