List of disproved mathematical ideas

2014-01-23T22:10:49Z

81.167.104.3: Removed factual inaccuracy.

The '''topological entanglement entropy'''{{ref|KitaevPreskill}} {{ref|LevinWen}}, usually denoted by ''γ'', is a number characterizing many-body states that possess [[topological order]].
The short form ''topological entropy'' is often used, although the same name in [[ergodic theory]] refers to an unrelated mathematical concept (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 [[topological order|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'':

:<math> S_L \; \longrightarrow \; \alpha L -\gamma +\mathcal{O}(L^{-\nu}) \; , \qquad \nu>0 \,\!</math>

''-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 ''Z''<sub>2</sub> fractionalized states, such as topologically ordered states of
''Z''<sub>2</sub> 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 ==
{{reflist}}
{{Nofootnotes|date=April 2008}}
===Introduction of the measure===
#{{note|KitaevPreskill}} Topological Entanglement Entropy, Alexei Kitaev and John Preskill, [http://link.aps.org/abstract/PRL/v96/e110404 Phys. Rev. Lett. '''96''', 110404 (2006)].
#{{note|LevinWen}} Detecting Topological Order in a Ground State Wave Function, Michael Levin and Xiao-Gang Wen, [http://link.aps.org/abstract/PRL/v96/e110405 Phys. Rev. Lett. '''96''', 110405 (2006)].

===Calculations for specific topologically ordered states===

* M. Haque, O. Zozulya and K. Schoutens; Phys. Rev. Lett. '''98''', 060401 (2007).
* S. Furukawa and G. Misguich, Phys. Rev. B '''75''', 214407 (2007).

{{physics-stub}}
[[Category:Condensed matter physics]]
[[Category:Statistical mechanics]]

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List of disproved mathematical ideas