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Title: A symmetry principle for topological quantum order

Abstract

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Kleinmore » spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev's Toric code model and Wen's plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.« less

Authors:
 [1];  [2]
  1. Department of Physics, Washington, University, Compton Hall, 1 Brookings Drive, St. Louis, MO 63160 (United States)
  2. Department of Physics, Indiana, University, Bloomington, IN 47405 (United States)
Publication Date:
OSTI Identifier:
21308047
Resource Type:
Journal Article
Journal Name:
Annals of Physics (New York)
Additional Journal Information:
Journal Volume: 324; Journal Issue: 5; Other Information: DOI: 10.1016/j.aop.2008.11.002; PII: S0003-4916(08)00171-1; Copyright (c) 2008 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; CONSERVATION LAWS; CORRELATION FUNCTIONS; DUALITY; ENTROPY; EXCITED STATES; GAUGE INVARIANCE; GRAPH THEORY; GROUND STATES; HAMILTONIANS; JAHN-TELLER EFFECT; QUANTUM ENTANGLEMENT; QUANTUM FIELD THEORY; QUASI PARTICLES; SELECTION RULES; SPIN EXCHANGE; SYMMETRY; TOPOLOGY; TRANSITION ELEMENTS

Citation Formats

Nussinov, Zohar, and Ortiz, Gerardo. A symmetry principle for topological quantum order. United States: N. p., 2009. Web. doi:10.1016/j.aop.2008.11.002.
Nussinov, Zohar, & Ortiz, Gerardo. A symmetry principle for topological quantum order. United States. https://doi.org/10.1016/j.aop.2008.11.002
Nussinov, Zohar, and Ortiz, Gerardo. Fri . "A symmetry principle for topological quantum order". United States. https://doi.org/10.1016/j.aop.2008.11.002.
@article{osti_21308047,
title = {A symmetry principle for topological quantum order},
author = {Nussinov, Zohar and Ortiz, Gerardo},
abstractNote = {We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. These symmetries may be actual symmetries of the Hamiltonian characterizing the system, or emergent symmetries. To this end, we introduce the concept of low-dimensional Gauge-like symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU(N) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev's Toric code model and Wen's plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We exactly recast some known non-local string correlators in terms of local correlation functions. We discuss relations to problems in graph theory.},
doi = {10.1016/j.aop.2008.11.002},
url = {https://www.osti.gov/biblio/21308047}, journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 5,
volume = 324,
place = {United States},
year = {2009},
month = {5}
}