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Bulk Entanglement and Boundary Spectra in Gapped Topological Phases

Zhu-Xi Luo, Brendan Pankovich, Yuting Hu, Yong-Shi Wu

Published Jun 20, 2018• 0 citations
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Abstract

We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold $\mathcal{M}$ with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as $\mathcal{M}$. Specifically, we focus on the case of generalized string-net models and discuss the cases where $\mathcal{M}$ is a disk or a cylinder. The correspondence turns out to be very different from that of the chiral topological orders. When $\mathcal{M}$ has the topology of a cylinder, different boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. Charge (smooth) and flux (rough) boundaries of toric code, more general $\mathbb{Z}_N$ models, and the simplest non-abelian case of doubled Fibonacci are demonstrated.

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We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold M\mathcal{M}M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as M\mathcal{M}M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M\mathcal{M}M is a disk or a cylinder. The correspondence turns out to be very different from that of the chiral topological orders. When M\mathcal{M}M has the topology of a cylinder, different boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. Charge (smooth) and flux (rough) boundaries of toric code, more general ZN\mathbb{Z}_NZN​ models, and the simplest non-abelian case of doubled Fibonacci are demonstrated.
We study the correspondence between boundary excitation distribution spectrum of nonchiral topological orders on an open surface M\mathcal{M}M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed surface. The closed surface is bipartitioned into two subsystems, one of which has the same topology as M\mathcal{M}M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M\mathcal{M}M is a disk or a cylinder. When M\mathcal{M}M has the topology of a cylinder, different combinations of boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. When both boundaries are charge (smooth) boundaries, the entanglement spectrum can be identified with the boundary excitation distribution spectrum at infinite temperature and constant fugacities. Examples of toric code, ZN{\mathbb{Z}}_{N}ZN​ theories, and the simple non-Abelian case of doubled Fibonacci are demonstrated.
We introduce the concept of boundary degeneracy, as the ground state degeneracy of topologically ordered states on a compact orientable spatial manifold with gapped boundaries. We emphasize that the boundary degeneracy provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of the fully gapped edge modes depends on boundary gapping conditions. By associating different types of boundary gapping conditions as different ways of particle or quasiparticle condensations on the boundary, we develop an analytic theory of gapped boundaries. By Chern-Simons theory, this allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which encodes more than the fusion algebra of fractionalized quasiparticles. We apply our theory to Kitaev's toric code and Levin-Wen string-net models. We predict that the Z2{Z}_{2}Z2​ toric code and Z2{Z}_{2}Z2​ double-semion model [more generally, the Zk{Z}_{k}Zk​ gauge theory and the $U{(1)}_{k}\ifmmode\times\else\texttimes\fi{}U{(1)}_{\ensuremath{-}k}$ nonchiral fractional quantum Hall state at even integer kkk] can be numerically and experimentally distinguished, by measuring their boundary degeneracy on an annulus or a cylinder.
Abstract We compute the entanglement entropy in a 2+1 dimensional topological order in the presence of gapped boundaries. Specifically, we consider entanglement cuts that cut through the boundaries. We argue that based on general considerations of the bulk- boundary correspondence, the “twisted characters” feature in the Renyi entropy, and the topological entanglement entropy is controlled by a “half-linking number” in direct analogy to the role played by the S-modular matrix in the absence of boundaries. We also construct a class of boundary states based on the half-linking numbers that provides a “closed-string” picture complementing an “open-string” computation of the entanglement entropy. These boundary states do not correspond to diagonal RCFT’s in general. These are illustrated in specific Abelian Chern-Simons theories with appropriate boundary conditions.
Phases of gapped quantum liquids are topologically ordered and have very interesting physical features that are completely robust against any local perturbation that do not close the bulk energy gap. These universal properties are hidden in the ground states of these systems, as different patterns of many-body long-range entanglement. In this thesis we study the universal properties of gapped quantum liquids from various perspectives. We propose the notion of Universal Wavefunction Overlap as a way of extracting almost complete information about the underlying entanglement structure in a system with topological order. We propose an efficient numerical methods to use these universal wavefunction overlaps as topological order parameters and demonstrate their usefulness with concrete numerical computations. In 2 + 1D these overlaps correspond to known quantities and contain information about anyonic particle excitations. We show that in 3 + 1D, these overlaps contain information about linked multi-string braiding processes, in particular three-string braiding. \n \nIn the second part of this thesis, we study boundary physics of systems with topological order. We investigate the correspondence between edge and entanglement spectra for non-chiral topological systems in general and with the presence of extra symmetries and dualities. We also show that by local deformations of the fixed-point wavefunction on non- chiral topological orders, all possible edge theories can be extracted from its entanglement Hamiltonian. Finally we introduce the notion of fermionic gapped boundaries and see how the phase diagram of the simplest topological orders get significantly enriched.
In many physical scenarios, close relations between the bulk properties of quantum systems and theories associated with 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. 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: a deformed AKLT model [I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Phys. Rev. Lett. 59, 799 (1987)], an Ising-type model [F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Phys. Rev. Lett. 96, 220601 (2006)], and Kitaev's toric code [A. Kitaev, Ann. Phys. 303, 2 (2003)], both in finite ladders and in infinite square lattices. In the second 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 nonlocal Hamiltonians. Because 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.
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.