Correspondence between bulk entanglement and boundary excitation spectra in two-dimensional gapped topological phases (original) (raw)
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In this paper we propose a Hamiltonian approach to gapped topological phases on open surfaces. Our setting is an extension of the Levin-Wen model to a 2d graph on an open surface, whose boundary is part of the graph. We systematically construct a series of boundary Hamiltonians such that each of them, when combined with the usual Levin-Wen bulk Hamiltonian, gives rise to a gapped energy spectrum which is topologically protected. It is shown that the corresponding wave functions are robust under changes of the underlying graph that maintain the spatial topology of the system. We derive explicit ground-state wavefunctions of the system on a disk as well as on a cylinder. For boundary quasiparticle excitations, we are able to construct their creation, annihilation, measuring and hopping operators etc. Given a bulk string-net theory, our approach provides a classification scheme of possible types of gapped boundary conditions by Frobenius algebras (modulo Morita equivalence) of the bulk fusion category; the boundary quasiparticles are characterized by bimodules of the pertinent Frobenius algebras. Our approach also offers a set of concrete tools for computations. We illustrate our approach by a few examples.
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Physical review research, 2022
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-onedimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.
Entanglement spectrum and boundary theories with projected entangled-pair states
2011
In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated to their boundaries have been observed. In this work, we provide an exact duality mapping between the bulk of a quantum spin system and its boundary using Projected Entangled Pair States (PEPS). This duality associates to every region a Hamiltonian on its boundary, in such a way that the entanglement spectrum of the bulk corresponds to the excitation spectrum of the boundary Hamiltonian. We study various specific models, like a deformed AKLT [1], an Ising-type , and Kitaev's toric code , both in finite ladders and infinite square lattices. In the latter case, some of those models display quantum phase transitions. We find that a gapped bulk phase with local order corresponds to a boundary Hamiltonian with local interactions, whereas critical behavior in the bulk is reflected on a diverging interaction length of the boundary Hamiltonian. Furthermore, topologically ordered states yield non-local Hamiltonians. As our duality also associates a boundary operator to any operator in the bulk, it in fact provides a full holographic framework for the study of quantum many-body systems via their boundary.
Topological Geometric Entanglement
Here we establish the connection between topological order and the geometric entanglement, as measured by the logarithm of the overlap between a given state and its closest product state of blocks. As happens for the entanglement entropy, we find that the geometric entanglement is the sum of two terms: a non-universal one obeying a boundary law times the number of blocks, and a universal one quantifying the underlying long-range entanglement of a topologically-ordered state. For simplicity we focus on the case of Kitaev's toric code model.
Topological quasiparticles and the holographic bulk-edge relation in 2+ 1D string-net models
String-net models allow us to systematically construct and classify 2+1D topologically ordered states which can have gapped boundaries. So we can use the simple ideal string-net wavefunctions to study all the universal properties of such topological orders. In this paper, we describe a finite computational method -Q-algebra module approach, that allows us to compute the non-Abelian statistics of the topological excitations [described by a modular tensor category (MTC)] from the string-net wavefunction [described by a unitary fusion category (UFC)]: MTC=Z(UFC), where Z is the functor that takes the Drinfeld center. We discuss several examples, including the twisted quantum double D α (G) phase. Our result can also be viewed from an angle of holographic bulkboundary relation. The 2+1D topological orders are classified by MTC plus the chiral central charge of the edge states, while the 1+1D anomalous topological orders (that appear on the edge of 2+1D gapped states) are classified by UFC. If we know an edge (described by a UFC) of a gapped 2+1D state, then our method allows us to compute the bulk topological order [described by a MTC=Z(UFC) with zero chiral central charge].
Physical Review B, 2015
In a system with chiral topological order, there is a remarkable correspondence between the edge and entanglement spectra: the low-energy spectrum of the system in the presence of a physical edge coincides with the lowest part of the entanglement spectrum (ES) across a virtual cut of the system into two parts, up to rescaling and shifting. This correspondence is believed to be due to the existence of protected gapless edge modes. In this paper, we explore whether the edge-entanglement spectrum correspondence extends to nonchiral topological phases, where there are no protected gapless edge modes. Specifically, we consider the Wen-plaquette model, which is equivalent to the Kitaev toric code model and has Z2 topological order (quantum double of Z2). The unperturbed Wen-plaquette model displays an exact correspondence: both the edge and entanglement spectra within each topological sector a (a = 1, • • • , 4) are flat and equally degenerate. Here, we show, through a detailed microscopic calculation, that in the presence of generic local perturbations: (i) the effective degrees of freedom for both the physical edge and the entanglement cut consist of a (spin-1/2) spin chain, with effective Hamiltonians H a edge and H a ent. , respectively, both of which have a Z2 symmetry enforced by the bulk topological order; (ii) there is in general no match between the low energy spectra of H a edge and H a ent. , that is, there is no edge-ES correspondence. However, if supplement the Z2 topological order with a global symmetry (translational invariance along the edge/entanglement cut), i.e., by considering the Wen-plaquette model as a symmetryenriched topological phase (SET), then there is a finite domain in Hamiltonian space in which both H a edge and H a ent. realize the critical Ising model, whose low-energy effective theory is the c = 1/2 Ising CFT. This is achieved because the presence of the global symmetry implies that the effective degrees of freedom of both the edge and entanglement cut are governed by Kramers-Wannier selfdual Hamiltonians, in addition to them being Z2 symmetric, which is imposed by the topological order. Thus, by considering the Wen-plaquette model as a SET, the topological order in the bulk together with the translation invariance of the perturbations along the edge/cut imply an edge-ES correspondence at least in some finite domain in Hamiltonian space.
2022
Topological quantum matter is typically associated with gapped phases and edge modes protected by the bulk gap. In contrast, recent work (Phys. Rev. B 104, 075132) proposed intrinsically gapless topological phases that, in one dimension, carry protected edge modes only when the bulk is a gapless Luttinger liquid. The edge modes of such a topological Luttinger liquid (TLL) descend from a nonlocal string order that is forbidden in gapped phases and whose precise form depends on the symmetry class of the system. In this work, we propose a powerful and unbiased entanglement-based smoking gun signature of the TLL. In particular, we show that the entanglement entropy profile of a TLL lacks Friedel oscillations that are invariably present in other gapless one dimensional phases such as ordinary Luttinger liquids, and argue that their absence is closely related to a long-ranged string order which is an intrinsic property of the TLL. Crucially, such a diagnostic is more robust against numeri...