Laboratoire de Physique Théorique
Toulouse - UMR 5152

In two recent articles [1,2], I have shown that a certain class of one-dimensional quantum and classical models called « exchange models » can be described by a spectral graph theory problem. More precisely, the eigendecompositions of the Hamiltonian in the quantum case and of the transition matrix in the classical stochastic case, can be deduced from the spectral properties of so called « Schreier graphs » associated with the permutation group. Using the well-studied mathematical properties of these graphs, I was able to derive completely general results on these systems. Besides, many questions remain open, and the large number of models that these graphs can describe — with their specific problems and methods — offer many promising perspectives.

In this presentation, I will focus on the graph description of a peculiar type of exchange models, namely, strongly repulsive fermionic mixtures confined one-dimensional continuous potentials. This is the model that I had studied during my thesis, and for which I first provided a graph description in [1]. If I have time, I will briefly describe my more recent extensions of this work [2], and some perspectives.

[1] J. Decamp et al., PRR 2, 023059 (2020).

[2] J. Decamp et al., PRR 2, 033297 (2020).