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arXiv:2506.09249

A non-semisimple Kitaev lattice model

Sebastian Halbig, Ulrich Krähmer

Published Jun 10, 2025• 0 citations
arXivHTMLPDF

Abstract

The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input of the model are ribbon graphs, that is, the combinatorial data of cellular decompositions of oriented closed surfaces. The other input are certain Hopf bimodules that are closely related to the coefficients in Hopf-cyclic homology. As in previous generalisations of the Kitaev model, a Yetter-Drinfeld module is constructed and shown to be a topological invariant of the surface with boundary that is obtained by "thickening" the graph. The generalisation of the protected space is defined using bitensor products of modules-comodules. Provided that the Hopf bimodule coefficients correspond to pairs in involution, this is shown to depend only on the genus of the graph. As examples, group algebras of finite groups and bosonisations of Nichols algebras are considered.

DOI
10.48550/arxiv.2506.09249

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The construction of the topologically protected code space of Kitaev's model for fault-tolerant quantum computation is extended from complex semisimple to arbitrary finite-dimensional Hopf algebras admitting pairs in involution. One input of the model are ribbon graphs, that is, the combinatorial data of cellular decompositions of oriented closed surfaces. The other input are certain Hopf bimodules that are closely related to the coefficients in Hopf-cyclic homology. As in previous generalisations of the Kitaev model, a Yetter-Drinfeld module is constructed and shown to be a topological invariant of the surface with boundary that is obtained by "thickening" the graph. The generalisation of the protected space is defined using bitensor products of modules-comodules. Provided that the Hopf bimodule coefficients correspond to pairs in involution, this is shown to depend only on the genus of the graph. As examples, group algebras of finite groups and bosonisations of Nichols algebras are considered.
We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double D( G) symmetry, where G is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of D( G) and combine this with D( G)-bimodule properties of open ribbon excitation spaces [Formula: see text] to show how open ribbons can be used to teleport information between their endpoints s0, s1. We give a self-contained account that builds on earlier work but emphasizes applications to quantum computing as surface code theory, including gates on D( S3). We show how the theory reduces to a simpler theory for toric codes in the case of [Formula: see text], including toric ribbon operators and their braiding. In the other direction, we show how our constructions generalize to D( H) models based on a finite-dimensional Hopf algebra H, including site actions of D( H) and partial results on ribbon equivariance even when the Hopf algebra is not semisimple.
In this thesis we study algebraic structures that are important for models of commuting-projector Hamiltonians which realize topological phases of matter. The Kitaev model is such a model, where the projectors are defined using the structure of a semisimple Hopf algebra. In the first part of the thesis we construct a Kitaev model based on more general Hopf-algebraic data – semisimple bicomodule algebras – thereby implementing defects and boundaries. In the second part of the thesis we find generalizations of the idempotents used in the standard Kitaev model to non-semisimple Hopf algebras. More precisely, in the first part of the thesis, we construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. Specifically, we show that one can consider cell decompositions of surfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeled by semisimple bicomodule algebras. We introduce an algebra whose representations label the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in the absence of defects. In this way we generalize the algebraic structure underlying the standard Kitaev model without defects or boundaries, where all 1-cells and 2-cells are labeled by a single Hopf algebra and where point defects are labeled by representations of its Drinfeld double. In the standard case, commuting local projectors are constructed using the Haar integral for semisimple Hopf algebras. A central insight we gain in this theis is that in the presence of defects and boundaries, the suitable generalization of the Haar integral is given by the unique symmetric separability idempotent for a semisimple (bi-)comodule algebra. This enables us to provide an explicit construction of a Kitaev model allowing for defects and boundaries. In the second part of the thesis we obtain representation-theoretic results. We study the isotypic decomposition of the regular module of a not necessarily semisimple, finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, it is known that the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. Here we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where we replace the Haar integral by the regular character of the dual Hopf algebra. For a large class of Hopf algebras we show that these form a complete set of orthogonal idempotents. Finally, we give an example which illustrates that the Chevalley property is crucial. In dieser Arbeit untersuchen wir algebraische Strukturen, die wichtig sind fur Modelle von Hamilton-Operatoren mit kommutierenden Projektoren, welche topologische Phasen der Materie realisieren. Das Kitaev-Modell ist ein solches Modell, bei dem die Projektoren mithilfe der Struktur einer halbeinfachen Hopf-Algebra definiert werden. Im ersten Teil der Arbeit konstruieren wir ein Kitaev-Modell, das auf allgemeineren Hopf-algebraischen Daten basiert -- halbeinfache Bikomodul-Algebren - und dabei Defekte und Rander implementiert. Im zweiten Teil der Arbeit finden wir Verallgemeinerungen der im Standard-Kitaev-Modell verwendeten Idempotenten fur nicht-halbeinfache Hopf-Algebren. Genauer gesagt konstruieren wir im ersten Teil der Arbeit ein Kitaev-Modell, das aus einem Hamilton-Operator besteht, der eine Summe von kommutierenden lokalen Projektoren ist, fur Flachen mit Randern und Defekten von Kodimension 0 und 1. Insbesondere zeigen wir, dass man Zellzerlegungen von Flachen betrachten kann, deren 2-Zellen mit halbeinfachen Hopf-Algebren und deren 1-Zellen mit halbeinfachen Bikomodul-Algebren dekoriert sind. Wir fuhren eine Algebra ein, deren Darstellungen die 0-Zellen dekorieren und die sich im Spezialfall ohne Defekte auf das Drinfeld-Doppel einer Hopf-Algebra reduziert. Auf diese Weise verallgemeinern wir die algebraische Struktur, die dem Standard-Kitaev-Modell ohne Defekte oder Rander zugrunde liegt, bei dem alle 1-Zellen und 2-Zellen mit einer einzigen Hopf-Algebra und Punkt-defekte mit Darstellungen seines Drinfeld-Doppels dekoriert werden. Im Standardfall werden kommutierende lokale Projektoren mithilfe des Haar-Integrals fur halbeinfache Hopf-Algebren konstruiert. Eine zentrale Einsicht dieser Arbeit ist, dass bei Vorhandensein von Defekten und Randern die geeignete Verallgemeinerung des Haar-Integrals durch die eindeutige symmetrische Separabilitatsidempotente einer halbeinfachen (Bi-)Komodul-Algebra gegeben ist. Dies ermoglicht es uns, eine explizite Konstruktion eines Kitaev-Modells anzugeben, welches Defekte und Rander zulasst. Im zweiten Teil der Arbeit erlangen wir darstellungstheoretische Resultate. Wir untersuchen die isotypische Zerlegung des regularen Moduls einer nicht unbedingt halbeinfachen, endlichdimensionalen Hopf-Algebra uber einem algebraisch abgeschlossenen Korper in Charakteristik 0. Fur eine halbeinfache Hopf-Algebra ist bekannt, dass die Idempotenten, die die isotypische Zerlegung realisieren, explizit durch Charaktere und das Haar-Integral ausgedruckt werden konnen. Hier untersuchen wir Hopf-Algebren mit der Chevalley-Eigenschaft, die nicht unbe-dingt halbeinfach sind. Wir finden explizite Ausdrucke fur Idempotente durch Hopf-algebraische Daten, wobei wir das Haar-Integral durch den regularen Charakter der dualen Hopf-Algebra ersetzen. Fur eine grose Klasse von Hopf-Algebren zeigen wir, dass diese einen vollstandigen Satz orthogonaler Idempotente bilden. Abschliesend geben wir ein Beispiel, das zeigt, dass die Chevalley-Eigenschaft von entscheidender Bedeutung ist.
We construct a Kitaev model, consisting of a Hamiltonian which is the sum of commuting local projectors, for surfaces with boundaries and defects of dimension 0 and 1. More specifically, we show that one can consider cell decompositions of surfaces whose 2-cells are labeled by semisimple Hopf algebras and 1-cells are labeled by semisimple bicomodule algebras. We introduce an algebra whose representations label the 0-cells and which reduces to the Drinfeld double of a Hopf algebra in the absence of defects. In this way we generalize the algebraic structure underlying the standard Kitaev model without defects or boundaries, where all 1-cells and 2-cells are labeled by a single Hopf algebra and where point defects are labeled by representations of its Drinfeld double. In the standard case, commuting local projectors are constructed using the Haar integral for semisimple Hopf algebras. A central insight we gain in this paper is that in the presence of defects and boundaries, the suitable generalization of the Haar integral is given by the unique symmetric separability idempotent for a semisimple (bi-)comodule algebra.
Abstract Kitaev’s quantum double model is a family of exactly solvable lattice models that realize two dimensional topological phases of matter. The model was originally based on finite groups, and was later generalized to semi-simple Hopf algebras. We rigorously define and study ribbon operators in the generalized quantum double model. These ribbon operators are important tools to understand quasi-particle excitations. It turns out that there are some subtleties in defining the operators in contrast to what one would naively think of. In particular, one has to distinguish two classes of ribbons which we call locally clockwise and locally counterclockwise ribbons. Moreover, we point out that the issue already exists in the original model based on finite non-abelian groups, but it seems to not have been noticed in the literature. We show how certain common properties would fail even in the original model if we were not to distinguish these two classes of ribbons. Perhaps not surprisingly, under the new definitions ribbon operators satisfy all properties that are expected. For instance, they create quasi-particle excitations only at the end of the ribbon, and the types of the quasi-particles correspond to irreducible representations of the Drinfeld double of the input Hopf algebra. However, the proofs of these properties are much more complicated than those in the case of finite groups. This is partly due to the complications in dealing with general Hopf algebras rather than group algebras.
Kitaev's lattice models are usually defined as representations of the Drinfeld quantum double $D(H)=H\bowtie H^{*\text{op}} $, as an example of a double cross product quantum group. We propose a new version based instead on M(H)=Hcop▶ ⁣ ⁣ ⁣◃HM(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HM(H)=Hcop▶◃H as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to D(H)D(H)D(H). Given a finite-dimensional Hopf algebra HHH, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group Hcop▶ ⁣ ⁣ ⁣◃HH^{\text{cop}}\blacktriangleright\!\!\!\triangleleft HHcop▶◃H. Even though the bicrossproduct has a more complicated and entangled coproduct, the construction of this new model is relatively natural as it relies on the use of the covariant Hopf algebra actions. Working locally, we obtain an exactly solvable Hamiltonian for the model and provide a definition of the ground state in terms of a tensor network representation.