GLT hidden structures in mean-field quantum spin systems
Christiaan J. F. van de Ven, Muhammad Faisal Khan, S. Serra-Capizzano
TL;DR
The paper develops a GLT-based framework to analyze structured matrix sequences arising from mean-field quantum spin systems, focusing on Curie–Weiss models. It proves that the normalized Curie–Weiss Hamiltonian sequence is a zero-distributed GLT sequence with symbol 0, while the restricted model yields a Berezin–Toeplitz GLT sequence with a nontrivial symbol on the sphere, both captured by explicit GLT generators. Numerical experiments validate the theoretical spectral distributions and illustrate how the spectrum lies within the GLT symbol's range and exhibits predictable extremal behavior. This GLT-centric approach provides a rigorous, scalable method for understanding large-spin limits and suggests avenues for extending to more complex, locally interacting quantum spin systems and higher-dimensional GLT-symbols.
Abstract
This work explores structured matrix sequences arising in mean-field quantum spin systems. We express these sequences within the framework of generalized locally Toeplitz (GLT) $*$-algebras, leveraging the fact that each GLT matrix sequence has a unique GLT symbol. This symbol characterizes both the asymptotic singular value distribution and, for Hermitian or quasi-Hermitian sequences, the asymptotic spectral distribution. Specifically, we analyze two cases of real symmetric matrix sequences stemming from mean-field quantum spin systems and determine their associated distributions using GLT theory. Our study concludes with visualizations and numerical tests that validate the theoretical findings, followed by a discussion of open problems and future directions.