Surrogate models for quantum spin systems based on reduced order modeling

The simulation of quantum spin models is an actively researched field. Albeit rather basic these many-body systems are inherently strongly correlated and as such feature a rich variety of phaenomena including involved patterns of ordering / discordering, topological order or varieties of phase changes. Furthermore these models often provide a good approximation to the low-temperature regime of real physical systems justifying their detailed study. One approach is to consider parametrised quantum spin models as a low-complexity proxy for real systems and use them to understand which parameter values (e.g. which spin coupling strengths) lead to interesting behaviours. From this one can deduce inversely how novel materials ought to be designed in order to probe and study these behaviours experimentally.

In a recent work my mentor Benjamin Stamm and myself teamed up with Stefan Wessel (RWTH physics department) and Matteo Rizzi (Universität Köln, Forschungszentrum Jülich) to work on cheap surrogate models for accelerating the study of such parametrised quantum spin models. Our key assumption is that the Hamiltonian of these models as well as the deduced quantities of interest (e.g. the structure factor) can be decomposed affinely in the parameters. For many standard models this is indeed the case. Exploiting the affine structure of the Hamiltonian our approach constructs a reduced-basis surrogate, which effectively represents the full problem in a basis of the exact solutions at a carefully chosen set of parameter values. As we demonstrate for two examples (a chain of Rydberg atoms as well as a sheet of coupled triangles) the information in relatively small reduced bases, which are orders of magnitude smaller than the dimensionality of the Hilbert space, sufficient information is accumulated by the reduced basis in order to reproduce key quantities of interest over the full parameter domain to an absolute error of 10⁻⁴ or less.

For me this was the first time working with quantum spin models. Even more so I enjoyed this interdisciplinary collaboration and the associated diving into a new subject in the discussions we had. Along the work on this paper we actually identified a number of possibilities for future work. In fact a number of the problems typically encountered when numerically modelling quantum spin models (e.g. due to highly degenerate ground states or issues with the iterative eigensolvers) are closely related to the challenges for modelling difficult quantum-chemical systems.

The full abstract of our paper reads

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 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 tantalising outlook of this work.

Michael F. Herbst, Stefan Wessel, Matteo Rizzi and Benjamin Stamm. Surrogate models for quantum spin systems based on reduced order modeling. Physical Review E, 105, 045303 (2022). DOI 10.1103/PhysRevE.105.045303