Singular Gauge Transformation Research Articles

Index theorem on magnetized blow-up manifold of T2/ZN

We investigate blow-up manifolds of ${T}^{2}/{\mathbb{Z}}_{N}(N=2,3,4,6)$ orbifolds with magnetic flux. First, we construct the blow-up manifolds and zero-mode wave functions on them more precisely. In particular, through an appropriate singular gauge transformation, winding numbers of wave functions on ${T}^{2}/{\mathbb{Z}}_{N}$ can be replaced with localized curvature and localized flux at orbifold fixed points. In addition, since the blow-up manifolds have no singularities, we apply the Atiyah-Singer index theorem to them; the chiral zero-mode number is given by the total magnetic flux. It can be also applied for ${T}^{2}/{\mathbb{Z}}_{N}$ orbifolds through the blow-up process, and then we find that it is consistent with the zero-mode counting formula in M. Sakamoto et al. [Zero-mode counting formula and zeros in orbifold compactifications, Phys. Rev. D 102, 025008 (2020).]. Furthermore, the Atiyah-Singer index theorem shows that an additional degree of freedom of localized flux gives new chiral zero modes. We study their wave functions and then we find that they correspond to localized modes at the orbifold singular points. We also calculate their Yukawa couplings.

Majorana bound states in vortex lattices on iron-based superconductors

Majorana quasi-particles may arise as zero-energy bound states in vortices on the surface of a topological insulator that is proximitized by a conventional superconductor. Such a system finds its natural realization in the iron-based superconductor FeTe0.55Se0.45 that combines bulk s-wave pairing with spin helical Dirac surface states, and which thus comprises the ingredients for Majorana modes in absence of an additional proximitizing superconductor. In this work, we investigate the emergence of Majorana vortex modes and lattices in such materials depending on parameters like the magnetic field strength and vortex lattice disorder. A simple 2D square lattice model here allows us to capture the basic physics of the underlying materials system. To address the problem of disordered vortex lattice, which occurs in real systems, we adopt the technique of the singular gauge transformation which we modify such that it can be used in a system with periodic boundary conditions. This approach allows us to go to larger vortex lattices than otherwise accessible, and is successful in replicating several experimental observations of Majorana vortex bound states in the FeTe0.55Se0.45 platform. Finally it can be related to a simple disordered Majorana lattice model that should be useful for further investigations on the role of interactions, and toward topological quantum computation.

Chern-Simons Origin of Superstring Integrability.

We derive the AdS_{5}×S^{5} Green-Schwarz superstring from four-dimensional Beltrami-Chern-Simons theory reduced on a manifold with singular boundary conditions. In this construction, the Lax connection and spectral parameter of the integrable superstring have a simple geometric origin in four dimensions as gauge connection and reduction coordinate. κ symmetry arises as a certain class of singular gauge transformations, while the world-sheet metric comes from complex-structure-changing Beltrami differentials. Our approach offers the possibility of investigating integrable holography using traditional field theory methods.

SO(5) Landau models and nested Nambu matrix geometry

The SO(5) Landau model is the mathematical platform of the 4D quantum Hall effect and provide a rare opportunity for a physical realization of the fuzzy four-sphere. We present an integrated analysis of the SO(5) Landau models and the associated matrix geometries through the Landau level projection. With the SO(5) monopole harmonics, we explicitly derive matrix geometry of a four-sphere in any Landau level: In the lowest Landau level the matrix coordinates are given by the generalized SO(5) gamma matrices of the fuzzy four-sphere satisfying the quantum Nambu algebra, while in higher Landau level the matrix geometry becomes a nested fuzzy structure realizing a pure quantum geometry with no counterpart in classical geometry. The internal fuzzy geometry structure is discussed in the view of an SO(4) Pauli-Schrödinger model and the SO(4) Landau model, where we unveil a hidden singular gauge transformation between their background non-Abelian field configurations. Relativistic versions of the SO(5) Landau model are also investigated and relationship to the Berezin-Toeplitz quantization is clarified. We finally discuss the matrix geometry of the Landau models in even higher dimensions.

Vortices and magnetic impurities

Ginzburg-Landau vortices in superconductors attract or repel depending on whether the value of the coupling constant is less than 1 or larger than 1. At critical coupling it was previously observed that a strongly localised magnetic impurity behaves very similarly to a vortex. This remains true for axially symmetric configurations away from critical coupling. In particular, a delta function impurity of a suitable strength is related to a vortex configuration without impurity by singular gauge transformation. However, the interaction of vortices and impurities is more subtle and depends not only on the coupling constant and the impurity strength, but also on how broad the impurity is. Furthermore, the interaction typically depends on the distance and may be attractive at short distances and repulsive at long distances. Numerical simulations confirm moduli space approximation results for the scattering of one and two vortices with an impurity. However, a double vortex will split up when scattering with an impurity, and the direction of the split depends on the sign of the impurity. Head-on collisions of a single vortex with different impurities away from critical coupling is also briefly discussed.

Magnetized orbifolds and localized flux

Magnetized orbifolds play an important role in compactifications of string theories and higher-dimensional field theories to four dimensions. Magnetic flux leads to chiral fermions, it can be a source of supersymmetry breaking and it is an important ingredient of moduli stabilization. Flux quantization on orbifolds is subtle due to the orbifold singularities. Generically, Wilson line integrals around these singularities are non-trivial, which can be interpreted as localized flux. As a consequence, flux densities on orbifolds can take the same values as on tori. We determine the transition functions for the flux vector bundle on the orbifold T2∕Z2 and the related twisted boundary conditions of zero-mode wave functions. We also construct “untwisted” zero-mode functions that are obtained for singular vector fields related to the Green’s function on a torus, and we discuss the connection between zeros of the wave functions and localized flux. Twisted and untwisted zero-mode functions are related by a singular gauge transformation.

Singular gauge transformation and the Erler–Maccaferri solution in bosonic open string field theory

We study candidates of the multiple-brane solutions of bosonic open string field theory. They are constructed by performing a singular gauge transformation $n$ times for the Erler-Maccaferri solution. We check the EOM in the strong sense, and find that it is satisfied only when we perform the gauge transformation once. We calculate the energy for that case and obtain a support that the solution is a multiple-brane solution. We also check the tachyon profile for a specific solution which we interpret as describing a D24-brane placed on a D25-brane.

Comments on lump solutions in SFT

We analyze a recently proposed scheme to construct analytic lump solutions in open SFT. We argue that in order for the scheme to be operative and to guarantee background independence it must be implemented in the same 2D conformal field theory in which SFT is formulated. We outline and discuss two different possible approaches. Next we reconsider an older proposal for analytic lump solutions and implement a few improvements. In the course of the analysis we formulate a distinction between regular and singular gauge transformations and advocate the necessity of defining a topology in the space of string fields.

20pAK-8 Singular Gauge Transformation and the Erler-Maccaferri Solution in Bosonic Open SFT

20pAK-8 Singular Gauge Transformation and the Erler-Maccaferri Solution in Bosonic Open SFT

The phantom term in open string field theory

We show that given any two classical solutions in open string field theory and a singular gauge transformation relating them, it is possible to write the second solution as a gauge transformation of the first plus a singular, projector-like state which describes the shift in the open string background between the two solutions. This is the "phantom term." We give some applications in the computation of gauge invariant observables.

Connecting solutions in open string field theory with singular gauge transformations

We show that any pair of classical solutions of open string field theory can be related by a formal gauge transformation defined by a gauge parameter U without an inverse. We investigate how this observation can be used to construct new solutions. We find that a choice of gauge parameter consistently generates a new solution only if the BRST charge maps the image of U into itself. When this occurs, we argue that U naturally defines a star algebra projector which describes a surface of string connecting the boundary conformal field theories of the classical solutions related by U. We also note that singular gauge transformations give the solution space of open string field theory the structure of a category, and we comment on the physical interpretation of this observation.

Degenerate Sklyanin algebras

Abstract New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.

FRACTIONAL QUANTUM HALL STATES IN GRAPHENE

We quantum mechanically analyze the fractional quantum Hall effect in graphene. This will be done by building the corresponding states in terms of a potential governing the interactions and discussing other issues. More precisely, we consider a system of particles in the presence of an external magnetic field and take into account of a specific interaction that captures the basic features of the Laughlin series [Formula: see text]. We show that how its Laughlin potential can be generalized to deal with the composite fermions in graphene. To give a concrete example, we consider the SU(N) wavefunctions and give a realization of the composite fermion filling factor. All these results will be obtained by generalizing the mapping between the Pauli–Schrödinger and Dirac Hamiltonian's to the interacting particle case. Meantime by making use of a gauge transformation, we establish a relation between the free and interacting Dirac operators. This shows that the involved interaction can actually be generated from a singular gauge transformation.

Dirac-Bogoliubov-deGennes quasiparticles in a vortex lattice

In the mixed state of an extreme type-II $d$-wave superconductor and within a broad regime of weak magnetic fields $({H}_{c1}⪡H⪡{H}_{c2})$, the low-energy Bogoliubov-deGennes quasiparticles can be effectively described as Dirac fermions moving in the field of singular scalar and vector potentials. Although the effective linearized Hamiltonian operator does not formally depend on the structure of vortex cores, a singular nature of the perturbation requires choosing a self-adjoint extension of the Hamiltonian by imposing additional boundary conditions at vortex locations. Each vortex is described by a single parameter $\ensuremath{\theta}$ that effectively represents all effects arising from the physics beyond linearization. With the value of $\ensuremath{\theta}$ properly fixed, the resulting density of states of Dirac Hamiltonian exhibits full invariance under arbitrary singular gauge transformations applied at vortex positions. We identify the self-adjoint extensions of the solutions found earlier, within the framework of the linearized Hamiltonian diagonalized by expansion in the plane wave basis, and analyze the relation between fully self-consistent formulation of the problem and the linearized model. In particular, we construct the low-field scaling form of the nodal quasiparticle spectra, which incorporates the self-adjoint extension parameter $\ensuremath{\theta}$ explicitly and generalizes the conventional Simon-Lee scaling. In a companion paper, we also present a detailed numerical study of the lattice $d$-wave superconductor model and examine its low-energy, low-magnetic-field behavior with an eye on determining the proper self-adjoint extension(s) of the linearized continuum limit. In general, we find that the density of quasiparticle states always vanishes at the chemical potential, either linearly or by virtue of a finite gap. The low-energy continuum limit is thus faithfully represented by Dirac-like fermions, which are either truly massless, massless at the linearized level (mass $\ensuremath{\sim}H$), or massive (mass $\ensuremath{\sim}{H}^{1∕2}$), depending on the mutual commensuration of magnetic length and lattice spacing.

Effective action for phase fluctuations in d-wave superconductors near a Mott transition

Phase fluctuations of a d-wave superconducting order parameter are theoretically studied in the context of high-T$_c$ cuprates. We consider the $t-J$ model describing layered compounds, where the Heisenberg interaction is decoupled by a d-wave order parameter in the particle-particle channel. Assuming first that the equilibirum state has long-range phase order, the effective action $\mathcal{S}_{eff}$ is derived perturbatively for small fluctuations within a path integral formalism, in the presence of the Coulomb and Hubbard interaction terms. In a second step, a more general derivation of $\mathcal{S}_{eff}$ is performed in terms of a gradient expansion which only assumes that the gradients of the order parameter are small whereas the value of the phase may be large. We show that in the phase-only approximation the resulting $\mathcal{S}_{eff}$ reduces in leading order in the field gradients to the perturbative one which thus allows to treat also the case without long-range phase order or vortices. Our result generalizes previous expressions for $\mathcal{S}_{eff}$ to the case of interacting electrons, is explicitly gauge invariant, and avoids problematic singular gauge transformations.

On different actions for the vacuum of bosonic string field theory

We study a family of kinetic operators in string field theory describing the theory around the closed string vacuum. Those operators are based on the analytical classical solutions of Takahashi and Tanimoto and are analogous to the pure ghost action usually referred to as "vacuum string field theory," but are much more general, and less singular than the pure ghost operator. The closed string vacuum is related to the D-brane vacuum by large, singular, gauge transformations or field redefinition, and all those different representations are related to each other by small gauge transformations. We try to clarify the nature of this singular gauge transformation. We also show that by choosing the Siegel gauge one recovers the propagator proposed in hep-th/0207266 that generates closed string surfaces.

Nonperturbative solution for BLOCH electrons in constant magnetic fields.

A general theoretical approach for the nonperturbative Bloch solution of Schrödinger's equation in the presence of a constant magnetic field is presented. Using a singular gauge transformation based on a lattice of magnetic flux lines, an equivalent quantum system with a periodic vector potential is obtained. For rational magnetic fields this system forms a magnetic superlattice for which Bloch's theorem then applies. Extensions of the approach to particles with spin and many-body systems and connections to the theory of magnetic translation groups are discussed.

D-wave quasiparticles in the tilted vortex lattice

We investigate the quasiparticle spectrum of a d-wave superconductor in the mixed state, focussing on the effects of varying the in-plane angular alignment (or tilting) of the vortex and crystal lattices. Our approach starts with a linearized Bogoliubov--de Gennes equation cast into a simple form by the Franz-Te\ifmmode \check{s}\else \v{s}\fi{}anovi\ifmmode \acute{c}\else \'{c}\fi{} singular gauge transformation. The zero-magnetic-field spectrum is characterized by the anisotropy ${\ensuremath{\alpha}}_{D}$ near the four Fermi-surface nodes; at large ${\ensuremath{\alpha}}_{D}$ the model simplifies into a one-dimensional form that provides asymptotic results. Numerical and analytical results are presented and compared with previous studies. We find that tilting can be understood in terms of the one-dimensional model at large ${\ensuremath{\alpha}}_{D}.$ We see a striking dependence on the choice of singular gauge, consistent with earlier results, but do not attempt to resolve the issue of which gauge best describes the microscopic physics.

Dirac quasiparticles and spin-lattice relaxation in the mixed state

We present the results of quantum-mechanical calculations, using the singular gauge transformation of Franz and Tesanovic, of the rate of planar Cu spin-lattice relaxation due to electron spin-flip scattering in the mixed state of high-Tc cuprate superconductors. The results show a non-monotonic temperature and frequency dependence that differs markedly from semiclassical Doppler-shifted results and challenges the assertion that recent experimental observations of the rate of planar Cu and O spin-lattice relaxation in the mixed state of YBCO point to antiferromagnetic spin fluctuations as a better candidate for the elementary excitations of the superconducting state.

Coulomb energy of quasiparticle excitations in Chern–Simons composite fermion states

The attachment of flux tubes to electrons by a Chern–Simons (CS) singular gauge transformation of the wavefunction opened up the field theoretical description of the fractional quantum Hall effect (FQHE). Nevertheless, in Jain's composite fermion (CF) theory, quasiparticles are believed to be vortices carrying a fractional charge in addition to the winding phase of the CS flux tubes. The different structure of the wavefunction in these two cases directly affects the excitation energy gaps. By using a simple ansatz we were able to evaluate analytically the Coulomb excitation energies for the mean-field level CS wavefunction, thus allowing a direct comparison with corresponding numerical results obtained from Jain's CF picture. The considerable difference between the excitation energies found in these two cases demonstrates in quantitative terms the very different impact that the internal structure of the wavefunction has in these two approaches, often used interchangeably to describe the FQHE.