A hierarchy of topological tensor network states

TL;DR

The paper develops a basis-independent, Hopf C*-algebra framework to extend Kitaev’s quantum double models beyond group algebras, formulating generalized D(H) lattice models whose ground states admit Hopf tensor-network representations. It introduces a hierarchical family of Hopf tensor network states, built from Hopf subalgebras, which captures charge condensation and yields distinct topological orders; the top level reproduces D(H) and exhibits a universal topological entanglement entropy γ = log|H|. The authors also establish an isometric entanglement-renormalization scheme that maps between graph realizations and hierarchy levels, and they connect Hopf tensor networks to PEPS while clarifying injectivity properties. Overall, the work provides a unified, algebraic toolkit for analyzing topological order, entanglement structure, and phase transitions in lattice models, with potential extensions to weak Hopf algebras and broader topological phases.

Abstract

We present a hierarchy of quantum many-body states among which many examples of topological order can be identified by construction. We define these states in terms of a general, basis-independent framework of tensor networks based on the algebraic setting of finite-dimensional Hopf C*-algebras. At the top of the hierarchy we identify ground states of new topological lattice models extending Kitaev's quantum double models [26]. For these states we exhibit the mechanism responsible for their non-zero topological entanglement entropy by constructing a renormalization group flow. Furthermore it is shown that those states of the hierarchy associated with Kitaev's original quantum double models are related to each other by the condensation of topological charges. We conjecture that charge condensation is the physical mechanism underlying the hierarchy in general.

Figures (7)

• Figure 1: Small graph representing $H\otimes H$. It affords a $\mathrm{D}(H)$-module structure at the site $(s,p)$ via the operators $A_a(s,p)$ and $B_f(s,p)$.
• Figure 2: A graph edge representing the Hopf algebra $H$ via the $H$-module structures $L_\pm$. At the same time it represents the Hopf algebra $X=(H^\mathrm{op})^*$ via the $X$-module structures $T_\pm$. These are related to each other by means of the antipode in \ref{['eq:L+-']} and \ref{['eq:T+-']}.
• Figure 3: Graph used in the proof of Theorem \ref{['thm:local_modules']}. The commutation relation and hence the structure of $\mathrm{D}(H)$ is determined on the intersection of the supports of $A_a(s,p)$ and $B_f(s,p)$.
• Figure 4: Ordinary faces vs. boundary “faces”. Wherever the boundary of the surface $M$ does not coincide with an edge of the graph $\Gamma$ it is displayed in grey. Upon smooth deformation of this boundary the grey vertices on either side can actually be identified with each other. This modifies $\Gamma$ and thus creates new faces whose preimages on the left-hand side we call boundary “faces”. Boundary edges are drawn in red.
• Figure 5: Diagram encoding the tensor trace $\mathop{\mathrm{ttr}}\nolimits(\{x_e\};\{f_p\}; \{y_e\};\{g_q\})$. While the interior degrees of freedom $\{x_e\} \subset H$ and $\{f_p\}\subset X$ are only shown partially the boundary degrees of freedom $\{y_e\}\subset H$ and $\{g_q\} \subset X$ are labelled in such a way that the ordering of the boundary is evident.

Theorems & Definitions (45)

• lemma 1
• proof
• definition 1
• theorem 1
• proof
• proposition 1
• lemma 2
• proof
• theorem 2: Generalized quantum double model
• proof