On the stability of topological order in tensor network states

Abstract

We construct a tensor network representation of the 3d toric code ground state that is stable to a generating set of uniform local tensor perturbations, including those that do not map to local operators on the physical Hilbert space. The stability is established by mapping the phase diagram of the perturbed tensor network to that of the 3d Ising gauge theory, which has a non-zero finite temperature transition. More generally, we find that the stability of a topological tensor network state is determined by the form of its virtual symmetries and the topological excitations created by virtual operators that break those symmetries. In particular, a dual representation of the 3d toric code ground state, as well as representations of the X-cube and cubic code ground states, for which point-like excitations are created by such operators, are found to be unstable.

arXiv comment8+10 pages, 1 figure; v2 published version

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• Dualities in one-dimensional quantum lattice models: symmetric Hamiltonians and matrix product operator intertwiners
• Long-range entanglement from measuring symmetry-protected topological phases
• Separability transitions in topological states induced by local decoherence
• Tensor network approach to electromagnetic duality in (3+1)d topological gauge models
• Topological fracton quantum phase transitions by tuning exact tensor network states
• On tensor network representations of the (3+1)d toric code
• Topological line in frustrated Toric code models
• Symmetry topological field theory and non-abelian Kramers-Wannier dualities of generalised Ising models
• Robustness of critical U(1) spin liquids and emergent symmetries in tensor networks
• Holographic duality between bulk topological order and boundary mixed-state order