The nonmodular topological phase and phase singularities

Ping Ao, Xiao-Mei Zhu

Michael Hartley Freedman

Ramanjit Sohal, Abhinav Prem

We propose a general framework for studying two-dimensional (2D) topologically ordered states subject to local correlated errors and show that the resulting mixed state can display (imTO)—topological order that is not expected to occur in the ground state of 2D local gapped Hamiltonians. Specifically, we show that decoherence, previously interpreted as anyon condensation in a doubled Hilbert space, is more naturally phrased as, and provides a physical mechanism for, “gauging out” anyons in the original Hilbert space. We find that gauging out anyons generically results in imTO, with the decohered mixed state strongly symmetric under certain anomalous 1-form symmetries. This framework lays bare a striking connection between the decohered density matrix and , which can appear as anomalous surface states of three-dimensional topological orders. Through a series of examples, we show that the decohered state can display a classical memory, encode logical qubits (i.e., exhibit a quantum memory), and even host chiral or nonmodular topological order. We argue that a partial classification of imTO is given in terms of nonmodular braided-fusion categories.

David S. Simon

Ching-Yu Huang, Feng-Li Lin

We study the connection between topological order and degeneracy of the singular value spectrum by explicitly solving the two-dimensional dimerized quantum Heisenberg model in the form of the tensor product state ansatz. Based on the ground-state solution, we find nonzero topological entanglement entropy at the frustrated regime. It indicates a possible topological phase. Furthermore, we find that the singular value spectrum associated with each link in the tensor product state is doubly degenerate only in this phase. Degeneracy of the singular value spectrum is robust against various types of perturbations, in accordance with our expectation for topological order. Our results support the connection among topological order, long-range entanglement, and the dominant degenerate singular values. In the context of the tensor product state ansatz, the numerical evaluation of the singular value spectrum costs far less computation power than that for topological entanglement entropy. Our results provide a more viable way to numerically identify the topological order for the generic frustrated systems.

Partha Ghose