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.
