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 Mathematics , 2014, DOI: 10.1088/1751-8113/48/12/12FT01 Abstract: Symmetry protected and symmetry enriched topological phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin-Wen model for spin/boson systems is an important rigorously solvable model for studying $2D$ topological phases. The input data for the Levin-Wen model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the Levin-Wen model, and show that the extended Levin-Wen model is a natural playground for the theoretical study of symmetry protected and symmetry enriched topological phases of matter.
 Liang Kong Mathematics , 2012, Abstract: We review the key steps of the construction of Levin-Wen type of models on lattices with boundaries and defects of codimension 1,2,3 in a joint work with Alexei Kitaev. We emphasize some universal properties, such as boundary-bulk duality and duality-defect correspondence, shared by all these models. New results include a detailed analysis of the local properties of a boundary excitation and a conjecture on the functoriality of the monoidal center.
 Mathematics , 2015, Abstract: In Levin-Wen (LW) models, a wide class of exactly solvable discrete models, for two dimensional topological phases, it is relatively easy to describe single fluxon excitations, but not the charge and dyonic as well as many-fluxon excitations. To incorporate charged and dyonic excitations in (doubled) topological phases, a generalization of the LW models is proposed in this paper. We first enlarge the Hilbert space with adding a tail on one of the edges of each trivalent vertex, to describe its internal charge degrees of freedom. Then we study the full dyon spectrum of generalized LW models, including both quantum numbers and wave functions for dyonic quasiparticle excitations. The local operators associated with the dyonic excitations are shown to form the so-called Tube algebra, whose representations (modules) form the quantum double (categoric center) of the input data (unitary fusion category). In physically relevant cases, the input data is from a finite or quantum group (with braiding $R$-matrices), we find that the elementary excitations (or dyon species), as well as any localized/isolated excited states, are characterized by three quantum numbers: charge, fluxon type, and twist. They provide a "complete basis" for many-body states in the enlarged Hilbert space. Concrete examples are presented and the relevance of our results to the electric-magnetic duality is discussed.
 Physics , 2010, DOI: 10.1088/1367-2630/13/6/065001 Abstract: Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice Hamiltonian description of doubled Chern-Simons theories. We relate the partition function of these theories to an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as Chain-Mail. This geometric construction gives physical interpretation of the Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining, and how two opposite chirality sectors of the doubled theory arise.
 Physics , 2011, DOI: 10.1103/PhysRevB.85.075107 Abstract: We study properties of topological phases by calculating the ground state degeneracy (GSD) of the 2d Levin-Wen (LW) model. Here it is explicitly shown that the GSD depends only on the spatial topology of the system. Then we show that the ground state on a sphere is always non-degenerate. Moreover, we study an example associated with a quantum group, and show that the GSD on a torus agrees with that of the doubled Chern-Simons theory, consistent with the conjectured equivalence between the LW model associated with a quantum group and the doubled Chern-Simons theory.
 Seung-Moon Hong Mathematics , 2009, Abstract: It is known that every ribbon category with unimodality allows symmetrized $6j$-symbols with full tetrahedral symmetries while a spherical category does not in general. We give an explicit counterexample for this, namely the category $\mathcal{E}$. We define the mirror conjugate symmetry of $6j$-symbols instead and show that $6j$-symbols of any unitary spherical category can be normalized to have this property. As an application, we discuss an exactly soluble model on a honeycomb lattice. We prove that the Levin-Wen Hamiltonian is exactly soluble and hermitian on a unitary spherical category.
 Spencer D. Stirling Mathematics , 2010, Abstract: We remark on the claim that the string-net model of Levin and Wen is a microscopic Hamiltonian formulation of the Turaev-Viro topological quantum field theory. Using simple counterexamples we indicate where interesting extra structure may be needed in the Levin-Wen model for this to hold (however we believe that some form of the correspondence is true). In order to be accessible to the condensed matter community we provide a brief and gentle introduction to the relevant concepts in category theory (relying heavily on analogy with ordinary group representation theory). Likewise, some physical ideas are briefly surveyed for the benefit of the more mathematical reader. The main feature of group categories under consideration is Turaev's unimodality. We pinpoint where unimodality should fit into the Levin-Wen construction, and show that the simplest example fails to be unimodal. Unimodality is straightforward to compute for group categories, and we provide a complete classification at the end of the paper.
 Physics , 2009, DOI: 10.1103/PhysRevB.80.155136 Abstract: We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows to carry over to these string net models the representation-theoretic classification of the excitations in quantum double models, as well as define them in arbitrary lattices, and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalising the map to broader classes of string nets is considered.
 Computer Science , 2010, Abstract: We consider boolean circuits computing n-operators f:{0,1}^n --> {0,1}^n. As gates we allow arbitrary boolean functions; neither fanin nor fanout of gates is restricted. An operator is linear if it computes n linear forms, that is, computes a matrix-vector product y=Ax over GF(2). We prove the existence of n-operators requiring about n^2 wires in any circuit, and linear n-operators requiring about n^2/\log n wires in depth-2 circuits, if either all output gates or all gates on the middle layer are linear.
 Journal of Computer and Communications (JCC) , 2016, DOI: 10.4236/jcc.2016.415015 Abstract: Systematic effort dedicated to the exploration of feasible ways how to permanently come up with even more space-efficient implementation of digital circuits based on conventional CMOS technology node may soon reach the ultimate point, which is mostly given by the constraints associated with physical scaling of fundamental electronic components. One of the possible ways of how to mitigate this problem can be recognized in deployment of multifunctional circuit elements. In addition, the polymorphic electronics paradigm, with its considerable independence on a parti- cular technology, opens a way how to fulfil this objective through the adoption of emerging semiconductor materials and advanced synthesis methods. In this paper, main attention is focused on the introduction of polymorphic operators (i.e. digital logic gates) that would allow to further increase the efficiency of multifunctional circuit synthesis techniques. Key aspect depicting the novelty of the proposed approach is primarily based on the intrinsic exploitation of components with ambi- polar conduction property. Finally, relevant models of the polymorphic operators are presented in conjunction with the experimental results.
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