Space-Time Geometry of Topological phases

Abstract

The 2+1 dimensional lattice models of Levin and Wen [PRB 71, 045110 (2005)] provide the most general known microscopic construction of topological phases of matter. Based heavily on the mathematical structure of category theory, many of the special properties of these models are not obvious. In the current paper, we present a geometrical space-time picture of the partition function of the Levin-Wen models which can be described as doubles (two copies with opposite chiralities) of underlying Anyon theories. Our space-time picture describes the partition function as a knot invariant of a complicated link, where both the lattice variables of the microscopic Levin-Wen model and the terms of the hamiltonian are represented as labeled strings of this link. This complicated link, previously studied in the mathematical literature, and known as Chain-Mail, can be related directly to known topological invariants of 3-manifolds such as the so called Turaev-Viro invariant and the Witten-Reshitikhin-Turaev invariant. We further consider quasi-particle excitations of the Levin-Wen models and we see how they can be understood by adding additional strings to the Chain-Mail link representing quasi-particle world-lines. Our construction gives particularly important new insight into how a doubled theory arises from these microscopic models.

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Cited by in corpus (16)

• Diagrammatics for Bose condensation in anyon theories
• Condensation of achiral simple currents in topological lattice models: a Hamiltonian study of topological symmetry breaking
• Topological Phase Transitions in the Golden String-Net Model
• Wilson Line Picture of Levin-Wen Partition Functions
• Dynamics at and near conformal quantum critical points
• Partition function of the Levin-Wen model
• Quantum Circuits for Measuring Levin-Wen Operators
• Kibble-Zurek Scaling and String-Net Coarsening in Topologically Ordered Systems
• Frustrated topological symmetry breaking: geometrical frustration and anyon condensation
• 3d Gravity as a random ensemble
• Fibonacci topological order from quantum nets
• 2D Quantum Double Models From a 3D Perspective
• Exactly Solvable Lattice Models with Crossing Symmetry
• Wegner-Wilson loops in string nets
• Magnification of signatures of topological phase transition by quantum zero point motion
• A Recipe for Constructing Frustration-Free Hamiltonians with Gauge and Matter Fields in One and Two Dimensions