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In physics, topological order [1] 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 [2] and quantized non-abelian geometric phases of degenerate ground states. [1]
An alternative algorithm for topological sorting is based on depth-first search.The algorithm loops through each node of the graph, in an arbitrary order, initiating a depth-first search that terminates when it hits any node that has already been visited since the beginning of the topological sort or the node has no outgoing edges (i.e., a leaf node):
Symmetry-protected topological (SPT) order [1] [2] is a kind of order in zero-temperature quantum-mechanical states of matter that have a symmetry and a finite energy gap. To derive the results in a most-invariant way, renormalization group methods are used (leading to equivalence classes corresponding to certain fixed points). [ 1 ]
In the branch of mathematics known as topology, the specialization (or canonical) preorder is a natural preorder on the set of the points of a topological space.For most spaces that are considered in practice, namely for all those that satisfy the T 0 separation axiom, this preorder is even a partial order (called the specialization order).
The left and right order topologies can be used to give counterexamples in general topology. For example, the left or right order topology on a bounded set provides an example of a compact space that is not Hausdorff. The left order topology is the standard topology used for many set-theoretic purposes on a Boolean algebra. [clarification needed]
It is also the simplest example of topological order—Z 2 topological order (first studied in the context of Z 2 spin liquid in 1991). [ 3 ] [ 4 ] The toric code can also be considered to be a Z 2 lattice gauge theory in a particular limit. [ 5 ]
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
The toric-code/ gauge theory in 2D is an example of the former, and the topological order in this phase can be diagnosed by Wilson loop operators. The topological order is destroyed in equilibrium at any finite temperature due to fluctuating vortices--- however, these can get localized by disorder, enabling glassy localization-protected ...