Half-quantum Vortices Research Articles

Numerical study of the stability regions for half-quantum vortices in superconducting Sr2RuO4

We numerically solve the coupled Landau-Ginzburg-Maxwell equations for a model of a spin triplet px + ipy superconductor in which whole or half-quanta of flux thread through a hole. We recover the pattern of stable and unstable regions for the half-flux quanta observed in a recent experiment.

Half-Fluxon on Long Josephson Junction with Chiral Superconductors

A half-fluxon on a long Josephson junction with a chiral p-wave superconductor is studied. The half-fluxon is similar to the half quantum vortex (half-fluxoid) proposed by Volovik and Mineev in that the half quantum vortex results from the cooperation between two kinds of phases provided by a superconducting order parameter: one is a global phase related to the symmetry \(U(1)\) and the other is the rotation phase of a \(\hat{\mathbf{d}}\) vector related to the symmetry \(\hat{D}^{(+)}_{(1)}\) in spin space. Each phase for the half-fluxon varies with \(\pi\) along a long Josephson junction. In contrast, each phase varies with \(2\pi(n+1/2)\) along a half-fluxoid. A half-fluxon and a half-fluxoid are accompanied by a spin current, which is induced by the spatial modulation of a \(\hat{\mathbf{d}}\) vector. This half-fluxon is different from a half-fluxon on a Josephson 0–\(\pi\)-junction in terms of the spin current. A spin current is parallel or antiparallel to a supercurrent. Therefore, the spin current ...

Mirror Majorana zero modes in spinful superconductors/superfluids Non-Abelian anyons in integer quantum vortices

It has been widely believed that half quantum vortices are indispensable to realize topological stable Majorana zero modes and non-Abelian anyons in spinful superconductors/superfluids. Contrary to this wisdom, we here demonstrate that integer quantum vortices in spinful superconductors can host topologically stable Majorana zero modes because of the mirror symmetry. The symmetry protected Majorana fermions may exhibit non-Abelian anyon braiding.

Disclination Classes, Fractional Excitations, and the Melting of Quantum Liquid Crystals

We consider how fractional excitations bound to a dislocation evolve as the dislocation is separated into a pair of disclinations. We show that some dislocation-bound excitations (such as Majorana modes and half-quantum vortices) are possible only if the elementary dislocation consists of two inequivalent disclinations, as is the case for stripes or square lattices but not for triangular lattices. The existence of multiple inequivalent disclination classes governs the two-dimensional melting of quantum liquid crystals (i.e., nematics and hexatics), determining whether superfluidity and orientational order can simultaneously vanish at a continuous transition.

Ground states, solitons and spin textures in spin-1 Bose-Einstein condensates

We present an overview of our recent theoretical studies on the quantum phenomena of the spin-1 Bose-Einstein condensates, including the phase diagram, soliton solutions and the formation of the topological spin textures. A brief exploration of the effects of spin-orbit coupling on the ground-state properties is given. We put forward proposals by using the transmission spectra of an optical cavity to probe the quantum ground states: the ferromagnetic and polar phases. Quasi-one-dimension solitons and ring dark solitons are studied. It is predicted that characteristics of the magnetic solitons in optical lattice can be tuned by controlling the long-range light-induced and static magnetic dipoledipole interactions; solutions of single-component magnetic and single-, two-, three-components polar solitons are found; ring dark solitons in spin-1 condensates are predicted to live longer lifetimes than that in their scalar counterparts. In the formation of spin textures, we have considered the theoretical model of a rapidly quenched and fast rotating trapped spin-1 Bose-Einstein condensate, whose dynamics can be studied by solving the stochastic projected Gross-Pitaevskii equations. Spontaneous generation of nontrivial topological defects, such as the hexagonal lattice skyrmions and square lattice of half-quantized vortices was predicted. In particular, crystallization of merons (half skyrmions) can be generated in the presence of spin-orbit coupling.

Half-flux quanta and vortex-antivortex pairs in two-band superconductors

Vortex states of two-band superconductors are investigated based on the microscopic Ginzburg-Landau model. We show that the nonlocal interband coupling can lead to the breaking of translation symmetry and the axial rotation symmetry of the conventional Abrikosov vortex states, resulting in a novel vortex structure comprising a mixture of triangle and square-like vortex lattices that have no counterparts in the single-band superconductors. Half-quantized vortices and stable vortex-antivortex pairs are found. These results may relate to the experimental observations of anomalous vortex structures in the two-band magnesium diboride superconductor.

Topological interface physics of defects and textures in spinor Bose-Einstein condensates

We provide a detailed description of our previously proposed scheme for topological interface engineering with constructed defects and textures perforating across coherent interfaces between different broken symmetries [Borgh and Ruostekoski, Phys. Rev. Lett. 109, 015302 (2012)]. We consider a spin-1 Bose-Einstein condensate, in which polar and ferromagnetic phases are prepared in spatially separated regions. We show that a stable coherent interface is established between the two phases, allowing defects of different topology to connect continuously across the boundary. We provide analytic constructions of interface-crossing defect solutions that could be experimentally phase imprinted using existing technology. By numerically minimizing the energy, we calculate the core structures of interface-crossing defect configurations. We demonstrate nontrivial core deformations to considerably more complex structures, such as the formation of an arch-shaped half-quantum line defect, an Alice arch, at the interface, with the topological charge of a point defect, whose emergence may be understood by the ``hairy ball'' theorem. Another example of an energetically stable object is the connection of a coreless vortex to a pair of half-quantum vortices. We show that rotation leads to spontaneous nucleation of defects in which a coreless vortex continuously transforms to a half-quantum vortex across the interface.

Appearance of Magnetization around a Pair of Half-Quantum Vortices in Chiralp-Wave Superconductors

We have investigated quasi-particle excitations around a pair of half-quantum vortices in chiral \(p\)-wave superconductors, using the Bogoliubov–de Gennes equations. We use the elliptic coordinates and the Mathieu function for numerical calculations. We found that the height of peak of a local density of states of the bound states for the half-quantum vortex core is a half of that for a singly quantized vortex core. Also, it was found that a magnetization appears along a pair of half-quantum vortices. Existence of half-quantum vortex might be confirmed by observation of this distribution of the magnetization.

Effect of thermal fluctuations in topologicalp-wave superconductors

We study the effect of thermal fluctuations on the topological stability of chiral p-wave superconductors. We consider two models of superconductors: spinless and spinful with a focus on topological properties and Majorana zero-energy modes. We show that proliferation of vortex-antivortex pairs above the Kosterlitz-Thouless temperature T_KT drives the transition from a thermal Quantum Hall insulator to a thermal metal/insulator, and dramatically modifies the ground-state degeneracy splitting. Therefore, in order to utilize 2D chiral p-wave superconductors for topological quantum computing, the temperature should be much smaller than T_KT. Within the spinful chiral p-wave model, we also investigate the interplay between half-quantum vortices carrying Majorana zero-energy modes and full-quantum vortices having trivial topological charge, and discuss topological properties of half-quantum vortices in the background of proliferating full-quantum vortices.

Energy of Stable Half-Quantum Vortex in Equal-Spin-Pairing

In the triplet equal-spin-pairing states of both 3He — A phase and Sr2RuO4 superconductor, existence of Half-Quantum Vortices (HQVs) are possible. The vortices carry half-integer multiples of magnetic quantum flux Φ0 = hc/2e. Our method is based on the explanation of the HQV in terms of a BCS—like wave function with a spin-dependent boots. To obtain equilibrium condition for such systems, one has to take into account not only weak interaction energy but also effects of Landau Fermi liquid. We have considered ℓ = 2 order effects of the Landau Fermi liquid. We have shown that the effects of Landau Fermi liquid interaction with ℓ = 2 are negligible. In stable HQV, an effective Zeeman field exists. In the thermodynamic stability state, the effective Zeeman field produces a non-zero spin polarization in addition to the polarization of external magnetic field.

Spin Diffusion Coefficient of A Phase of Liquid 3He at Low Temperatures and Stability of Half Quantum Vortex

We theoretically investigate the spin diffusion coefficient tensor in the A phase of liquid 3He in term of quasiparticle life-time by using the Kubo formula approach at low temperatures. In general, the coefficient is a fourth rank tensor for the anisotropic states and can be defined as a function of both normal component of spin-current and magnetization. The quasiparticle life-time is obtained by using the Boltzmann equation. We find that components of the spin diffusion coefficient are proportional to T−2 at low temperatures. The normal components of spin current, hence, are strongly diffusive and one can ignore the contribution of these components to the stability of half quantum vortices (HQVs) in the equal-spin-pairing of 3He — A state. Hence to make a stable HQV, it is enough for one to consider weak interaction plus the effects of Landau Fermi liquid.

Quasi-particle structures around a pair of half-quantum vortices in chiral p-wave super conductors

Quasi-particle structures around a pair of half-quantum vortices (HQVs) and two singly quantized vortices are investigated, using the Bogoliubov-de Gennes equation. For this purpose, a new numerical method, which incorporates the two phase singularity points, is developed, using elliptic coordinates and the (modified) Mathieu functions. We investigate the local density of states around two singly quantized vortices and a pair of HQVs in chiral p-wave superconductors. As a result, we found that there is a difference between them: Quasiparticle bound states interfere with each other for two singly quantized vortices case and do not for a pair of HQVs case.

Crossover from Fulde-Ferrell state to Larkin-Ovchinnikov state in cold fermion gases

We study an effect of rotation on the two component Fermi superfluid gases with population imbalance in a toroidal trap. We investigate how the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is changed by the rotation using the Bogoliubov-de Gennes equation in the quasi-one-dimensional regime. We find that two novel FFLO phases, i.e., the half quantum vortex state and the intermediate state of Fulde-Ferrell (FF) state and Larkin-Ovchinnikov (LO) state, are stabilized by the rotation. The phase diagram for the FF state, LO state, intermediate state, and half quantum vortex state is shown in T-Φ plane. We show the order parameter of these states.

Majorana Fermions Bound at Vortices and Surface of Superfluid 3He

We investigate the Majorana fermions bound at vortices and specular surface of superfluid 3He A- and B-phases. It is demonstrated that the statistics of the integer vortex which is competitive to the half-quantum vortex is sensitive to the orientation of the magnetic field. We also discuss the surface Andreev bound state in the B-phase, which is expected to behave as Majorana fermions at the zero field limit. Based on the quasiclassical and Bogoliubov-de Gennes theories, we clarify that in the presence of the magnetic field parallel to the surface, the stable configuration of the -vector, which reflects the spin-orbit symmetry breaking, traces the point at which the surface state has the maximum gap.

Half-Quantum Vortices in Thin Film of Superfluid3He

Stability of a half-quantum vortex (HQV) in superfluid $^3$He has been discussed recently by Kawakami, Tsutsumi and Machida in Phys. Rev. B {\bf 79}, 092506 (2009). We further extend this work here and consider the A$_2$ phase of superfluid $^3$He confined in thin slab geometry and analyze the HQV realized in this setting. Solutions of HQV and singly quantized singular vortex are evaluated numerically by solving the Ginzburg-Landau (GL) equation and respective first critical angular velocities are obtained by employing these solutions. We show that the HQV in the A$_2$ phase is stable near the boundary between the A$_2$ and A$_1$ phases. It is found that temperature and magnetic field must be fixed first in the stable region and subsequently the angular velocity of the system should be increased from zero to a sufficiently large value to create a HQV with sufficiently large probability. A HQV does not form if the system starts with a fixed angular velocity and subsequently the temperature is lowered down to the A$_2$ phase. It is estimated that the external magnetic field with strength on the order of 1 T is required to have a sufficiently large domain in the temperature-magnetic field phase diagram to have a stable HQV.

Skyrmionic state and stable half-quantum vortices in chiralp-wave superconductors

Observability of half-quantum vortices and skyrmions in $p$-wave superconductors is an outstanding open question. Under the most common conditions, fractional flux vortices are not thermodynamically stable in bulk samples. Here we show that in chiral $p$-wave superconductors, there is a regime where, in contrast, lattices of integer-flux vortices are not thermodynamically stable. Instead, skyrmions made of spatially separated half-quantum vortices are the topological defects produced by an applied external field.

Energetically stable singular vortex cores in an atomic spin-1 Bose-Einstein condensate

We analyze the structure and stability of singular singly quantized vortices in a rotating spin-1 Bose-Einstein condensate. We show that the singular vortex can be energetically stable in both the ferromagnetic and polar phases despite the existence of a lower-energy nonsingular coreless vortex in the ferromagnetic phase. The spin-1 system exhibits an energetic hierarchy of length scales resulting from different interaction strengths and we find that the vortex cores deform to a larger size determined by the characteristic length scale of the spin-dependent interaction. We show that in the ferromagnetic phase the resulting stable core structure, despite apparent complexity, can be identified as a single polar core with axially symmetric density profile which is nonvanishing everywhere. In the polar phase, the energetically favored core deformation leads to a splitting of a singly quantized vortex into a pair of half-quantum vortices that preserves the topology of the vortex outside the extended core region, but breaks the axial symmetry of the core. The resulting half-quantum vortices exhibit nonvanishing ferromagnetic cores.

Half-quantum vortex state in a spin-orbit-coupled Bose-Einstein condensate

We investigate theoretically the condensate state and collective excitations of a two-component Bose gas in two-dimensional harmonic traps subject to isotropic Rashba spin-orbit coupling. In the weakly interacting regime when the inter-species interaction is larger than the intra-species interaction ($g_{\uparrow\downarrow}>g$), we find that the condensate ground state has a half-quantum-angular-momentum vortex configuration with spatial rotational symmetry and skyrmion-type spin texture. Upon increasing the interatomic interaction beyond a threshold $g_{c}$, the ground state starts to involve higher-order angular momentum components and thus breaks the rotational symmetry. In the case of $g_{\uparrow\downarrow}<g$, the condensate becomes unstable towards the superposition of two degenerate half-quantum vortex states. Both instabilities (at $g>g_{c}$ and $g_{\uparrow\downarrow}<g$) can be determined by solving the Bogoliubov equations for collective density oscillations of the half-quantum vortex state, and by analyzing the softening of mode frequencies. We present the phase diagram as functions of the interatomic interactions and the spin-orbit coupling. In addition, we directly simulate the time-dependent Gross-Pitaevskii equation to examine the dynamical properties of the system. Finally, we investigate the stability of the half-quantum vortex state against both the trap anisotropy and anisotropy in the spin-orbit coupling term.

Observation of Topologically Stable 2D Skyrmions in an Antiferromagnetic Spinor Bose-Einstein Condensate

We present the creation and time evolution of two-dimensional Skyrmion excitations in an antiferromagnetic spinor Bose-Einstein condensate. Using a spin rotation method, the Skyrmion spin textures were imprinted on a sodium condensate in a polar phase, where the two-dimensional Skyrmion is topologically protected. The Skyrmion was observed to be stable on a short time scale of a few tens of ms but to dynamically deform its shape and eventually decay to a uniform spin texture. The deformed spin textures reveal that the decay dynamics involves breaking the polar phase inside the condensate without having topological charge density flow through the boundary of the finite-sized sample. We discuss the possible formation of half-quantum vortices in the deformation process.

Vortex configuration and vortex–vortex interaction in nano-structured superconductors

We study the vortex structures and quasi-particle structures in nano-structured superconductors. We used the Bogoliubov–de Gennes equation and the finite element method and obtained stable magnetic flux structures and the quasi-particle states. We found the vortex configurations are affected by the interference of the quasi-particle bound states around the vortices. In order to clarify the interference between the quasi-particle wave-functions around two vortices we have developed a numerical method using the elliptic coordinates and the Mathieu functions. We apply this method to two singly quantized vortex state in a conventional s-wave superconductor and a pair of half-quantum vortices in a chiral p-wave superconductor.