Semidual Kitaev lattice model and tensor network representation

TL;DR

This paper introduces a new version of Kitaev's lattice model based on bicrossproduct quantum groups, providing a natural construction, an exactly solvable Hamiltonian, and a tensor network ground state representation.

Contribution

It proposes a novel Kitaev lattice model using bicrossproduct quantum groups, expanding the framework beyond the traditional Drinfeld double approach.

Findings

Constructs a representation of the bicrossproduct quantum group on quadrangulated surfaces.

Provides an exactly solvable Hamiltonian for the new model.

Defines the ground state using a tensor network representation.

Abstract

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 $M(H)=H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$ as an example of Majid's bicrossproduct quantum group, related by semidualisation or `quantum Born reciprocity' to $D(H)$. Given a finite-dimensional Hopf algebra $H$, we show that a quadrangulated oriented surface defines a representation of the bicrossproduct quantum group $H^{\text{cop}}\blacktriangleright\!\!\!\triangleleft H$. 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.