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arXiv:2008.11793 · Physical review. B./Physical review. B

Entanglement bootstrap approach for gapped domain walls

Bowen Shi, Isaac H. Kim

Published Mar 26, 2021• 28 citations
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Abstract

We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector---referred to as the parton sector---that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground-state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020). Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.

DOI
10.1103/physrevb.103.115150

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We develop a theory of gapped domain wall between topologically ordered systems in two spatial dimensions. We find a new type of superselection sector---referred to as the parton sector---that subdivides the known superselection sectors localized on gapped domain walls. Moreover, we introduce and study the properties of composite superselection sectors that are made out of the parton sectors. We explain a systematic method to define these sectors, their fusion spaces, and their fusion rules, by deriving nontrivial identities relating their quantum dimensions and fusion multiplicities. We propose a set of axioms regarding the ground-state entanglement entropy of systems that can host gapped domain walls, generalizing the bulk axioms proposed in B. Shi, K. Kato, and I. H. Kim, Ann. Phys. 418, 168164 (2020). Similar to our analysis in the bulk, we derive our main results by examining the self-consistency relations of an object called information convex set. As an application, we define an analog of topological entanglement entropy for gapped domain walls and derive its exact expression.
In (2+1)(2+1)(2+1)-dimensional topologically ordered many-body states, transparent (topologically deformable) domain walls are invisible to local topological probes, yet can modify the ground state degeneracy (GSD) and transmute anyons transported across them. Here, we develop an entanglement-bootstrap framework using information convex sets (ICSs) on local and noncontractible annuli to extract information about transparent domain walls directly from ground state wavefunctions at fixed points of Abelian topological phases on a torus, without taking categorical defect data as input. We derive fusion rules governing the action of anyons on extreme points of ICSs on noncontractible annuli and determine their quantum dimensions. Extreme points invariant under transport around the complementary cycle correspond one-to-one to minimum entropy states (MESs), and their number equals the GSD. The maximal topological entanglement entropy (TEE), γLWmax⁡=log⁡(D/dα)γ_{\rm LW}^{\max}=\log(\mathcal{D}/d_α)γLWmax​=log(D/dα​), probes the net effect of walls crossing the chosen annulus, where D\mathcal{D}D is the total quantum dimension and dαd_αdα​ is the quantum dimension of an extreme point of its ICS. In contrast, the maximal entanglement asymmetry is ΔSXmax⁡=log⁡GSDΔS_X^{\max}=\log\mathrm{GSD}ΔSXmax​=logGSD. This value is the same for both annulus orientations and reflects the combined effect of the transparent domain walls. We further show that transparent domain walls can give rise to symmetries supported jointly on the two chosen fundamental cycles that cannot be decomposed into a product of two 111-form symmetry operators, one supported on each cycle. By relating their action on MESs to anyon tunneling and the fusion rules, we clarify how these symmetries connect distinct ground states. Additionally, we apply the framework to Wen's plaquette model, the anisotropic dipolar toric code, and the rank-2 toric code.
We investigate domain walls between topologically ordered phases in two spatial dimensions. We present a method which allows for the determination of the superselection sectors of excitations of such walls and which leads to a unified description of the kinematics of a wall and the two phases to either side of it. This incorporates a description of scattering processes at domain walls which can be applied to questions of transport through walls. In addition to the general formalism, we give representative examples including domain walls between the Abelian and non-Abelian topological phases of Kitaev's honeycomb lattice model in a magnetic field, as well as recently proposed domain walls between spin polarized and unpolarized non-Abelian fractional quantum Hall states at different filling fractions.
We establish a direct connection between the entanglement bootstrap and the Doplicher--Haag--Roberts theory of superselection sectors for two-dimensional quantum spin systems. Under the entanglement-bootstrap assumptions, we prove Haag duality and establish a one-to-one correspondence between simple entanglement-bootstrap anyon types and irreducible superselection sectors. For each fusion channel, we further construct an affine bijection from density matrices on the entanglement-bootstrap fusion space to normalized positive intertwiners in the corresponding operator-algebraic fusion channel. These results connect the entropic and operator-algebraic descriptions of anyons directly, without passing through exactly solvable lattice models.
We extend the entanglement bootstrap program to (3+1)-dimensions. We study knotted excitations of (3+1)-dimensional liquid topological orders and exotic fusion processes of loops. As in previous work in (2+1)-dimensions [Ann. Phys. 418, 168164 (2020), Phys. Rev. B 103, 115150 (2021)], we define a variety of superselection sectors and fusion spaces from two axioms on the ground state entanglement entropy. In particular, we identify fusion spaces associated with knots. We generalize the information convex set to a new class of regions called immersed regions, promoting various theorems to this new context. Examples from solvable models are provided; for instance, a concrete calculation of knot multiplicity shows that the knot complement of a trefoil knot can store quantum information. We define spiral maps that allow us to understand consistency relations for torus knots as well as spiral fusions of fluxes.
Gapped domain walls, as topological line defects between (2+1)D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which (2+1)D topological orders must have gapless edge modes, namely, which (1+1)D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix W, whose entries are the fusion-space dimensions W(ia), to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.