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APS

Entanglement Spectra of Stabilizer Codes: A Window into Fracton Order and Sub-System Symmetry Protected Topological States

Albert T. Schmitz, Sheng-Jie Huang, Abhinav Prem

Published Jan 1, 2019• 0 citations
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Entanglement spectra of stabilizer codes: A window into gapped quantum phases of matter

Albert T. Schmitz, Sheng-Jie Huang, Abhinav Prem

The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that nonlocal surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal nonlocal features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems---fracton and SSPT phases---additionally show a nontrivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasilocal entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.

arXiv:1901.10486•Physical review. B./Physical review. B•May 8, 2019•31 citations

Distilling Fractons from Layered Subsystem-Symmetry Protected Phases

Albert T. Schmitz

It is well-known that 3D Type-I fracton models can be obtained from the condensations of stacked layers of 2D anyons. It is less obvious if 3D Type-II fractons can be understood from a similar perspective. In this paper, we affirm that this is the case: we produce the paradigm Type-II fracton model, Haah's cubic code, from a 2D layer construction. However, this is not a condensation of 2D anyons, but rather we start with stacks of 2D subsystem-symmetry protected topological states (SSPT). As this parent model is not topologically ordered in the strict sense, whereas the final state is, we refer to this process as a distillation as we are forming a long-range entangled (LRE) state from several copies of a short-range entangled (SRE) state. We also show that Type-I fracton topological order can also be distilled from SSPT states in the form of the cluster-cube model which we introduce here. We start by introducing the Brillouin-Wigner perturbation theory of distillation. However, a more detailed analysis from which the perturbation theory follows, is also performed using linear gauge structures and an extension we introduce here referred to as gauge substructures. This allows us to rigorously define distillation as well as understand the process of obtaining LRE from SRE. We can diagnose the source of LRE as the distillation of subsystem symmetries into robust long-range ground state degeneracy as characterized by logical operators of the resulting stabilizer code. Furthermore, we find which Hamiltonian terms are necessary for selecting the ground state which results from the perturbation. This leads to a protocol for realizing a fracton quantum error correcting code initialized in a chosen fiducial state using only finite-depth circuits and local measurements.

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Topological entanglement entropy of fracton stabilizer codes

Han Ma, Albert T. Schmitz, S. A. Parameswaran +2 more

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

arXiv:1710.01744•Physical review. B./Physical review. B•Mar 1, 2018•111 citations
The entanglement spectrum (ES) provides a barometer of quantum entanglement and encodes physical information beyond that contained in the entanglement entropy. In this paper, we explore the ES of stabilizer codes, which furnish exactly solvable models for a plethora of gapped quantum phases of matter. Studying the ES for stabilizer Hamiltonians in the presence of arbitrary weak local perturbations thus allows us to develop a general framework within which the entanglement features of gapped topological phases can be computed and contrasted. In particular, we study models harboring fracton order, both type-I and type-II, and compare the resulting ES with that of both conventional topological order and of (strong) subsystem symmetry protected topological (SSPT) states. We find that nonlocal surface stabilizers (NLSS), a set of symmetries of the Hamiltonian which form on the boundary of the entanglement cut, act as purveyors of universal nonlocal features appearing in the entanglement spectrum. While in conventional topological orders and fracton orders, the NLSS retain a form of topological invariance with respect to the entanglement cut, subsystem symmetric systems---fracton and SSPT phases---additionally show a nontrivial geometric dependence on the entanglement cut, corresponding to the subsystem symmetry. This sheds further light on the interplay between geometric and topological effects in fracton phases of matter and demonstrates that strong SSPT phases harbour a measure of quasilocal entanglement beyond that encountered in conventional SPT phases. We further show that a version of the edge-entanglement correspondence, established earlier for gapped two-dimensional topological phases, also holds for gapped three-dimensional fracton models.
It is well-known that 3D Type-I fracton models can be obtained from the condensations of stacked layers of 2D anyons. It is less obvious if 3D Type-II fractons can be understood from a similar perspective. In this paper, we affirm that this is the case: we produce the paradigm Type-II fracton model, Haah's cubic code, from a 2D layer construction. However, this is not a condensation of 2D anyons, but rather we start with stacks of 2D subsystem-symmetry protected topological states (SSPT). As this parent model is not topologically ordered in the strict sense, whereas the final state is, we refer to this process as a distillation as we are forming a long-range entangled (LRE) state from several copies of a short-range entangled (SRE) state. We also show that Type-I fracton topological order can also be distilled from SSPT states in the form of the cluster-cube model which we introduce here. We start by introducing the Brillouin-Wigner perturbation theory of distillation. However, a more detailed analysis from which the perturbation theory follows, is also performed using linear gauge structures and an extension we introduce here referred to as gauge substructures. This allows us to rigorously define distillation as well as understand the process of obtaining LRE from SRE. We can diagnose the source of LRE as the distillation of subsystem symmetries into robust long-range ground state degeneracy as characterized by logical operators of the resulting stabilizer code. Furthermore, we find which Hamiltonian terms are necessary for selecting the ground state which results from the perturbation. This leads to a protocol for realizing a fracton quantum error correcting code initialized in a chosen fiducial state using only finite-depth circuits and local measurements.
Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.