Exact results on the entanglement entropy of Abelian and non-Abelian topological states

Bo Han, Chi Yan Jeffrey Teo

Bei Zeng

Hong Yao, Xiao-Liang Qi

In this letter, we obtain an exact formula for the entanglement entropy of the ground state and all excited states of the Kitaev model. Remarkably, the entanglement entropy can be expressed in a simple separable form S = SG+SF, with SF the entanglement entropy of a free Majorana fermion system and SG that of a Z2 gauge field. The Z2 gauge field part contributes to the universal "topological entanglement entropy" of the ground state while the fermion part is responsible for the nonlocal entanglement carried by the Z2 vortices (visons) in the non-Abelian phase. Our result also enables the calculation of the entire entanglement spectrum and the more general Renyi entropy of the Kitaev model. Based on our results we propose a new quantity to characterize topologically ordered states--the capacity of entanglement, which can distinguish the st ates with and without topologically protected gapless entanglement spectrum.

Paul Fendley, Matthew P. A. Fisher, Chetan Nayak

Sarath Sankar, Eran Sela, Cheolhee Han

Topological entanglement entropy (TEE) is a key diagnostic of topological order, allowing one to detect the presence of Abelian or non-Abelian anyons. However, there are currently no experimentally feasible protocols to measure TEE in condensed matter systems. Here, we propose a scheme to measure the TEE of chiral topological phases, carrying protected edge states, based on a nontrivial connection with the thermodynamic entropy change occurring in a quantum point contact (QPC) as it pinches off the topological liquid into two. We show how this entropy change can be extracted using Maxwell relations from charge detection of a nearby quantum dot. We demonstrate this explicitly for the Abelian Laughlin states, using an exact solution of the sine-Gordon model describing the universal crossover in the QPC. Our approach might open a new thermodynamic detection scheme of topological states also with non-Abelian statistics.

Suk Bum Chung, Hong Yao, Taylor L. Hughes +1 more

We study the finite-temperature nature of a quantum phase transition between an Abelian and a non-Abelian topological phase in an exactly solvable model of the chiral spin liquid of Yao and Kivelson [Phys. Rev. Lett. 99, 247203 (2007)]. As it is exactly solvable, this model can serve as a testbed for developing better measures for describing topological quantum phase transitions. We use the global flux and entanglement entropy to characterize this phase transition, and we discuss to what extent the existence of a topologically ordered ground state with non-Abelian excitations is revealed at finite temperature.