On Quantum Entanglement in Topological Phases on a Torus

TL;DR

This paper investigates how non-trivial spatial topology, specifically on a torus, influences quantum entanglement in topologically ordered systems, providing a general formula for entanglement entropy and demonstrating it with concrete examples.

Contribution

It introduces a general formula for entanglement entropy in topological phases on a torus, accounting for boundary degrees of freedom and global quantum numbers, extending the concept of minimally entangled states.

Findings

Derived a formula for reduced density matrices in topological systems on a torus.

Validated the formula with numerical data from finite groups and modular tensor categories.

Showed the significance of global quantum numbers and boundary decompositions in entanglement.

Abstract

In this paper we study the effect of non-trivial spatial topology on quantum entanglement by examining the degenerate ground states of a topologically ordered system on torus. Using the string-net (fixed-point) wave-function, we propose a general formula of the reduced density matrix when the system is partitioned into two cylinders. The cylindrical topology of the subsystems makes a significant difference in regard to entanglement: a global quantum number for the many-body states comes into play, together with a decomposition matrix $M$ which describes how topological charges of the ground states decompose into boundary degrees of freedom. We obtain a general formula for entanglement entropy and generalize the concept of minimally entangled states to minimally entangled sectors. Concrete examples are demonstrated with data from both finite groups and modular tensor categories (i.e., Fibonacci, Ising, etc.), supported by numerical verification.