Bachmann, Sven and Michalakis, Spyridon and Nachtergaele, Bruno and Sims, Robert (2012) Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems. Communications in Mathematical Physics, 309 (3). pp. 835871. ISSN 00103616. http://resolver.caltech.edu/CaltechAUTHORS:20120314075459587

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Abstract
Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on s Є [0,1], such that for each s, H(s) has a nonvanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that ’belong to the same phase’ are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an sdependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasiadiabatic evolution’ technique, of which we give a proof extended to infinitedimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasilocal structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a LiebRobinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a cocycle of automorphisms of the algebra of quasilocal observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H(s), 0 ≤ s ≤ 1.
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Additional Information:  © 2011 by the authors. This paper may be reproduced, in its entirety, for noncommercial purposes. Received: 11 February 2011. Accepted: 9 June 2011. Published online: 17 November 2011. Communicated by M. Salmhofer. This work was supported by the National Science Foundation and the Department of Energy: S.B. under Grant DMS0757581, B.N. under grant DMS1009502, and R.S. under Grant DMS0757424. S.M. received support from NSF DMS0757581 and PHY0803371, and DOE Contract DEAC5206NA25396. B.N. gratefully acknowledges the kind hospitality of the Institute MittagLeffler (Djursholm, Sweden) during Fall 2010 where part of the work reported here was carried out and of the Department of Mathematics at the University of Arizona where it was completed.  
Group:  IQIM, Institute for Quantum Information and Matter  
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Record Number:  CaltechAUTHORS:20120314075459587  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20120314075459587  
Official Citation:  Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems 835871 Sven Bachmann, Spyridon Michalakis, Bruno Nachtergaele and Robert Sims  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  29714  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  14 Mar 2012 16:20  
Last Modified:  26 Dec 2012 14:57 
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