Topological entropy in physics

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The topological entanglement entropy^[1]^[2]^[3] or topological entropy, usually denoted by $\gamma$ , is a number characterizing many-body states that possess topological order.

A non-zero topological entanglement entropy reflects the presence of long range quantum entanglements in a many-body quantum state. So the topological entanglement entropy links topological order with pattern of long range quantum entanglements.

Given a topologically ordered state, the topological entropy can be extracted from the asymptotic behavior of the Von Neumann entropy measuring the quantum entanglement between a spatial block and the rest of the system. The entanglement entropy of a simply connected region of boundary length L, within an infinite two-dimensional topologically ordered state, has the following form for large L:

S_{L}\;\longrightarrow \;\alpha L-\gamma +{\mathcal {O}}(L^{-\nu })\;,\qquad \nu >0\,\!

where $-\gamma$ is the topological entanglement entropy.

The topological entanglement entropy is equal to the logarithm of the total quantum dimension of the quasiparticle excitations of the state.

For example, the simplest fractional quantum Hall states, the Laughlin states at filling fraction 1/m, have γ = ½log(m). The Z₂ fractionalized states, such as topologically ordered states of Z₂ spin-liquid, quantum dimer models on non-bipartite lattices, and Kitaev's toric code state, are characterized γ = log(2).

References

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^ Hamma, Alioscia; Ionicioiu, Radu; Zanardi, Paolo (2005). "Ground state entanglement and geometric entropy in the Kitaev model". Physics Letters A. 337 (1–2): 22–28. arXiv:quant-ph/0406202. doi:10.1016/j.physleta.2005.01.060. S2CID 118924738.
^ Kitaev, Alexei; Preskill, John (24 March 2006). "Topological Entanglement Entropy". Physical Review Letters. 96 (11): 110404. arXiv:hep-th/0510092. Bibcode:2006PhRvL..96k0404K. doi:10.1103/physrevlett.96.110404. ISSN 0031-9007. PMID 16605802. S2CID 18480266.
^ Levin, Michael; Wen, Xiao-Gang (24 March 2006). "Detecting Topological Order in a Ground State Wave Function". Physical Review Letters. 96 (11): 110405. arXiv:cond-mat/0510613. Bibcode:2006PhRvL..96k0405L. doi:10.1103/physrevlett.96.110405. ISSN 0031-9007. PMID 16605803. S2CID 206329868.

Calculations for specific topologically ordered states

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Haque, Masudul; Zozulya, Oleksandr; Schoutens, Kareljan (6 February 2007). "Entanglement Entropy in Fermionic Laughlin States". Physical Review Letters. 98 (6): 060401. arXiv:cond-mat/0609263. Bibcode:2007PhRvL..98f0401H. doi:10.1103/physrevlett.98.060401. ISSN 0031-9007. PMID 17358917. S2CID 5731929.
Furukawa, Shunsuke; Misguich, Grégoire (5 June 2007). "Topological entanglement entropy in the quantum dimer model on the triangular lattice". Physical Review B. 75 (21): 214407. arXiv:cond-mat/0612227. Bibcode:2007PhRvB..75u4407F. doi:10.1103/physrevb.75.214407. ISSN 1098-0121. S2CID 118950876.

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Calculations for specific topologically ordered states