Low-Energy Spectrum of Toeplitz Operators with a Miniwell
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Communications in
Mathematical Physics
Low-Energy Spectrum of Toeplitz Operators with a Miniwell Alix Deleporte CNRS, IRMA UMR 7501, Université de Strasbourg, 67000 Strasbourg, France. E-mail: [email protected] Received: 5 March 2018 / Accepted: 18 July 2019 Published online: 28 August 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We study the concentration properties of low-energy states for quantum systems in the semiclassical limit, in the setting of Toeplitz operators, which include quantum spin systems as a large class of examples. We establish tools proper to Toeplitz quantization to give a general subprincipal criterion for localisation. In addition, we build up symplectic normal forms in two particular settings, including a generalisation of Helffer–Sjöstrand miniwells, in order to prove asymptotics for the ground state and estimates on the number of low-lying eigenvalues. 1. Introduction 1.1. Quantum selection. The computation of ground states for quantum systems is an ubiquitous problem of great difficulty in the non-integrable case, such as antiferromagnetic spin models on lattices in several dimensions. On those systems, approaches in the large spin limit are commonly used [11,26,35,39], in an effort to reduce the problem to the study of the minimal set of the classical energy. A general procedure of semiclassical order by disorder was proposed by Douçot and Simon [15], in situations where this classical minimal set is not discrete. In the mathematical setting of Schrödinger operators in the semiclassical limit, a general study of ground state properties was done by Helffer and Sjöstrand [22,23], including situations where the minimal set of the potential is a smooth submanifold. The classical phase space of spin systems, a product of spheres, is compact. In particular, spin systems are neither Schrödinger operators nor given by Weyl quantization. However, spin operators are example of Toeplitz operators, which allows to understand the large spin limit as a semiclassical limit. In a previous article [13], we studied semiclassical concentration of ground states in the context of semiclassical Toeplitz operators, when the minimal set of the classical energy (or symbol) is a finite set of non-degenerate This work was supported by Grant ANR-13-BS01-0007-01.
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points, with results analogous to the Schrödinger case [22]. To this end we introduced the Melin value (see Definition 2.17 in the present article) associated with a quadratic form on R2n . In frustrated antiferromagnetic spin systems, such as on the Kagome lattice, the minimal set of the classical energy does not form a smooth submanifold. The goal of this article is to not only to extend the degenerate case [23] to Toeplitz quantization, but also to generalise the geometrical conditions on the zero set of the classical energy. We prove several results of quantum selection: not all points of classical phase space where the energy is minimal are equivalent for quantum systems;
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