In
quantum many-body physics
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
, topological degeneracy is a phenomenon in which the
ground state of a
gapped many-body Hamiltonian becomes degenerate in the
limit of large system size such that the degeneracy cannot be lifted by any
local perturbations.
Applications
Topological degeneracy can be used to protect qubits which allows
topological quantum computation
A topological quantum computer is a theoretical quantum computer proposed by Russian-American physicist Alexei Kitaev in 1997. It employs quasiparticles in two-dimensional systems, called anyons, whose world lines pass around one another to form ...
. It is believed that topological degeneracy implies
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 ...
(or long-range entanglement ) in the ground state.
Many-body states with topological degeneracy are described by
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 ...
at low energies.
Background
Topological degeneracy was first introduced to physically define topological order.
In two-dimensional space, the topological degeneracy depends on the topology of space, and the topological degeneracy on high genus Riemann surfaces encode all information on the
quantum dimensions and the fusion algebra of the quasiparticles. In particular, the topological degeneracy on torus is equal to the number of quasiparticles types.
The topological degeneracy also appears in the situation with topological defects (such as vortices, dislocations, holes in 2D sample, ends of a 1D sample, etc.), where the topological degeneracy depends on the number of defects. Braiding those topological defect leads to topologically protected non-Abelian
geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the ...
, which can be used to perform topologically protected
quantum computation.
Topological degeneracy of topological order can be defined on a closed space or an open space with gapped boundaries or gapped domain
walls,
including both Abelian topological orders
and non-Abelian topological orders.
The application of these types of systems for
quantum computation has been proposed.
In certain generalized cases, one can also design the systems with topological interfaces enriched or extended by global or gauge symmetries.
The topological degeneracy also appear in non-interacting fermion systems (such as p+ip superconductors) with trapped defects (such as vortices). In non-interacting fermion systems, there is only one type of topological degeneracy
where number of the degenerate states is given by
, where
is the number of the defects (such as the number of vortices).
Such topological degeneracy is referred as "Majorana zero-mode" on the defects.
In contrast, there are many types of topological degeneracy for interacting systems.
A systematic description of topological degeneracy is given by tensor category (or
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left a ...
) theory.
See also
*
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 ...
*
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 associa ...
*
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 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 ...
*
Majorana fermion
References
{{Reflist
Quantum phases
Condensed matter physics