Levin-Wen Model Research Articles

Ground-state degeneracy in the Levin-Wen model for topological phases

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.

Galois conjugates of topological phases

Galois conjugation relates unitary conformal field theories (CFTs) and topological quantum field theories (TQFTs) to their non-unitary counterparts. Here we investigate Galois conjugates of quantum double models, such as the Levin-Wen model. While these Galois conjugated Hamiltonians are typically non-Hermitian, we find that their ground state wave functions still obey a generalized version of the usual code property (local operators do not act on the ground state manifold) and hence enjoy a generalized topological protection. The key question addressed in this paper is whether such non-unitary topological phases can also appear as the ground states of Hermitian Hamiltonians. Specific attempts at constructing Hermitian Hamiltonians with these ground states lead to a loss of the code property and topological protection of the degenerate ground states. Beyond this we rigorously prove that no local change of basis can transform the ground states of the Galois conjugated doubled Fibonacci theory into the ground states of a topological model whose Hermitian Hamiltonian satisfies Lieb-Robinson bounds. These include all gapped local or quasi-local Hamiltonians. A similar statement holds for many other non-unitary TQFTs. One consequence is that the "Gaffnian" wave function cannot be the ground state of a gapped fractional quantum Hall state.

Microscopic models of interacting Yang–Lee anyons

Collective states of interacting non-Abelian anyons have recently been studied mostly in the context of certain fractional quantum Hall states, such as the Moore–Read state proposed to describe the physics of the quantum Hall plateau at filling fraction ν=5/2. In this paper, we further expand this line of research and present non-unitary generalizations of interacting anyon models. In particular, we introduce the notion of Yang–Lee anyons, discuss their relation to the so-called ‘Gaffnian’ quantum Hall wave function and describe an elementary model for their interactions. A one-dimensional (1D) version of this model—a non-unitary generalization of the original golden chain model—can be fully understood in terms of an exact algebraic solution and numerical diagonalization. We discuss the gapless theories of these chain models for general su(2)k anyonic theories and their Galois conjugates. We further introduce and solve a 1D version of the Levin–Wen model for non-unitary Yang–Lee anyons.

Space–time geometry of topological phases

The 2 + 1 dimensional lattice models of Levin and Wen (2005) [1] 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.