that topological degeneracy implies topological order (or long-range entanglement ) in the ground state. Many-body states with topological degeneracy are...

a phase transition. Various topologically ordered states have interesting properties, such as (1) topological degeneracy and fractional statistics or...

by the singular set SY of a topological space Y. Here SYn consists of all the continuous maps from the standard topological n-simplex to Y. The singular...

of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how...

In quantum mechanics, the Kramers' degeneracy theorem states that for every energy eigenstate of a time-reversal symmetric system with half-integer total...

mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. While...

tensor gauge theories. Gapped fracton models often feature a topological ground state degeneracy that grows exponentially and sub-extensively with system...

states (see Quantum phase transition, Quantum critical point, Topological degeneracy, Topological order) and where the overall system structure is independent...

material. But in a topological insulator, these bands are, in an informal sense, "twisted", relative to a trivial insulator. The topological insulator cannot...

polygon do not consider the digon to be a proper polygon because of its degeneracy in the Euclidean case. A digon as a face of a polyhedron is degenerate...

entanglement see topological order, which is not related to the famous EPR paradox). Since short-range entangled states have only trivial topological orders we...

identifying 3+1‑dimensional topological orders. The multi-loop/string-braiding statistics of 3+1‑dimensional topological orders can be captured by the...

no degeneracy at all, low, intermediate, or high degeneracy. For example, the number of molecule atoms and the molecular weights are high degeneracy descriptors...

insensitive to noise and small extra features, robust to arbitrary topological degeneracies, and discriminating of shape differences at many scales. 3D search...

numbers | n ⟩ {\displaystyle |n\rangle } are referred to as topological vacua. Topological vacua are not candidate vacuum states of Yang–Mills theories...

quantum double models. It is also the simplest example of topological order—Z2 topological order (first studied in the context of Z2 spin liquid in 1991)...

(2015), "Topological Insulators, Topological Dirac semimetals, Topological Crystalline Insulators, and Topological Kondo Insulators", Topological Insulators...

intense gravitational attraction of the compact mass overcomes the electron degeneracy pressure and causes electron capture to occur within the star. The result...

mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex (resp. CW complex) refers to...

A graph H is called a topological minor of a graph G if a subdivision of H is isomorphic to a subgraph of G. Every topological minor is also a minor....

rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was...

topologically nontrivial phase of matter, together with Helium-3 A superfluid phase, that broadens the topological classification beyond topological insulators...

the geometric and topological information belonging to a 3-manifold. Thus, there is an interplay between group theory and topological methods. 3-manifolds...

vertices. embedding A graph embedding is a topological representation of a graph as a subset of a topological space with each vertex represented as a point...

In particular, πU is irreducible if and only if U is irreducible. Non-degeneracy of a representation π of Cc(G) on a Hilbert space Hπ means that { π (...

corresponding topological phase transitions. Topological phases are quantum phases of matter that are characterized by robust ground state degeneracy and quantized...

set. In certain categories of topological spaces, such as first-countable spaces, sequences characterize most topological properties, but this is not true...

notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks (the faces) whose pairwise...

, the chromatic number equals the degeneracy plus one. For these graphs, the greedy algorithm with the degeneracy ordering is always optimal. Every β...

Topological censorship Topological conjugacy Topological defect Topological degeneracy Topological entropy Topological entropy in physics Topological...