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arXiv:2502.10139 · Physical Review Letters

Energy Diffusion in the Long-Range Interacting Spin Systems

Hideaki Nishikawa, Keiji Saito

Published Sep 10, 2025• 2 citations
arXivHTMLPDF

Abstract

We investigate energy diffusion in long-range interacting spin systems, where the interaction decays algebraically as V(r)∝r^{-α} with the distance r between the sites. We consider prototypical spin systems, the transverse Ising model, and the XYZ model in the D-dimensional lattice with a finite exponent α>D which guarantees the thermodynamic extensivity. In one dimension, both normal and anomalous diffusion are observed, where the anomalous diffusion is attributed to anomalous enhancement of the amplitude of the equilibrium current correlation. We prove the power-law clustering property of arbitrary orders of joint cumulants in general dimensions. Applying this theorem to equal-time current correlations, we further prove several theorems leading to the statement that the sufficient condition for normal diffusion in one dimension is α>3/2 regardless of the models. The fluctuating hydrodynamics approach consistently explains Lévy diffusion for α<3/2, which implies that the condition is optimal. In higher dimensions of D≥2, normal diffusion is indicated as long as α>D.

DOI
10.1103/hsbt-c46n

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We investigate energy diffusion in long-range interacting spin systems, where the interaction decays algebraically as V(r)∝r^{-α} with the distance r between the sites. We consider prototypical spin systems, the transverse Ising model, and the XYZ model in the D-dimensional lattice with a finite exponent α>D which guarantees the thermodynamic extensivity. In one dimension, both normal and anomalous diffusion are observed, where the anomalous diffusion is attributed to anomalous enhancement of the amplitude of the equilibrium current correlation. We prove the power-law clustering property of arbitrary orders of joint cumulants in general dimensions. Applying this theorem to equal-time current correlations, we further prove several theorems leading to the statement that the sufficient condition for normal diffusion in one dimension is α>3/2 regardless of the models. The fluctuating hydrodynamics approach consistently explains Lévy diffusion for α<3/2, which implies that the condition is optimal. In higher dimensions of D≥2, normal diffusion is indicated as long as α>D.
We consider heat transfer in one-dimensional systems with long-range interactions. It is known that typical short-range interacting systems shows anomalous behavior in heat transport when total momentum is conserved, whereas momentum-nonconserving systems do not exhibit anomaly. In this study, we focus on the effect of long-range interaction. We propose an exactly solvable model that reduces to the so-called momentum-exchange model in the short-range interaction limit. We exactly calculate the asymptotic time decay in the energy current correlation function, which is related to the thermal conductivity via the Green-Kubo formula. From the time decay of the current correlation, we show three qualitatively crucial results. First, the anomalous exponent in the time-decay continuously changes as a function of the index of the long-range interaction. Second, there is a regime where the current correlation diverges with increasing the system size with fixed time, and hence, the exponent of the time decay cannot be defined. Third, even momentum-nonconserving systems can show the anomalous exponent indicating anomalous heat transport. Higher dimensions are also considered, and we found that long-range interaction can induce the anomalous exponent even in three-dimensional systems.
Quantum thermalization of closed systems -- i.e. the relaxation of local properties to their thermodynamic value along the Hamiltonian evolution -- is generically expected in non-integrable systems at finite energy density. In this work we show that quantum spin systems with at least one spin component coupled through long-range ferromagnetic interactions, and a second spin component coupled to a field, cannot in general thermalize to a paramagnetic state whenever they have a finite, negative energy per spin. The long-range interactions we consider decay as a power law of the distance, ∼r−α\sim r^{-α}∼r−α with α<dα< dα<d, where ddd is the number of spatial dimensions of the system. The lack of thermalization is generic to an infinite family of Hamiltonians, and it is not associated to any form of integrability. It can instead be understood by the anomalous features of the thermodynamics of the system in the paramagnetic phase, which cannot be reflected in the properties of the eigenstates. This leads to a breakdown of eigenstate typicality and of the eigenstate thermalization hypothesis. Interestingly, the non-thermal states emerging at long times in the dynamics display long-range multipartite entanglement, elevating the breakdown of thermalization to a resource.
We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension D with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent α≥2D, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to α at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.
In this thesis we study the time evolution of correlation functions in quantum lattice models in the presence of long-range interactions or hopping decaying asymptotically as a power law.For a large class of initial conditions, exact analytic results are obtained in arbitrary lattice dimension for the longrange Ising model.In contrast to the nearest-neighbour case, we find that correlations decay like stretched or compressed exponentials in time.Provided the long-range character of the interactions is sufficiently strong, pronounced prethermalization plateaus are observed and relaxation timescales are widely separated.Starting from uncorrelated states that are easily prepared in experiments, we show the dynamical emergence of correlations and entanglement in these far-from-equilibrium interacting quantum systems.We characterize these correlations by the entanglement entropy, concurrence, and squeezing, which are inequivalent measures of entanglement corresponding to different quantum resources.I am grateful for financial support form NITheP over the past six years, not only for the yearly stipend, but also for supporting me in attending two international workshops, one in Boulder Colorado and the other in Trieste Italy.With the support from the NITheP staff I have explored the frontiers of quantum physics, however they also taught me how to communicate scientific ideas and results in the best possible way, for this I am extremely grateful.A word of thanks is due to Dawid Storch and Kaden Hazzard for many helpful discussions.Lastly, but most importantly, I would like to thank my friends,
During this Ph.D, we studied one-dimensional systems with long-range couplings. In the first part, we demonstrate that power-law couplings lead to an algebraic decay of correlations at long distances in disordered quantum wires. In the second chapter, we analyzed an extended Hubbard model where particles interact via a finite-range potential that induces frustration and new exotic phases. In the third chapter, we demonstrated that restoring energy extensivity has an influence on the low-energy properties of quantum model in the thermodynamic limit. Finally, we provide preliminary results on the modification of Anderson localization due to the coupling to a cavity mode.
We study the low-energy properties of the long-range random transverse-field Ising chain with ferromagnetic interactions decaying as a power of the distance. Using variants of the strong-disorder renormalization group method, the critical behavior is found to be controlled by a strong-disorder fixed point with a finite dynamical exponent . Approaching the critical point, the correlation length diverges exponentially. In the critical point, the magnetization shows an -independent logarithmic finite-size scaling and the entanglement entropy satisfies the area law. These observations are argued to hold for other systems with long-range interactions, even in higher dimensions.