non-Abelian Berry connection
E911212
concept in geometric phase theory
concept in quantum mechanics
gauge field
geometric connection
matrix-valued Berry connection
The non-Abelian Berry connection is a gauge-theoretic generalization of the Berry phase that describes how degenerate quantum states transform under adiabatic evolution, leading to matrix-valued geometric phases and phenomena such as the Yang monopole.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
concept in geometric phase theory
ⓘ
concept in quantum mechanics ⓘ gauge field ⓘ geometric connection ⓘ matrix-valued Berry connection ⓘ |
| appliesTo | degenerate quantum states ⓘ |
| associatedWith |
degenerate eigenspaces
ⓘ
vector bundles of degenerate states ⓘ |
| captures |
non-Abelian holonomy
ⓘ
path-dependent unitary transformations ⓘ |
| contrastsWith | Abelian Berry connection NERFINISHED ⓘ |
| definedFor | families of Hamiltonians with degeneracies ⓘ |
| definedFrom | basis of instantaneous degenerate eigenstates ⓘ |
| definedOn | parameter space of a Hamiltonian ⓘ |
| describes |
adiabatic evolution of degenerate subspaces
ⓘ
holonomy in parameter space ⓘ non-Abelian geometric phases ⓘ parallel transport of degenerate eigenstates ⓘ |
| enables |
description of Yang monopole
ⓘ
matrix-valued geometric phases ⓘ |
| gaugeGroup |
U(N)
NERFINISHED
ⓘ
unitary group ⓘ |
| generalizationOf |
Abelian Berry phase
NERFINISHED
ⓘ
Berry connection ⓘ |
| hasMathematicalForm | matrix-valued one-form on parameter space ⓘ |
| hasProperty |
adiabatic
ⓘ
gauge-dependent ⓘ geometric ⓘ matrix-valued ⓘ non-commutative ⓘ |
| introducedInContextOf | adiabatic theorem ⓘ |
| relatedTo |
Berry curvature
ⓘ
Wilczek–Zee phase NERFINISHED ⓘ Yang monopole ⓘ fiber bundles ⓘ gauge theory ⓘ holonomy group ⓘ non-Abelian Berry curvature ⓘ |
| requires | degeneracy of energy levels ⓘ |
| takesValuesIn |
Lie algebra of a unitary group
ⓘ
u(N) ⓘ |
| transformsUnder | non-Abelian gauge transformations ⓘ |
| usedIn |
adiabatic quantum computation
ⓘ
holonomic quantum computation NERFINISHED ⓘ non-Abelian anyons NERFINISHED ⓘ quantum Hall systems ⓘ topological insulators ⓘ topological phases of matter ⓘ topological superconductors ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.