A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions

Abstract

State sum constructions, such as Kuperberg's algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights, to different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space-time dimensions. The matter fields introduces new weights to the vertices and they correspond to Potts spin configurations described by an $\mathcal{A}$-module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain the transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDM) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as ${\mathbb{Z}}_2$ gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in "confined Walker-Wang" models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper.

arXiv comment52 pages, 58 figures. This paper is an extension of arXiv:1310.8483 [cond-mat.str-el] with the inclusion of matter fields. This version includes substantial changes with a connection made to confined Walker-Wang models along the lines of arXiv:1208.5128 and subsequent works. Accepted for publication in JPhysA

References in corpus (24)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• String-net condensation: A physical mechanism for topological phases
• 2D Gravity and Random Matrices
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Fractional topological insulators
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge
• Classification and analysis of two dimensional abelian fractional topological insulators
• Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order
• Stability of Frustration-Free Hamiltonians
• Three-dimensional topological lattice models with surface anyons
• Symmetry Enforced Non-Abelian Topological Order at the Surface of a Topological Insulator
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• A hierarchy of topological tensor network states
• Space-Time Geometry of Topological phases
• The dual of pure non-Abelian lattice gauge theory as a spin foam model
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• Space group symmetry fractionalization in a family of exactly solvable models with Z2 topological order
• String flux mechanism for fractionalization in topologically ordered phases
• Exactly soluble lattice models for non-abelian states of matter in 2 dimensions

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