Surrogate models for quantum spin systems based on reduced-order modeling

Physical review. E, vol. 105, pp. 045303

Abstract

We present a methodology to investigate phase diagrams of quantum models based on the principle of the reduced basis method (RBM). The RBM is built from a few ground-state snapshots, i.e., lowest eigenvectors of the full system Hamiltonian computed at well-chosen points in the parameter space of interest. We put forward a greedy strategy to assemble such a small-dimensional basis, i.e., to select where to spend the numerical effort needed for the snapshots. Once the RBM is assembled, physical observables required for mapping out the phase diagram (e.g., structure factors) can be computed for any parameter value with a modest computational complexity, considerably lower than the one associated to the underlying Hilbert space dimension. We benchmark the method in two test cases, a chain of excited Rydberg atoms and a geometrically frustrated antiferromagnetic two-dimensional lattice model, and illustrate the accuracy of the approach. In particular, we find that the ground-state manifold can be approximated to sufficient accuracy with a moderate number of basis functions, which increases very mildly when the number of microscopic constituents grows-in stark contrast to the exponential growth of the Hilbert space needed to describe each of the few snapshots. A combination of the presented RBM approach with other numerical techniques circumventing even the latter big cost, e.g., tensor network methods, is a tantalizing outlook of this work.

Authors 4

1. Michael F. Herbst Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

2. Benjamin Stamm Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

3. Stefan Weßel Aachen Institute for Theoretical Solid State Physics Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Theoretical Solid State Physics, RWTH Aachen University, Otto-Blumenthal-Straße 26, 52074 Aachen, Germany

   Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany

4. Matteo Rizzi Aachen Department of Mathematics

   Forschungszentrum Jülich · University of Cologne · RWTH Aachen University

   Affiliation as printed

   Forschungszentrum Jülich GmbH, Institute of Quantum Control, Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany and Institute for Theoretical Physics, University of Cologne, D-50937 Köln, Germany

   Department of Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany