Abstract
We demonstrate that a large class of one-dimensional quantum and classical exchange models can be described by the same type of graphs, namely Cayley graphs of the permutation group. Their well-studied spectral properties allow us to derive crucial information about those models of fundamental importance in both classical and quantum physics, and to completely characterize their algebraic structure. Notably, we prove that the spectral gap can be obtained in polynomial computational time, which has strong implications in the context of adiabatic quantum computing with quantum spin-chains. This quantity also characterizes the rate to stationarity of some important classical random processes such as interchange and exclusion processes. Reciprocally, we use results derived from the celebrated Bethe ansatz to obtain original mathematical results about these graphs in the unweighted case. We also discuss extensions of this unifying framework to other systems, such as asymmetric exclusion processes -- a paradigmatic model in non-equilibrium physics, or the more exotic non-Hermitian quantum systems.
Highlights
Consider the following situations: (a) a card shuffling, where at each step two randomly chosen adjacent cards of the deck are being switched; (b) a cold atom experiment involving strongly interacting 173Yb atoms confined in one dimension; (c) the quantum Heisenberg X X X spin chain; and (d) the protein synthesis on RNA
One-dimensional (1D) exchange models are ubiquitous in both quantum and classical physics, and their study has been associated with important theoretical breakthroughs
We have shown that a wide range of classical and quantum 1D exchange models are described by the same theoretical object, namely, the Laplacian matrix of a Schreier graph associated with the permutation group
Summary
Consider the following situations: (a) a card shuffling, where at each step two randomly chosen adjacent cards of the deck are being switched; (b) a cold atom experiment involving strongly interacting 173Yb atoms confined in one dimension; (c) the quantum Heisenberg X X X spin chain; and (d) the protein synthesis on RNA. Extracting any physically relevant information from this method is still strenuous, and, more importantly, the Bethe ansatz can no longer be applied if the system is inhomogeneous, as one could expect in a realistic experimental situation [17] In this case, the study of such strongly correlated systems is greatly challenged by their computational complexity. We show that results derived from the Bethe ansatz may be applied in order to study the spectra of Cayley graphs of the permutation group with large number of elements. This approach should attract the attention of mathematicians. As further explained in the main text, the results we present here are significantly stronger and universal
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