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Doubled-Semion Model

Steven H. Simon

Published Sep 29, 2023• 0 citations
Publisher

Abstract

Abstract The Doubled-Semion model is another simple example of generalizing the toric code. Here we start with a planar diagram algebra that is the d=-1, Z2 loop gas. I.e, here a loop is given the value of -1. This is also known as the nontrivial cocyle of Z2. This model can be thought of as the simplest case of a twisted Kitaev Quantum Double, and it is also the simplest example of the Drinfeld double of a modular anyon theory (the double of the semion theory). In this case we construct a Hamiltonian such again we have a loop gas, but now flipping over a plaqutte incurs a sign in the wavefunction if we change the parity of the number of loops. We again turn to the tube algebra to identify the excitations of the model, which are both right- and left-handed semions. We run into a problem related to the nontrivial Frobenius-Schur indicator of the input diagram algebra. However, here we can resolve the issue by choosing a convenient gauge.

DOI
10.1093/oso/9780198886723.003.0032

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Abstract The Doubled-Semion model is another simple example of generalizing the toric code. Here we start with a planar diagram algebra that is the d=-1, Z2 loop gas. I.e, here a loop is given the value of -1. This is also known as the nontrivial cocyle of Z2. This model can be thought of as the simplest case of a twisted Kitaev Quantum Double, and it is also the simplest example of the Drinfeld double of a modular anyon theory (the double of the semion theory). In this case we construct a Hamiltonian such again we have a loop gas, but now flipping over a plaqutte incurs a sign in the wavefunction if we change the parity of the number of loops. We again turn to the tube algebra to identify the excitations of the model, which are both right- and left-handed semions. We run into a problem related to the nontrivial Frobenius-Schur indicator of the input diagram algebra. However, here we can resolve the issue by choosing a convenient gauge.
Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.
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 construct two dimensional lattice models from transfer matrices of lattice gauge theories with discrete gauge groups. These transfer matrices are built out of local operators acting on links, vertices and plaquettes and are parametrized by the center of the gauge group algebra and its dual. They contain the familiar 2D quantum double models for a particular choice of parameters, which includes the well studied example of the toric code. However for more general choices of parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase towards a paramagnetic phase. These perturbations can be thought of as magnetic fields added to the system which destroy the exact solvability of the quantum double model. Nevertheless from the same transfer matrix with perturbations we exhibit exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. This is shown by working with the groups Zn and S3 for the Abelian and non-Abelian cases respectively. The quantum phases are found by studying the excitations of these systems. The fusion rules and the statistics of these anyons indicate the quantum phases of these models. The implementation of these models can possibly improve the use of quantum double models for fault tolerant quantum computation. We then construct theories which arise from transfer matrices that are not the transfer matrices of lattice gauge theories. In particular we show that for the Z2 case this contains the double semion model. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.
The toric code can be constructed as a gauge theory of finite groups on oriented two dimensional lattices. Here we construct analogous models with the gauge fields belonging to groupoids, which are categories where every morphism has an inverse. We show that a consistent system can be constructed for an arbitrary groupoid and analyze the simplest example that can be seen as the analog of the Abelian Z2\mathbb{Z}_2Z2​ toric code. We find several exactly solvable models that have fracton-like features which include an extensive ground state degeneracy and excitations that are either immobile or have restricted mobility. Among the possibilities we study in detail the one where the ground state degeneracy scales as 2×2Nv2\times 2^{N_v}2×2Nv​, where NvN_vNv​ is the number of vertices in the lattice. The origin of this degeneracy can be traced to loop operators supported on both contractible and non-contractible loops. In particular, different non-contractible loops, along the same direction on a torus, result in different ground states. This is an exponential increase in the number of logical qubits that can be encoded in this code. Moreover the face excitations in this system are deconfined, free to move without an energy cost along certain directions of the lattice, whereas in certain other directions their movement incurs an energy cost. This places a restriction on the types of loop operators that contribute to the ground state degeneracy. The vertex excitations are immobile. The results are also extended to the groupoid analogs of Abelian ZN\mathbb{Z}_NZN​ toric codes.
Abstract Ideas similar to that of the Kauffman bracket introduced in chapter 2 can be used to generate valid anyon theories. We begin with a planar diagram algebra defined only by the value of a loop (the loop weight). One wants to build new particle types by grouping together multiple strands of loops. However, to do so one needs so-called Jones-Wenzl projectors to orthogonalize between species resulting in the so-called Temperly-Lieb Algebra. We demonstrate this construction by building a planar diagram algebra corresponding to the Z2 loop gas (bosons, fermions, semions), and then Ising Anyons. We show how this generalizes to a much bigger set of possible theories. We show how these theories can be made unitary, and how the F-matrices may be calculated. Finally we discuss how the R-matrices can be simply obtained by returning to the Kauffman bracket rules.
We construct a Pauli stabilizer model for every two-dimensional Abelian topological order that admits a gapped boundary.Our primary example is a Pauli stabilizer model on four-dimensional qudits that belongs to the double semion (DS) phase of matter.The DS stabilizer Hamiltonian is constructed by condensing an emergent boson in a Z 4 toric code, where the condensation is implemented at the level of the ground states by two-body measurements.We rigorously verify the topological order of the DS stabilizer model by identifying an explicit finite-depth quantum circuit (with ancillary qubits) that maps its ground-state subspace to that of a DS string-net model.We show that the construction of the DS stabilizer Hamiltonian generalizes to all twisted quantum doubles (TQDs) with Abelian anyons.This yields a Pauli stabilizer code on composite-dimensional qudits for each such TQD, implying that the classification of topological Pauli stabilizer codes extends well beyond stacks of toric codes-in fact, exhausting all Abelian anyon theories that admit a gapped boundary.We also demonstrate that symmetry-protected topological phases of matter characterized by type-I and type-II cocycles can be modeled by Pauli stabilizer Hamiltonians by gauging certain 1-form symmetries of the TQD stabilizer models.