A Caltech Library Service

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

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. 835-871. ISSN 0010-3616.

PDF - Submitted Version
See Usage Policy.


Use this Persistent URL to link to this item:


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 non-vanishing 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 s-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasi-adiabatic evolution’ technique, of which we give a proof extended to infinite-dimensional 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 quasi-local structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local 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.

Item Type:Article
Related URLs:
URLURL TypeDescription
Additional Information:© 2011 by the authors. This paper may be reproduced, in its entirety, for non-commercial 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 DMS-0757581, B.N. under grant DMS-1009502, and R.S. under Grant DMS-0757424. S.M. received support from NSF DMS-0757581 and PHY-0803371, and DOE Contract DE-AC52-06NA25396. B.N. gratefully acknowledges the kind hospitality of the Institute Mittag-Leffler (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
Funding AgencyGrant Number
Department of Energy (DOE)DE-AC52-06NA25396
Record Number:CaltechAUTHORS:20120314-075459587
Persistent URL:
Official Citation:Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems 835-871 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
Deposited By: Ruth Sustaita
Deposited On:14 Mar 2012 16:20
Last Modified:26 Dec 2012 14:57

Repository Staff Only: item control page