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arXiv:2605.02039 · Journal of Physics A Mathematical and Theoretical

Lattice realization of twist defects in a Z 2 × Z 2 Topological Order

Gustavo M. Yoshitome

Published Aug 7, 2026• 0 citations
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Abstract

Abstract In this work, we explore a microscopic realization of three types of anyonic symmetries in a Z 2 × Z 2 topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.

DOI
10.1088/1751-8121/ae96e2

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Journal of Mathematical Physics•Nov 1, 2023•7 citations
Abstract In this work, we explore a microscopic realization of three types of anyonic symmetries in a Z 2 × Z 2 topological order, corresponding to a double toric code. These symmetries act as nontrivial permutations on the anyon labels of the parent state. We consider a setup consisting of two decoupled Wen plaquette models stacked on top of each other and introduce dislocations that modify the Hamiltonian, giving rise to localized twist defects, eventually inducing interactions between the layers. In this context, branch cuts act as sources of anyon permutations when they cross it. We characterize the defects by calculating their quantum dimensions, and we also consider double loop operators around them that allow us to determine the non-Abelian fusion rules between the defects, including when they carry different anyon permutations.
The global eee-mmm exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.
There exists a close relationship between topological quantum error-correcting codes and topological order in condensed matter systems. Indeed, a topological stabilizer code can be regarded as the ground state of a suitable Hamiltonian model, so that "wrong" syndromes correspond to excitations. These excitations are anyons, quasiparticles that carry a topological charge and exhibit exotic statistics. Anyon models can be symmetric under some permutations of their topological charges. One can then conceive topological defects that, under monodromy, transform anyons according to a symmetry. We call these defects twists. Twists give rise to new topological degrees of freedom in the ground state, useful as a quantum memory. Moreover, twists can be braided to perform topologically protected gates on these topological qubits. Thus, twists provide a new way to encode and compute with topological codes through code deformations. Because the properties of twists depend on the anyon model, codes with different anyon content give rise to different computational capabilities. E.g., in the well-known toric code a process where suitable twists are braided and fused has the same outcome as if they were Ising anyons. These are non-abelian anyons: braiding produces non-trivial gates on encoded qubits.
The global eee-mmm exchange symmetry of the toric code is realized locally through an exactly solvable Hamiltonian on a two dimensional lattice which has no lattice dislocations and their associated defect line. The Hamiltonian is still changed locally in selected sites where we wish to realize this anyon symmetry. We refer to these selected sites as defect sites in analogy with the usual lattice defects. The operators on the defect sites condense dyons of the toric code and are shown to support states which obey the fusion rules of Ising anyons just as the lattice dislocations thus achieving the transition to the Ising phase from the toric code phase. They can also be introduced in an entire region leading to an idea of non-localized defects. The method leads to a natural generalization for other Abelian groups where they help realize all the anyon exchange symmetries locally.
Non-Abelian excitations are sought after because of the promise they hold for topological quantum computation. However, in spite of many proposals of systems in which they can occur intrinsically, observation has proved to be elusive. In this thesis, we study ways in which we can engineer such excitations by combining two parent Abelian theories, and as a result allow for defects which host non-Abelian zero modes. We first formulate the symmetries of the anyonic excitations of Abelian topological states. Then, we study properties of semiclassical point defects, called twist defects, which permute orbiting anyons by their symmetries. We see how to realize these defects at domain walls on the gapped edges of non-chiral fractional quantum Hall states. We have distinct gapped edge phases in one to one correspondence with the anyonic symmetries. Non-Abelian twist defects appear between different gapped phases. We study the twist defects for a variety of bosonic ADE fractional quantum spin Hall systems for which the anyonic symmetries turn out to be given by their Dynkin diagrams. We also look at superconducting heterostructures with fermionic quantum Hall states and show the presence of anyonic symmetries and twist defects which physically correspond to a fermion parity flip. This can result in Majorana bound states at such defects instead of more exotic parafermions as was believed before. This is because the presence of the superconductor necessitates the consideration of a larger bosonic topological order. We then investigate the twist defects further by promoting the anyonic symmetries from global to local. This deconfines the defects and results in non-Abelian topological states called twist liquids. Finally, we study the emergence of new topological order from two constituent systems driven by the condensation of a composite boson. This allows us to study a variety of systems with non-Abelian excitations.
Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2+1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.
Topological quantum error correction based on the manipulation of the anyonic defects constitutes one of the most promising frameworks towards realizing fault-tolerant quantum devices. Hence, it is crucial to understand how these defects interact with external defects such as boundaries or domain walls. Motivated by this line of thought, in this work, we study the fusion events between anyons in the bulk and at the boundary in fixed-point models of 2 + 1-dimensional non-chiral topological order defined by arbitrary fusion categories. Our construction uses generalized tube algebra techniques to construct a bi-representation of bulk and boundary defects. We explicitly derive a formula to calculate the fusion multiplicities of a bulk-to-boundary fusion event for twisted quantum double models and calculate some exemplary fusion events for Abelian models and the (twisted) quantum double model of S3, the simplest non-Abelian group-theoretical model. Moreover, we use the folding trick to study the anyonic behavior at non-trivial domain walls between twisted S3 and twisted Z2 as well as Z3 models. A recurring theme in our construction is an isomorphism relating twisted cohomology groups to untwisted ones. The results of this work can directly be applied to study logical operators in two-dimensional topological error correcting codes with boundaries described by a twisted gauge theory of a finite group.