Condensed Matter Seminar - "The tenfold way on a finite Hilbert space: topology of entanglement for qubits and fermions"

The tenfold classification of topological matter applies equally to finite-dimensional Hilbert spaces, where it classifies the generators of entanglement. For two qubits, time-reversal-symmetric Hamiltonians split into five disconnected phases (ℤ₅), with Bell states as zero-dimensional edge states; for two fermions in graphene vacancy modes, particle–hole symmetry plays the same role and adds orbital entanglement with no qubit analog. A sum rule relates the winding number of any gate to measurable Berry phases. Topology is necessary, but not sufficient, for entanglement.