1,708 research outputs found
Topological gauge theory, symmetry fractionalization, and classification of symmetry-enriched topological phases in three dimensions
Symmetry plays a crucial role in enriching topological phases of matter.
Phases with intrinsic topological order that are symmetric are called
symmetry-enriched topological phases (SET). In this paper, we focus on SETs in
three spatial dimensions, where the intrinsic topological orders are described
by Abelian gauge theory and the symmetry groups are also Abelian. As a series
work of our previous research [Phys. Rev. B 94, 245120 (2016);
(arXiv:1609.00985)], we study these topological phases described by twisted
gauge theories with global symmetry and consider all possible topologically
inequivalent "charge matrices". Within each equivalence class, there is a
unique pattern of symmetry fractionalization on both point-like and string-like
topological excitations. In this way, we classify Abelian topological order
enriched by Abelian symmetry within our field-theoretic approach. To
illustrate, we concretely calculate many representative examples of SETs and
discuss future directions
Anomalous boundary correspondence of topological phases
Topological phases protected by crystalline symmetries and internal
symmetries are shown to enjoy fascinating one-to-one correspondence in
classification. Here we investigate the physics content behind the abstract
correspondence in three or higher-dimensional systems. We show correspondence
between anomalous boundary states, which provides a new way to explore the
quantum anomaly of symmetry from its crystalline equivalent counterpart. We
show such correspondence directly in two scenarios, including the anomalous
symmetry-enriched topological orders (SET) and critical surface states. (1)
First of all, for the surface SET correspondence, we demonstrate it by
considering examples involving time-reversal symmetry and mirror symmetry. We
show that one 2D topological order can carry the time reversal anomaly as long
as it can carry the mirror anomaly and vice versa, by directly establishing the
mapping of the time reversal anomaly indicators and mirror anomaly indicators.
Besides, we also consider other cases involving continuous symmetry, which
leads us to introduce some new anomaly indicators for symmetry from its
counterpart. (2) Furthermore, we also build up direct correspondence for (near)
critical boundaries. Again taking topological phases protected by time reversal
and mirror symmetry as examples, the direct correspondence of their (near)
critical boundaries can be built up by coupled chain construction that was
first proposed by Senthil and Fisher. The examples of critical boundary
correspondence we consider in this paper can be understood in a unified
framework that is related to \textit{hierarchy structure} of topological
nonlinear sigma model, that generalizes the Haldane's derivation of
sigma model from spin one-half system.Comment: 17 pages, 5 figure
Exactly solvable model for two-dimensional topological superconductors
In this paper, we present an exactly solvable model for two-dimensional topological superconductors with helical Majorana edge modes protected by time-reversal symmetry. Our construction is based on the idea of decorated domain walls and makes use of the Kasteleyn orientation on a two-dimensional lattice, which was used for the construction of the symmetry protected fermion phase with Z_2 symmetry by Tarantino et al. and Ware et al. By decorating the time-reversal domain walls with spinful Majorana chains, we are able to construct a commuting projector Hamiltonian with zero correlation length ground state wave function that realizes a strongly interacting version of the two-dimensional topological superconductor. From our construction, it can be seen that the T_2 = −1 transformation rule for the fermions is crucial for the existence of such a nontrivial phase; with T_2 = 1, our construction does not work
Generating Pairing-friendly Parameters for the CM Construction of Genus 2 Curves over Prime Fields
We present two contributions in this paper. First, we give a quantitative analysis of the scarcity of pairing-friendly genus 2 curves. This result is an improvement relative to prior work which estimated the density of pairing-friendly genus 2 curves heuristically. Second, we present a method for generating pairing-friendly parameters for which , where is a measure of efficiency in pairing-based cryptography. This method works by solving a system of equations given in terms of coefficients of the Frobenius element. The algorithm is easy to understand and implement
Semi-local convergence of Cordero's sixth-order method
In this paper, the semi-local convergence of the Cordero's sixth-order iterative method in Banach space was proved by the method of recursion relation. In the process of proving, the auxiliary sequence and three increasing scalar functions can be derived using Lipschitz conditions on the first-order derivatives. By using the properties of auxiliary sequence and scalar function, it was proved that the iterative sequence obtained by the iterative method was a Cauchy sequence, then the convergence radius was obtained and its uniqueness was proven. Compared with Cordero's process of proving convergence, this paper does not need to ensure that is continuously differentiable in higher order, and only the first-order Fréchet derivative was used to prove semi-local convergence. Finally, the numerical results showed that the recursion relationship is reasonable
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