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arXiv:1705.09300 · Physical review. B./Physical review. B

Deciphering the nonlocal entanglement entropy of fracton topological orders

Bowen Shi, Yuan-Ming Lu

Published Apr 12, 2018• 68 citations
arXivHTMLPDF

Abstract

The authors discuss a lower bound of nonlocal entanglement entropy (a generalization of topological entanglement entropy). The lower bound applies to Abelian conventional topological order as well as to type-I and type-II fracton models. It provides the physical intuition that each type of topological excitation, created by a deformable string, membrane, or fractal operator, gives a contribution to the nonlocal entanglement entropy. For fracton models, the lower bound is geometry dependent; it can take extensive values.

DOI
10.1103/physrevb.97.144106

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The authors discuss a lower bound of nonlocal entanglement entropy (a generalization of topological entanglement entropy). The lower bound applies to Abelian conventional topological order as well as to type-I and type-II fracton models. It provides the physical intuition that each type of topological excitation, created by a deformable string, membrane, or fractal operator, gives a contribution to the nonlocal entanglement entropy. For fracton models, the lower bound is geometry dependent; it can take extensive values.
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.
We study entanglement entropy of unusual ZN{\mathbb{Z}}_{N}ZN​ topological stabilizer codes which admit fractional excitations with restricted mobility constraint in a manner akin to fracton topological phases. It is widely known that the subleading term of the entanglement entropy of a disk geometry in conventional topologically ordered phases is related to the total number of the quantum dimension of the fractional excitations. We show that, in our model, such a relation does not hold, i.e., the total number of the quantum dimension varies depending on the system size, whereas the subleading term of the entanglement entropy takes a constant number irrespective to the system size. We give a physical interpretation of this result in the simplest case of the model. More thorough analysis on the entanglement entropy of the model on generic lattices is also presented.
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.
Abstract We compute the topological entanglement entropy for a large set of lattice models in d-dimensions. It is well known that many such quantum systems can be constructed out of lattice gauge models. For dimensionality higher than two, there are generalizations going beyond gauge theories, which are called higher gauge theories and rely on higher-order generalizations of groups. Our main concern is a large class of d-dimensional quantum systems derived from Abelian higher gauge theories. In this paper, we derive a general formula for the bipartition entanglement entropy for this class of models, and from it we extract both the area law and the sub-leading terms, which explicitly depend on the topology of the entangling surface. We show that the entanglement entropy SA in a sub-region A is proportional to log (GSDA~) \left({GSD}_{\tilde{A}}\right) (GSDA~​) GSD A ˜ , where GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ is the ground state degeneracy of a particular restriction of the full model to A. The quantity GSDA~ {GSD}_{\tilde{A}} GSDA~​ GSD A ˜ can be further divided into a contribution that scales with the size of the boundary ∂A and a term which depends on the topology of ∂A. There is also a topological contribution coming from A itself, that may be non-zero when A has a non-trivial homology. We present some examples and discuss how the topology of A affects the topological entropy. Our formalism allows us to do most of the calculation for arbitrary dimension d. The result is in agreement with entanglement calculations for known topological models.
Excitations in (3+1)-dimensional [(3+1)D] topologically ordered phases have very rich structures. (3+1)D topological phases support both pointlike and stringlike excitations, and in particular the loop (closed string) excitations may admit knotted and linked structures. In this work, we ask the following question: How do different types of topological excitations contribute to the entanglement entropy or, alternatively, can we use the entanglement entropy to detect the structure of excitations, and further obtain the information of the underlying topological order? We are mainly interested in (3+1)D topological order that can be realized in Dijkgraaf-Witten (DW) gauge theories, which are labeled by a finite group GGG and its group 4-cocycle $\ensuremath{\omega}\ensuremath{\in}{\mathcal{H}}^{4}[G;\text{U}(1)]$ up to group automorphisms. We find that each topological excitation contributes a universal constant lndiln{d}_{i}lndi​ to the entanglement entropy, where di{d}_{i}di​ is the quantum dimension that depends on both the structure of the excitation and the data $(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})$. The entanglement entropy of the excitations of the linked/unlinked topology can capture different information of the DW theory $(G,\phantom{\rule{0.16em}{0ex}}\ensuremath{\omega})$. In particular, the entanglement entropy introduced by Hopf-link loop excitations can distinguish certain group 4-cocycles $\ensuremath{\omega}$ from the others.
In this work, we develop a coupled layer construction of fracton topological orders in d=3d=3d=3 spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple d=2d=2d=2 topological phases and from stacks of d=3d=3d=3 fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.