Topological entropy in physics

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

For the mathematical concept in ergodic theory, see topological entropy.

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:

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

-γ 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).

See also

• Quantum topology
• Topological defect
• Topological order
• Topological quantum field theory
• Topological quantum number
• Topological string theory

References

1. Hamma, Alioscia; Ionicioiu, Radu; Zanardi, Paolo (28 March 2004). "Ground state entanglement and geometric entropy in the Kitaev model". Physics Letter A. Elsevier. 337 (1–2). arXiv:quant-ph/0406202.
2. Kitaev, Alexei; Preskill, John (24 March 2006). "Topological Entanglement Entropy". Physical Review Letters. American Physical Society (APS). 96 (11): 110404. arXiv:hep-th/0510092. doi:10.1103/physrevlett.96.110404. ISSN 0031-9007. PMID 16605802.
3. Levin, Michael; Wen, Xiao-Gang (24 March 2006). "Detecting Topological Order in a Ground State Wave Function". Physical Review Letters. American Physical Society (APS). 96 (11): 110405. arXiv:cond-mat/0510613. doi:10.1103/physrevlett.96.110405. ISSN 0031-9007. PMID 16605803.

Calculations for specific topologically ordered states

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