Higher structures in matrix product states
Shuhei Ohyama, Shinsei Ryu
Abstract
For a parameterized family of invertible states (short-range-entangled states) in $(1+1)$ dimensions, we discuss a generalization of the Berry phase. Using translationally invariant, infinite matrix product states (MPSs), we introduce a gerbe structure, a higher generalization of complex line bundles, as an underlying mathematical structure describing topological properties of a parameterized family of MPSs. Furthermore, we introduce a generalization of a quantum mechanical inner product, which we call the ``triple inner product,'' defined for three matrix product states. The triple inner product proves to extract a topological invariant, the Dixmier-Douady class over the parameter space.
Topics & Concepts
Parameterized complexityInvertible matrixGeneralizationInvariant (physics)Pure mathematicsMathematicsMatrix multiplicationProduct (mathematics)Matrix (chemical analysis)Topology (electrical circuits)QuantumPhysicsMathematical physicsMathematical analysisCombinatoricsQuantum mechanicsGeometryComposite materialMaterials scienceTopological Materials and PhenomenaQuantum many-body systemsAlgebraic structures and combinatorial models