The topological entanglement entropy
or ''topological entropy'', usually denoted by
, is a number characterizing many-body states that possess
topological order
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ge ...
.
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
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ge ...
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
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density mat ...
measuring the
quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state o ...
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'':
:
where
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''
2 fractionalized states, such as topologically ordered states of
''Z''
2 spin-liquid,
quantum dimer models on non-bipartite lattices, and Kitaev's
toric code The toric code is a topological quantum error correcting code, and an example of a stabilizer code, defined on a two-dimensional spin lattice. It is the simplest and most well studied of the quantum double models. It is also the simplest example o ...
state, are characterized ''γ'' = log(2).
See also
*
Quantum topology
Quantum topology is a branch of mathematics that connects quantum mechanics with low-dimensional topology.
Dirac notation provides a viewpoint of quantum mechanics which becomes amplified into a framework that can embrace the amplitudes associat ...
*
Topological defect
A topological soliton occurs when two adjoining structures or spaces are in some way "out of phase" with each other in ways that make a seamless transition between them impossible. One of the simplest and most commonplace examples of a topological ...
*
Topological order
In physics, topological order is a kind of order in the zero-temperature phase of matter (also known as quantum matter). Macroscopically, topological order is defined and described by robust ground state degeneracy and quantized non-Abelian ge ...
*
Topological quantum field theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants.
Although TQFTs were invented by physicists, they are also of mathe ...
*
Topological quantum number
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are ...
*
Topological string theory
In theoretical physics, topological string theory is a version of string theory. Topological string theory appeared in papers by theoretical physicists, such as Edward Witten and Cumrun Vafa, by analogy with Witten's earlier idea of topological ...
References
Calculations for specific topologically ordered states
*
*
Condensed matter physics
Statistical mechanics
Entropy
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