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arXiv:2302.11468 · Physical review. B./Physical review. B

Entanglement entropy of topological phases with multipole symmetry

Hiromi Ebisu

Published Jul 25, 2024• 3 citations
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Abstract

We study entanglement entropy of unusual ${\mathbb{Z}}_{N}$ 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.

DOI
10.1103/physrevb.110.045146

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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.
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 models with fractonlike order based on Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories with subsystem symmetries in two and three spatial dimensions. The three-dimensional (3D) model reduces to the 3D toric code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase. Although not topologically protected, its ground-state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows that fractonlike phases are also present in more usual lattice gauge theories. We calculate the entanglement entropy SA{S}_{A}SA​ of these models in a subregion AAA of the lattice and show that it is equal to the logarithm of the ground-state degeneracy of a particular restriction of the full model to AAA.
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.
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.
We discuss entanglement entropy of gapped ground states in different dimensions, obtained on partitioning space into two regions. For trivial phases without topological order, we argue that the entanglement entropy may be obtained by integrating an ``entropy density'' over the partition boundary that admits a gradient expansion in the curvature of the boundary. This constrains the expansion of entanglement entropy as a function of system size and points to an even-odd dependence on dimensionality. For example, in contrast to the familiar result in two dimensions, a size-independent constant contribution to the entanglement entropy can appear for trivial phases in any odd spatial dimension. We then discuss phases with topological entanglement entropy (TEE) that cannot be obtained by adding local contributions. We find that in three dimensions there is just one type of TEE, as in two dimensions, that depends linearly on the number of connected components of the boundary (the ``zeroth Betti number''). In D>3D>3D>3 dimensions, new types of TEE appear which depend on the higher Betti numbers of the boundary manifold. We construct generalized toric code models that exhibit these TEEs and discuss ways to extract TEE in $D\ensuremath{\ge}3$.
Motivated by recent interest in fracton topological phases, we explore the interplay between gapped 2D ZN{\mathbb{Z}}_{N}ZN​ topological phases which admit fractional excitations with restricted mobility and geometry of the lattice on which such phases are placed. We investigate the properties of the phases in a geometric context---graph theory. By placing the phases on a 2D lattice consisting of two arbitrary connected graphs, ${G}_{x}\ensuremath{\boxtimes}{G}_{y}$, we study the behavior of fractional excitations of the phases. We derive the formula of the ground-state degeneracy of the phases, which depends on invariant factors of the Laplacian.