Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

By S. Bachmann, Spyridon Michalakis, Bruno Nachtergaele, Robert Sims. Communications in Mathematical Physics, 2011

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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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s\in [0,1]}$$\end{document}, 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.

Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a22c8392a84c9a2cb4e3330bdac38819d0bd925f

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Citations in the corpus

Provably efficient machine learning for quantum many-body problems (2021, methodology): “The smoothness condition can be rigorously established as an upper bound on the average magnitude of the gradient of tr O r x ð Þ ½ (cid:4) using quasi-adiabatic evolution ( 53 , 54 ), assuming that the spectral gap of H ( x ) is bounded below by a nonzero constant throughout the parameter space.”
Stability of Frustration-Free Hamiltonians (2011, methodology): “We make extensive use of Lieb-Robinson bounds [28], both for the evolution of operators according to the Hamiltonians under consideration [15,19,30–34,39,40], as well as Lieb-Robinson bounds on the quasi-adiabatic evolution, or spectral flow, of gapped eigenspaces [3,17,20].”