Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two-dimensional lattices are Hamiltonian versions of three-dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three-dimensional geometrical interpretation are given.
Abstract The Levin-Wen model is a general construction that uses an input planar diagram algebra to construct a Hamiltonian with a topologically ordered ground state (the Drinfeld double of the input category). The ground state (code) space consists of a sum over all diagrams weighted by the evaluation of the diagram. We work through the example of the doubled Fibonacci model. Here the input category is the Fibonacci fusion rules and F-matrices. We work with continuum diagrams and construct the full tube algebra of the model. We identify the quasiparticle types to include both left- and right-handed Fibonacci anyons. In the case that the input category admits a modular braiding, much of the analyis simplifies and the output is two copies of the input category of opposite chiralities (both left- and right-handed). We finally turn to the construction of the explicit Hamiltonian on the lattice, and construction of quasiparticle string (or ribbon) operators. We return to the reconsider the toric code and doubled semion models as simple examples.
We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2
+ 1 dimensions in terms of doubled Chern-Simons theories.
Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two‐dimensional lattices are Hamiltonian versions of three‐dimensional topological quantum field theories described by the Turaev‐Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well‐known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three‐dimensional geometrical interpretation are given.
Framing anomaly is a key property of
(2+1)d chiral topological orders, for it reveals that the chirality is an intrinsic bulk property of the system, rather than a property of the boundary between two systems. Understanding framing anomaly in lattice models is particularly interesting, as concrete, solvable lattice models of chiral topological orders are rare. In a recent work, we defined and solved the
U(1) Chern-Simons-Maxwell theory on spacetime lattice, showing its chiral edge mode and the associated gravitational anomaly on boundary. In this work, we show its framing anomaly in the absence of boundary, by computing the expectation of a lattice version of the modular
T operator in the ground subspace on a spatial torus, from which we extract that
⟨T⟩ has a universal phase of
−2π/12 as expected:
−2π/8 from the Gauss-Milgram sum of the topological spins of the ground states, and
2π/24 from the framing anomaly; we can also extract the
2π/24 framing anomaly phase alone from the full spectrum of
T in the ground subspace by computing
⟨Tm⟩. This pins down the last and most crucial property required for a valid lattice definition of
U(1) Chern-Simons theory.