Universal graph description for one-dimensional exchange models

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 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 nonequilibrium physics-or the more exotic non-Hermitian quantum systems.

Hypergrid subgraphs and the origin of scarred quantum walks in many-body Hilbert space

“…19. More generally, there has been growing interest in using graph theory to describe universal properties of quantum many-body systems [20,21] and their thermalization dynamics [22]. Intriguingly, even the simple quantum systems, such as a free particle hopping on a 1D lattice, require careful tuning of the hopping amplitudes in order to sustain perfect quantum state transfer [17].…”

Disorder enhanced quantum many-body scars in Hilbert hypercubes

“…Finally, exploiting the mapping of the present model to the Heisenberg spin-1/2 XX chain [78], we identify the recently proposed Onsager scars [65,79] with scars corresponding to sparse eigenstates residing at the "edge" of the Hilbert space. This provides intriguing connections between QMBS and single-particle scars and highlights the utility of a graph-theoretical approach to many-body dynamics, which has been advocated also in the studies of quantum chaos [80][81][82][83][84], integrability [85], QMBS [86], and fermionic and exchange models [87,88].…”

“…This provides a number of interesting openings, such as the interpretation of the disordered Heisenberg XXZ spin chain as that of an Anderson model on a hypercubic lattice, which is relevant to the ongoing discussion about the scaling of the Thouless time in many-body systems [98,99]. It would also be interesting to explore the role of sparse eigenvectors, which play an important role in various applications, such as in the signal analysis of networks [100,101], in the context of many-body Hamiltonians and their graph-theoretic representations [85,87,88,102]. Finally, to describe the entangling dynamics between the motional and internal degrees of freedom, new approaches, such as the variational ansatz based on non-Gaussian states [103], need to be investigated.…”

Hypergrid subgraphs and the origin of scarred quantum walks in the many-body Hilbert space

“…19. More generally, there has been growing interest in using graph theory to describe universal properties of quantum many-body systems [20,21] and their thermalization dynamics [22]. Intriguingly, even the simple quantum systems, such as the free particle hopping on a 1D lattice, require careful tuning of the hopping amplitudes in order to sustain perfect quantum state transfer [17].…”