Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.Comment: 32 pages, 6 figure

Unconventional Fusion and Braiding of Topological Defects in a Lattice Model

We demonstrate the semiclassical nature of symmetry twist defects that differ from quantum deconfined anyons in a true topological phase by examining non-abelian crystalline defects in an abelian lattice model. An underlying non-dynamical ungauged S3-symmetry labels the quasi-extensive defects by group elements and gives rise to order dependent fusion. A central subgroup of local Wilson observables distinguishes defect-anyon composites by species, which can mutate through abelian anyon tunneling by tuning local defect phase parameters. We compute a complete consistent set of primitive basis transformations, or F-symbols, and study braiding and exchange between commuting defects. This suggests a modified spin-statistics theorem for defects and non-modular group structures unitarily represented by the braiding S and exchange T matrices. Non-abelian braiding operations in a closed system represent the sphere braid group projectively by a non-trivial central extension that relates the underlying symmetry.Comment: 44 pages, 43 figure

Braiding Statistics and Congruent Invariance of Twist Defects in Bosonic Bilayer Fractional Quantum Hall States

We describe the braiding statistics of topological twist defects in abelian bosonic bilayer (mmn) fractional quantum Hall (FQH) states, which reduce to the Z_n toric code when m=0. Twist defects carry non-abelian fractional Majorana-like characteristics. We propose local statistical measurements that distinguish the fractional charge, or species, of a defect-quasiparticle composite. Degenerate ground states and basis transformations of a multi-defect system are characterized by a consistent set of fusion properties. Non-abelian unitary exchange operations are determined using half braids between defects, and projectively represent the sphere braid group in a closed system. Defect spin statistics are modified by equating exchange with 4\pi rotation. The braiding S matrix is identified with a Dehn twist (instead of a \pi/2 rotation) on a torus decorated with a non-trivial twofold branch cut, and represents the congruent subgroup \Gamma_0(2) of modular transformations.Comment: 6 pages, 3 figure

From Dirac semimetals to topological phases in three dimensions: a coupled wire construction

Weyl and Dirac (semi)metals in three dimensions have robust gapless electronic band structures. Their massless single-body energy spectra are protected by symmetries such as lattice translation, (screw) rotation and time reversal. In this manuscript, we discuss many-body interactions in these systems. We focus on strong interactions that preserve symmetries and are outside the single-body mean-field regime. By mapping a Dirac (semi)metal to a model based on a three dimensional array of coupled Dirac wires, we show (1) the Dirac (semi)metal can acquire a many-body excitation energy gap without breaking the relevant symmetries, and (2) interaction can enable an anomalous Weyl (semi)metallic phase that is otherwise forbidden by symmetries in the single-body setting and can only be present holographically on the boundary of a four dimensional weak topological insulator. Both of these topological states support fractional gapped (gapless) bulk (resp. boundary) quasiparticle excitations.Comment: 29 pages, 19 figures. This version has an expanded 'Summary of Results' and 'Conclusion and Discussion' section to make it more accessible to a broader audienc