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arXiv:cond-mat/0505209 · Physical Review E

Topological nonconnectivity threshold in long-range spin systems

F. Borgonovi, G. L. Celardo, Alessandro Musesti, R. Trasarti–Battistoni, Pavel Váchal

Published Feb 16, 2006• 12 citations
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Abstract

We demonstrate the existence of a topological disconnection threshold, recently found by Borgonovi [J. Stat. Phys. 116, 1435 (2004)], for generic 1-d anisotropic Heisenberg models interacting with an interparticle potential R(-alpha) when 0<alpha<1(here R is the distance among spins). We also show that alpha if is greater than the embedding dimension then the ratio between the disconnected energy region and the total energy region goes to zero when the number of spins becomes very large. On the other hand, numerical simulations in d=2,3 for the long-range case support the conclusion that such a ratio remains finite for large N values. The disconnection threshold can thus be thought of as a distinctive property of anisotropic long-range interacting systems.

DOI
10.1103/physreve.73.026116

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We demonstrate the existence of a topological disconnection threshold, recently found by Borgonovi [J. Stat. Phys. 116, 1435 (2004)], for generic 1-d anisotropic Heisenberg models interacting with an interparticle potential R(-alpha) when 0<alpha<1(here R is the distance among spins). We also show that alpha if is greater than the embedding dimension then the ratio between the disconnected energy region and the total energy region goes to zero when the number of spins becomes very large. On the other hand, numerical simulations in d=2,3 for the long-range case support the conclusion that such a ratio remains finite for large N values. The disconnection threshold can thus be thought of as a distinctive property of anisotropic long-range interacting systems.
A quantum Heisenberg model with anisotropic coupling and all-to-all interaction has been analyzed using the Bose-Einstein statistics. In F. Borgonovi, G. L. Celardo, M. Maianti, and E. Pedersoli, J. Stat. Phys. 116, 1435 (2004), the existence of a disconnection in the classical phase space has been proved analytically. Such topological disconnection occurs when the energy exceeds a suitable threshold value, called the topological nonconnectivity threshold. We address here the problem of finding quantum signatures of such a threshold. An independent definition of the quantum nonconnectivity threshold, motivated by considerations strictly valid in the quantum regime, is given. We also discuss the dynamical relevance of the quantum border with respect to the quantum magnetic reversal. Contrary to the classical case the magnetization can flip even below the classical threshold through macroscopic quantum tunneling. We evaluate the time scale for magnetic reversal from statistical and spectral properties for a small number of particles and in the semiclassical limit.
Topological phase space disconnection has been recently found to be a general phenomenon in many-body spin system with anisotropic interaction. We show that the power law divergence of magnetic reversal time at the energy signaling such disconnection is generic for long-range interacting systems with an exponent proportional to the number of particles. We also study the modifications induced putting the system in contact with a thermal bath. Using the canonical formalism we analyze the magnetic reversal times at any temperature. Moreover, due to the divergence of reversal time at the energy disconnection threshold we can recover, using saddle point approximation, a simple exponential dependence on the inverse temperature showing the explicit relevance of the energy disconnection threshold for finite many-body interacting systems at finite temperature. This sets a general framework to understand the emergence of ferromagnetism in finite magnetic systems starting from microscopic models without phenomenological on-site barriers.
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 investigate the existence of N\'eel long-range order in a spatially anisotropic antiferromagnetic Heisenberg model. As the anisotropical coupling ${J}_{\ensuremath{\perp}}$ increases from 000 to 1,1,1, the model interpolates between an infinite number of decoupled one-dimensional gapped spin chains and a two-dimensional gapless isotropic Heisenberg antiferromagnet. By applying a method based on reflection positivity [F. J. Dyson, E. H. Lieb, and B. Simon, J. Stat. Phys. 18,18,18, 335 (1978)], we determine the upper bound of the critical values of ${J}_{\ensuremath{\perp}}^{c}$ such that when ${J}_{\ensuremath{\perp}}>~{J}_{\ensuremath{\perp}}^{c},$ the ground state of this model has an antiferromagnetic long-range order. For the case of spin S=1,S=1,S=1, our result confirms and strengthens previous results derived by numerical calculations on finite-size samples, while for other values of S, our results are new.
Dynamical properties of lattice systems with long-range pair interactions, decaying like 1/r(α) with the distance r, are investigated, in particular the time scales governing the relaxation to equilibrium. Upon varying the interaction range α, we find evidence for the existence of a threshold at α=d/2, dependent on the spatial dimension d, at which the relaxation behavior changes qualitatively and the corresponding scaling exponents switch to a different regime. Based on analytical as well as numerical observations in systems of vastly differing nature, ranging from quantum to classical, from ferromagnetic to antiferromagnetic, and including a variety of lattice structures, we conjecture this threshold and some of its characteristic properties to be universal.