Nontrivial Topological Charge Research Articles

Localized Fermions on Superconducting Domain Walls and Extended Supersymmetry with Non-trivial Topological Charges

In this letter we demonstrate that the fermionic zero modes on a superconducting domain wall can be associated to an one dimensional \(N=6\) supersymmetry that contains non-trivial topological charges. In addition, the system also possesses three distinct \(N=4\) supersymmetries with non-trivial topological charges and we also study some duality transformations of the supersymmetric algebras.

Fermions in a Reissner–Nordström–anti-de Sitter black hole background and N = 4 supersymmetry with nontrivial topological charges

We demonstrate that the fermions in Reissner–Nordström–anti-de Sitter black hole background in the chiral limit m = 0, are related to an N = 4 extended one-dimensional supersymmetry with nontrivial topological charges. We also show that the N = 4 extended supersymmetry is unbroken and we extend the two N = 2 one-dimensional supersymmetries that the system also possesses, to have a trivial central charge. The implications of the trivial central charge on the Witten index are also discussed.

Superconducting cosmic strings and one dimensional extended supersymmetric algebras

In this article we study in detail the supersymmetric structures that underlie the system of fermionic zero modes around a superconducting cosmic string. Particularly, we extend the analysis existing in the literature on the one dimensional N=2 supersymmetry and we find multiple N=2, d=1 supersymmetries. In addition, compact perturbations of the Witten index of the system are performed and we find to which physical situations these perturbations correspond. More importantly, we demonstrate that there exists a much more rich supersymmetric structure underlying the system of fermions with Nf flavors and these are N-extended supersymmetric structures with non-trivial topological charges, with “N” depending on the fermion flavors.

Gauge-spin-space rotation-invariant vortices in spin-orbit-coupled Bose-Einstein condensates

We revisit ground states of spinor Bose-Einstein condensates with a Rashba spin-orbit coupling, and find that votices show up as a direct consequence of spontaneous symmetry breaking into a combined gauge, spin, and space rotation symmetry, which determines the vortex-core spin state at the rotating center. For the continuous combined symmetry, the total spin rotation about the rotating axis is restricted to $2\pi$, whereas for the discrete combined symmetry, we further need 2F quantum numbers to characterize the total spin rotation for the spin-$F$ system. For lattice phases we find that in the ground state the topological charge for each unit cell vanishes. However, we find two types of highly symmetric lattices with a nontrivial topological charge in the spin-$\frac{1}{2}$ system based on the symmetry classification, and show that they are skyrmion crystals.

Topological classification and stability of Fermi surfaces.

In the framework of the Cartan classification of Hamiltonians, a kind of topological classification of Fermi surfaces is established in terms of topological charges. The topological charge of a Fermi surface depends on its codimension and the class to which its Hamiltonian belongs. It is revealed that six types of topological charges exist, and they form two groups with respect to the chiral symmetry, with each group consisting of one original charge and two descendants. It is these nontrivial topological charges which lead to the robust topological protection of the corresponding Fermi surfaces against perturbations that preserve discrete symmetries.

Stable Vortex–Bright-Soliton Structures in Two-Component Bose-Einstein Condensates

We report the numerical realization of robust two-component structures in 2D and 3D Bose-Einstein condensates with nontrivial topological charge in one component. We identify a stable symbiotic state in which a higher-dimensional bright soliton exists even in a homogeneous setting with defocusing interactions, due to the effective potential created by a stable vortex in the other component. The resulting vortex-bright-solitons, generalizations of the recently experimentally observed dark-bright solitons, are found to be very robust both in the homogeneous medium and in the presence of external confinement.

Is a color superconductor topological?

A fully gapped state of matter, whether insulator or superconductor, can be asked if it is topologically trivial or nontrivial. Here we investigate topological properties of superconducting Dirac fermions in 3D having a color superconductor as an application. In the chiral limit, when the pairing gap is parity even, the right-handed and left-handed sectors of the free space Hamiltonian have nontrivial topological charges with opposite signs. Accordingly, a vortex line in the superconductor supports localized gapless right-handed and left-handed fermions with the dispersion relations E=+/-vp_z (v is a parameter dependent velocity) and thus propagating in opposite directions along the vortex line. However, the presence of the fermion mass immediately opens up a mass gap for such localized fermions and the dispersion relations become E=+/-v(m^2+p_z^2)^(1/2). When the pairing gap is parity odd, the situation is qualitatively different. The right-handed and left-handed sectors of the free space Hamiltonian in the chiral limit have nontrivial topological charges with the same sign and therefore the presence of the small fermion mass does not open up a mass gap for the fermions localized around the vortex line. When the fermion mass is increased further, there is a topological phase transition at m=(\mu^2+\Delta^2)^(1/2) and the localized gapless fermions disappear. We also elucidate the existence of gapless surface fermions localized at a boundary when two phases with different topological charges are connected. A part of our results is relevant to the color superconductivity of quarks.

Self-dual hopfions

We construct static and time-dependent exact soliton solutions with non-trivial Hopf topological charge for a field theory in 3+1 dimensions with the target space being the two dimensional sphere S**2. The model considered is a reduction of the so-called extended Skyrme-Faddeev theory by the removal of the quadratic term in derivatives of the fields. The solutions are constructed using an ansatz based on the conformal and target space symmetries. The solutions are said self-dual because they solve first order differential equations which together with some conditions on the coupling constants, imply the second order equations of motion. The solutions belong to a sub-sector of the theory with an infinite number of local conserved currents. The equation for the profile function of the ansatz corresponds to the Bogomolny equation for the sine-Gordon model.

Chiral magnetic effect at low temperature

We investigate the chiral magnetic effect (CME) under a strong magnetic field B = B_0 x_3 at low temperature T < T^chi_c. For this purpose, we employ the instanton vacuum configuration with the finite instanton-number fluctuation Delta, which relates to the nontrivial topological charge Q_t. We compute the vacuum expectation values of the local chiral density <rho_chi>, chiral charge density <n_chi> and induced electromagnetic current <j_mu>, which signal the CME, as functions of T and B_0. We observed that the longitudinal EM current is much larger than the transverse one, |j_perp/j_parallel| ~ Q_t, and the <n_chi> equals to the |<j_{3,4}>|. It also turns out that the CME becomes insensitive to the magnetic field as T increases, according to the decreasing instanton, i.e. tunneling effect. Within our framework, the instanton contribution to the CME becomes almost negligible beyond T~300 MeV.

Compact baby Skyrmions

For the baby Skyrme model with a specific potential, compacton solutions, i.e., configurations with a compact support and parabolic approach to the vacuum, are derived. Specifically, in the nontopological sector, we find spinning $Q$-balls and $Q$-shells, as well as peakons. Moreover, we obtain compact baby skyrmions with nontrivial topological charge. All these solutions may form stable multisoliton configurations provided they are sufficiently separated.

Pullback of the volume form, integrable models in higher dimensions and exotic textures

A procedure allowing for the construction of Lorentz invariant integrable models living in d+1 dimensional space time and with an n dimensional target space is provided. Here, integrability is understood as the existence of the generalized zero curvature formulation and infinitely many conserved quantities. A close relation between the Lagrange density of the integrable models and the pullback of the pertinent volume form on target space is established. Moreover, we show that the conserved currents are Noether currents generated by the volume-preserving diffeomorphisms. Further, we show how such models may emerge via Abelian projection of some gauge theories. Then we apply this framework to the construction of integrable models with exotic textures. Particularly, we consider integrable models providing exact suspended Hopf maps, i.e., solitons with a nontrivial topological charge of π4(S3)≅Z2. Finally, some families of integrable models with solitons of πn(Sn) type are constructed. Infinitely many exact solutions with arbitrary value of the topological index are found. In addition, we demonstrate that they saturate a Bogomolny bound.

Spinning Hopf solitons onS3× Bbb R

We consider a field theory with target space being the two dimensional sphere S^2 and defined on the space-time S^3 x R. The Lagrangean is the square of the pull-back of the area form on S^2. It is invariant under the conformal group SO(4,2) and the infinite dimensional group of area preserving diffeomorphisms of S^2. We construct an infinite number of exact soliton solutions with non-trivial Hopf topological charges. The solutions spin with a frequency which is bounded above by a quantity proportional to the inverse of the radius of S^3. The construction of the solutions is made possible by an ansatz which explores the conformal symmetry and a U(1) subgroup of the area preserving diffeomorphism group.

Exact time dependent Hopf solitons in 3+1 dimensions

We construct an infinite number of exact time dependent soliton solutions, carrying non-trivial Hopf topological charges, in a 3+1 dimensional Lorentz invariant theory with target space S^2. The construction is based on an ansatz which explores the invariance of the model under the conformal group SO(4,2) and the infinite dimensional group of area preserving diffeomorphisms of S^2. The model is a rare example of an integrable theory in four dimensions, and the solitons may play a role in the low energy limit of gauge theories.

Remnant index theorem and low-lying eigenmodes for twisted mass fermions

We analyze the low-lying spectrum and eigenmodes of lattice Dirac operators with a twisted mass term. The twist term expels the eigenvalues from a strip in the complex plane and all eigenmodes obtain a non-vanishing matrix element with γ5. For a twisted Ginsparg–Wilson operator the spectrum is located on two arcs in the complex plane. Modes due to non-trivial topological charge of the underlying gauge field have their eigenvalues at the edges of these arcs and obey a remnant index theorem. For configurations in the confined phase we find that the twist mainly affects the zero modes, while the bulk of the spectrum is essentially unchanged.

A model for Hopfions on the space–time S3×R

We construct static and time dependent exact soliton solutions for a theory of scalar fields taking values on a wide class of two dimensional target spaces, and defined on the four dimensional space–time S3×R. The construction is based on an ansatz built out of special coordinates on S3. The requirement for finite energy introduce boundary conditions that determine an infinite discrete spectrum of frequencies for the oscillating solutions. For the case where the target space is the sphere S2, we obtain static soliton solutions with nontrivial Hopf topological charges. In addition, such Hopfions can oscillate in time, preserving their topological Hopf charge, with any of the frequencies belonging to that infinite discrete spectrum.

Non-linear σ-models in noncommutative geometry: fields with values in finite spaces

We study σ-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space [Formula: see text].

Instantons in the saturation environment

We show that instanton calculations in QCD become theoretically well defined in the gluon saturation environment which suppresses large-size instantons. The effective cutoff scale is determined by the inverse of the saturation scale. We concentrate on two most important cases: the small- x tail of a gluon distribution of a high-energy hadron or a large nucleus and the central rapidity region in a high-energy hadronic or heavy-ion collision. In the saturation regime the gluon density in a single large ultrarelativistic nucleus is high and gluonic fields are given by the classical solutions of the equations of motion. We show that these strong classical fields do not affect the density of instantons in the nuclear wave function compared to the instanton density in the vacuum. A classical solution with nontrivial topological charge is found for the gluon field of a single nucleus at the lowest order in the instanton perturbation theory. In the case of ultrarelativistic heavy-ion collisions a strong classical gluonic field is produced in the central rapidity region. We demonstrate that this field introduces a suppression factor of exp{− cρ 4 Q s 4/[8 α s 2 N c( Q s τ) 2]} in the instanton-size distribution, where Q s is the saturation scale of both (identical) nuclei, τ is the proper time and c≈1 is the gluon liberation coefficient. This factor suggests that gluonic saturation effects at the early stages of nuclear collisions regulate the instanton-size distribution in the infrared region and make the instanton density finite by suppressing large-size instantons.

Dyonic integrable models

A class of non-abelian affine Toda models arising from the axial gauged two-loop WZW model is presented. Their zero curvature representation is constructed in terms of a graded Kac–Moody algebra. It is shown that the discrete multivacua structure of the potential together with non-abelian nature of the zero grade subalgebra allows soliton solutions with non-trivial electric and topological charges. The dressing transformation is employed to explicitly construct one and two soliton solutions and their bound states in terms of the tau functions. A discussion of the classical spectra of such solutions and the time delays are given in detail.

New instanton solutions at finite temperature

We discuss the newly found exact instanton solutions at finite temperature with a nontrivial Polyakov loop at infinity. They can be described in terms of monopole constituents and we discuss in this context an old result due to Taubes how to make out of monopoles configurations with non-trivial topological charge, with possible applications to abelian projection.

Quantum tunneling for the sine-Gordon potential: Energy band structure and Bogomolny-Fateyev relation.

A path-integral calculation of the energy band structure for the sine-Gordon potential with the help of the instanton method is presented. The periodic pseudoparticle configuration obtained recently is seen to be responsible for tunneling at a finite energy which leads to the level splitting of excited states. The lowest band structure due to the tunneling of vacuum instantons is recovered from our result in the low energy limit. The above result is obtained by considering a half period of the classical solution as a kinklike configuration (with nontrivial topological charge). On the other hand if a full period of the solution is treated as a nontopological (bouncelike) pseudoparticle configuration, the imaginary parts of energy eigenvalues can be calculated. We then obtain the Bogomolny-Fateyev relation which is well known in perturbation theory at large order. The agreement of our results with those of WKB calculations also resolves a controversy concerning the method: In fact this method yields the same correct result as the method of complex paths.