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arXiv:1511.07867 · Physical review. B./Physical review. B

Robust topological degeneracy of classical theories

Mohammad-Sadegh Vaezi, Gerardo Ortíz, Zohar Nussinov

Published May 9, 2016• 5 citations
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Abstract

We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display nonlocal entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is ``holographic'', i.e., extensive in the system boundary. We find that clock and $U(1)$ gauge theories display topological (in addition to gauge) degeneracies.

DOI
10.1103/physrevb.93.205112

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We challenge the hypothesis that the ground states of a physical system whose degeneracy depends on topology must necessarily realize topological quantum order and display nonlocal entanglement. To this end, we introduce and study a classical rendition of the Toric Code model embedded on Riemann surfaces of different genus numbers. We find that the minimal ground state degeneracy (and those of all levels) depends on the topology of the embedding surface alone. As the ground states of this classical system may be distinguished by local measurements, a characteristic of Landau orders, this example illustrates that topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and, as shown, it applies to many other models. We also demonstrate that certain lattice realizations of these models, and other theories, display a ground state entropy (and those of all levels) that is ``holographic'', i.e., extensive in the system boundary. We find that clock and U(1)U(1)U(1) gauge theories display topological (in addition to gauge) degeneracies.
We study various nontrivial facets of Ҥegeneracyӭ a concept of paramount importance in numerous physical systems. In the first part of this thesis, we challenge the folklore that if the ground state degeneracy of a physical system depends on topology then this system must necessarily realize an unconventional, so-called Ҵopological quantumӬ order. To this end, we introduce a classical rendition of the Toric Code model that displays such a topological degeneracy yet exhibits conventional Landau order. As the ground states of this classical system may be distinguished by local measurements, this example illustrates that, on its own, topological degeneracy is not a sufficient condition for topological quantum order. This conclusion is generic and applies to many other models. In the second part of this thesis, we prove that under fairly modest conditions, all ҤualitiesӠare conformal. This general result has enormous practical consequences. For example, one can establish that weak- and strong-coupling series expansions of arbitrarily large finite size systems are trivially related. As we explain, this relation partially solves or, equivalently, localizes the computational complexity of evaluating the series expansions to only a subset of those coefficients. The coefficients in the strong-coupling series expansions are related to the degeneracy of the system. Thus, our results may facilitate the computation of the degeneracies of the various levels. We end this thesis by establishing a unified framework for studying general disordered systems with either discrete or continuous coupling distributions. We introduce a ҢinomialӠspin glass wherein the couplings are the sum of ҭӠidentically distributed Bernoulli random variables. We demonstrate that for short-range Ising models on d-dimensional hypercubic lattices, the ground-state entropy density for N spins is bounded from above by ( sqrt(d/2m)+1/N )ln2. This confirms the long hand suspicion that the degeneracy of real (finite dimensional) spin glasses with Gaussian couplings is not extensive. Exact calculations reveal the presence of a crossover length scale L*(m) below which the binomial spin glass is indistinguishable from the Gaussian system. Our analytical and numerical results underscore the non-commutativity of the thermodynamic and continuous coupling limits.
Topological entanglement entropy, a measure of the long-ranged entanglement, is related to the degeneracy of the ground state on a higher genus surface. The exact relation depends on the details of the topological theory. We consider a class of holographic models where such relation might be similar to the one exhibited by Chern-Simons theory in a certain large N limit. Both the non-vanishing topological entanglement entropy and the ground state degeneracy in these holographic models are consequences of the topological Gauss-Bonnet term in the dual gravitational description. A soft wall holographic model of confinement is used to generate finite correlation length but keep the disk topology of the entangling surface in the bulk, necessary for nonvanishing topological entanglement entropy.
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.
Topological entanglement entropy, a measure of the long-ranged entanglement, is related to the degeneracy of the ground state on a higher genus surface. The exact relation depends on the details of the topological theory. We consider a class of holographic models where such relation might be similar to the one exhibited by Chern-Simons theory in a certain large N limit. Both the non-vanishing topological entanglement entropy and the ground state degeneracy in these holographic models are consequences of the topological Gauss-Bonnet term in the dual gravitational description. A soft wall holographic model of confinement is used to generate finite correlation length but keep the disk topology of the entangling surface in the bulk, necessary for nonvanishing topological entanglement entropy.
Topologically ordered phases in 2 + 1 dimensions are generally characterized by three mutually related features: fractionalized (anyonic) excitations, topological entanglement entropy, and robust ground state degeneracy that does not require symmetry protection or spontaneous symmetry breaking. Such a degeneracy is known as topological degeneracy and can be usually seen under the periodic boundary condition regardless of the choice of the system sizes L1 and L2 in each direction. In this work, we introduce a family of extensions of the Kitaev toric code to N level spins (N ≥ 2). The model realizes topologically ordered phases or symmetry-protected topological phases depending on the parameters in the model. The most remarkable feature of topologically ordered phases is that the ground state may be unique, depending on L1 and L2, despite that the translation symmetry of the model remains unbroken. Nonetheless, the topological entanglement entropy takes the nontrivial value. We argue that this behavior originates from the nontrivial action of translations permuting anyon species.
The entanglement properties of a class of topological stabilizer states, the so-called topological color codes defined on a two-dimensional lattice or 2-colex, are calculated. The topological entropy is used to measure the entanglement of different bipartitions of the 2-colex. The dependency of the ground-state degeneracy on the genus of the surface shows that the color code can support a topological order, and the contribution of the color in its structure makes it interesting to compare with Kitaev's toric code. While a qubit is maximally entangled with the rest of the system, two qubits are no longer entangled showing that the color code is genuinely multipartite entangled. For a convex region, it is found that entanglement entropy depends only on the degrees of freedom living on the boundary of two subsystems. The boundary scaling of entropy is supplemented with a topological subleading term, which for a color code defined on a compact surface, is twice over the toric code. From the entanglement entropy we construct a set of bipartitions in which the diverging term arising from the boundary term is washed out, and the remaining nonvanishing term will have a topological nature. Besides the color code on the compact surface, we also analyze the entanglement properties of a version of color code with a border, i.e., triangular color code.