[Paper Review] Topological entanglement entropy relations for multi phase systems with interfaces

F.A. Bais, J. K. Slingerland|arXiv (Cornell University)|Jun 10, 2010

Quantum many-body systems3 citations

TL;DR

This paper derives exact relations for topological entanglement entropy (TEE) in two-dimensional topological phases with interfaces, particularly at phase transitions driven by Bose condensation. It introduces a quantum embedding index $ q = D_A / D_U $, showing that the change in TEE between phases is $ \gamma_A - \gamma_U = \log q $, providing a precise measure of topological order reduction during symmetry-breaking transitions.

ABSTRACT

We study the change in topological entanglement entropy that occurs when a two-dimensional system in a topologically ordered phase undergoes a transition to another such phase due to the formation of a Bose condensate. We also consider the topological entanglement entropy of systems with domains in different topological phases, and of phase boundaries between these domains. We calculate the topological entropy of these interfaces and derive two fundamental relations between the interface topological entropy and the bulk topological entropies on both sides of the interface.

Motivation & Objective

• To understand how topological entanglement entropy (TEE) changes across phase boundaries in 2D topological systems with domain walls.
• To characterize the TEE of an interface formed between two topological phases connected by a Bose condensate.
• To derive fundamental relations between the TEE of the interface and the bulk TEEs of the two phases.
• To establish a quantitative measure, the quantum embedding index $ q $, that captures the loss of topological order during topological symmetry breaking.

Proposed method

• Uses a three-level framework: the original topological phase $ A $, the intermediate fusion algebra $ T $ describing the interface, and the condensed phase $ U $.
• Applies restriction and lift maps between fusion sectors of $ A $, $ T $, and $ U $, with integer coefficients $ n^t_a $ and $ n^a_t $, ensuring consistency with fusion and quantum dimension conservation.
• Defines the quantum embedding index $ q = \sum_a n^a_t d_a / d_t $, which is proven independent of the choice of $ t \in T $, and shown to satisfy $ q = D_A / D_U $.
• Uses Perron-Frobenius eigenvectors of fusion matrices to prove that $ B\omega_A = q\omega_T $, where $ B $ is the restriction map, establishing $ q = D_A^2 / D_T^2 $.
• Derives the key relation $ D_T^2 = D_A D_U $, which, combined with $ q = D_A^2 / D_T^2 $, yields $ q = D_A / D_U $, confirming the entropy change $ \gamma_A - \gamma_U = \log q $.
• Employs physical intuition and elementary algebraic techniques, avoiding advanced category theory, to derive the entropy relations.

Experimental results

Research questions

• RQ1 How does the topological entanglement entropy change when a topological phase undergoes a transition via Bose condensation?
• RQ2 What is the precise value of the topological entanglement entropy at an interface between two topological phases separated by a domain wall?
• RQ3 How is the interface TEE related to the bulk TEEs of the two phases involved in the transition?
• RQ4 Can a universal quantity be defined that captures the reduction in topological order during such phase transitions?
• RQ5 Is the quantum embedding index $ q $, defined via fusion coefficients and quantum dimensions, independent of the choice of anyon sector at the interface?

Key findings

• The change in topological entanglement entropy between the original phase $ A $ and the condensed phase $ U $ is exactly $ \gamma_A - \gamma_U = \log q $, where $ q = D_A / D_U $.
• The quantum embedding index $ q $, defined as $ q = \sum_a n^a_t d_a / d_t $, is independent of the anyon sector $ t $ chosen at the interface.
• The relation $ D_T^2 = D_A D_U $ holds, where $ D_T $ is the quantum dimension of the intermediate fusion algebra $ T $, linking the three phases.
• The quantity $ q $ satisfies $ q = D_A^2 / D_T^2 $, and this is derived using the Perron-Frobenius eigenvector structure of fusion matrices.
• The interface TEE is fully determined by the bulk TEEs and the quantum embedding index $ q $, with no additional free parameters.
• The result provides a robust, perturbation-invariant measure of topological order reduction during topological symmetry breaking transitions in 2D systems.