The Drinfeld center fusion category $\mathcal{Z}(\mathrm{Vec}_G)$ famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that $\mathcal{Z}(\mathrm{Vec}_G)$ reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group $G$ .The Drinfeld center fusion category $\mathcal{Z}(\mathrm{Vec}_G)$ famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that $\mathcal{Z}(\mathrm{Vec}_G)$ reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group $G$.

cond-mat.str-elCross-listedarXiv:2603.22029Mar 24, 20260 citations

Bloch Hamiltonians and topologically ordered states

Luigi MARTINA, A. P. Protogenov, V. Verbus

We consider a universal representation for the Hamiltonian of systems in topologically ordered phase states. We show that for strongly correlated electronic systems, the Hamiltonian expressed in terms of projectors of the Temperley–Lieb algebra on the spin singlet state has the form of a two-dimensional Bloch matrix in the case of doubly linked excitation world lines; in this case, different topological orderings are separated by a quantum critical point where the matrix elements of the Hamiltonian vanish.We consider a universal representation for the Hamiltonian of systems in topologically ordered phase states. We show that for strongly correlated electronic systems, the Hamiltonian expressed in terms of projectors of the Temperley–Lieb algebra on the spin singlet state has the form of a two-dimensional Bloch matrix in the case of doubly linked excitation world lines; in this case, different topological orderings are separated by a quantum critical point where the matrix elements of the Hamiltonian vanish.

Jan 1, 20090 citations

Topological phases in Fock-state lattices

Han Cai, Dawei Wang

Sep 30, 20190 citations

Topological order on the Bloch sphere

Rotem Liss, Tal Mor, Román Orús

Abstract A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [Boyer et al 2017 PRA 95 032 308]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev’s toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ‘trivial’ states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.Abstract A Bloch sphere is the geometrical representation of an arbitrary two-dimensional Hilbert space. Possible classes of entanglement and separability for the pure and mixed states on the Bloch sphere were suggested by [Boyer et al 2017 PRA 95 032 308]. Here we construct a Bloch sphere for the Hilbert space spanned by one of the ground states of Kitaev’s toric code model and one of its closest product states. We prove that this sphere contains only one separable state, thus belonging to the fourth class suggested by the said paper. We furthermore study the topological order of the pure states on its surface and conclude that, according to conventional definitions, only one state (the toric code ground state) seems to present non-trivial topological order. We conjecture that most of the states on this Bloch sphere are neither ‘trivial’ states (namely, they cannot be generated from a product state using a trivial circuit) nor topologically ordered. In addition, we show that the whole setting can be understood in terms of Grover rotations with gauge symmetry, akin to the quantum search algorithm.

arXiv:1812.00671New Journal of PhysicsMar 1, 20200 citations

A symmetry principle for topological quantum order

Zohar Nussinov, Gerardo Ortíz

arXiv:cond-mat/0702377Annals of PhysicsDec 11, 2008311 citations

The Drinfeld center fusion category

\mathcal{Z}(\mathrm{Vec}_G)

famously models anyons in certain lattice models. Here we demonstrate how its fusion rules may also describe topological order in fractional topological insulator materials, in the vicinity of point defects in the Brillouin zone. Concretely, we prove that

\mathcal{Z}(\mathrm{Vec}_G)

reflects, locally over a punctured disk in the Brillouin zone, the monodromy (topological order) of gapped quantum states over the parameter space of Bloch Hamiltonians whose classifying space has fundamental group

G