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arXiv:cond-mat/0007422

Universal behavior in the static and dynamic properties of the $α$-XY model

Andrea Giansanti, Daniele Moroni, Alessandro Campa

Published Jul 26, 2000• 0 citations
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Abstract

The $α$-XY model generalizes, through the introduction of a power-law decaying potential, a well studied mean-field hamiltonian model with attractive long-range interactions. In the $α$-model, the interaction between classical rotators on a lattice is gauged by the exponent $α$ in the couplings decaying as $r^α$, where $r$ are distances between sites. We review and comment here a few recent results on the static and dynamic properties of the $α$-model. We discuss the appropriate $α$ dependent rescalings that map the canonical thermodynamics of the $α$-model into that of the mean field model. We also show that the chaotic properties of the model, studied as a function of $α$ display an universal behaviour.

DOI
10.48550/arxiv.cond-mat/0007422

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Universal behavior in the static and dynamic properties of the ααα-XY model
The ααα-XY model generalizes, through the introduction of a power-law decaying potential, a well studied mean-field hamiltonian model with attractive long-range interactions. In the ααα-model, the interaction between classical rotators on a lattice is gauged by the exponent ααα in the couplings decaying as rαr^αrα, where rrr are distances between sites. We review and comment here a few recent results on the static and dynamic properties of the ααα-model. We discuss the appropriate ααα dependent rescalings that map the canonical thermodynamics of the ααα-model into that of the mean field model. We also show that the chaotic properties of the model, studied as a function of ααα display an universal behaviour.
We discuss the critical behaviour of the 1-dimensional classical XY model with power law decay long range interactions by using Monte Carlo simulation. It has been predicted by renormalization group calculation that the critical behaviour changes with the power of interactions. From the simulation result of temperature dependence of the specific heat, we confirm that the system shows the critical behaviour of the mean field type and the non-trivial long range order. We also estimate the critical exponent η for the system with the inverse square long range interactions, and by using the phenomenological renormalization group plot, we confirm the 'Berezinskii-Kosterlitz-Thouless (BKT)-like' transition which has been predicted by the spin wave calculations.
There are problems with defining the thermodynamic limit of systems with long-range interactions; as a result, the thermodynamic behavior of these types of systems is anomalous. In the present work, we review some concepts from both extensive and nonextensive thermodynamic perspectives. We use a model, whose Hamiltonian takes into account spins ferromagnetically coupled in a chain via a power law that decays at large interparticle distance rrr as $1∕{r}^{\ensuremath{\alpha}}$ for $\ensuremath{\alpha}\ensuremath{\geqslant}0$. Here, we review old nonextensive scaling. In addition, we propose a Hamiltonian scaled by $2\frac{{(N∕2)}^{1\ensuremath{-}\ensuremath{\alpha}}\ensuremath{-}1}{1\ensuremath{-}\ensuremath{\alpha}}$ that explicitly includes symmetry of the lattice and dependence on the size NNN of the system. The approach enabled us to improve upon previous results. A numerical test is conducted through Monte Carlo simulations. In the model, periodic boundary conditions are adopted to eliminate surface effects.
We investigate the ground-state properties of the spin-1/21/21/2 XXZ model with power-law-decaying ($1/{r}^{\ensuremath{\alpha}}$) interactions, which describe spins interacting with long-range transverse (XX) ferromagnetic interactions and longitudinal (Z) antiferromagnetic interactions, or hard-core bosons with long-range repulsion and hopping. The long-range nature of the couplings allows us to quantitatively study the spectral, correlation, and entanglement properties of the system by making use of linear spin-wave theory, supplemented with density-matrix renormalization group in one-dimensional systems. Our most important prediction is the existence of three distinct coupling regimes, depending on the decay exponent $\ensuremath{\alpha}$ and number of dimensions ddd: (1) a short-range regime for $\ensuremath{\alpha}>d+{\ensuremath{\sigma}}_{c}$ (where ${\ensuremath{\sigma}}_{c}=1$ in the gapped N\'eel antiferromagnetic phase exhibited by the XXZ model, and ${\ensuremath{\sigma}}_{c}=2$ in the gapless XY ferromagnetic phase), sharing the same properties as those of finite-range interactions ($\ensuremath{\alpha}=\ensuremath{\infty}$); (2) a long-range regime $\ensuremath{\alpha}<d$, sharing the same properties as those of the infinite-range interactions ($\ensuremath{\alpha}=0$) in the thermodynamic limit; and (3) a most intriguing medium-range regime for $d<\ensuremath{\alpha}<d+{\ensuremath{\sigma}}_{c}$, continuously interpolating between the finite-range and the infinite-range behavior. The latter regime is characterized by elementary excitations with a long-wavelength dispersion relation $\ensuremath{\omega}\ensuremath{\approx}{\mathrm{\ensuremath{\Delta}}}_{g}+c{k}^{z}$ in the gapped phase, and $\ensuremath{\omega}\ensuremath{\sim}{k}^{z}$ in the gapless phase, exhibiting a continuously varying dynamical exponent $z=(\ensuremath{\alpha}\ensuremath{-}d)/{\ensuremath{\sigma}}_{c}$. In the gapless phase of the model the zzz exponent is found to control the scaling of fluctuations, the decay of correlations, and a universal subdominant term in the entanglement entropy, leading to a very rich palette of behaviors for ground-state quantum correlations beyond what is known for finite-range interactions.
Motivated by recent experiments with ultracold matter, we derive a new bound on the propagation of information in D-dimensional lattice models exhibiting 1/r^{α} interactions with α>D. The bound contains two terms: One accounts for the short-ranged part of the interactions, giving rise to a bounded velocity and reflecting the persistence of locality out to intermediate distances, whereas the other contributes a power-law decay at longer distances. We demonstrate that these two contributions not only bound but, except at long times, qualitatively reproduce the short- and long-distance dynamical behavior following a local quench in an XY chain and a transverse-field Ising chain. In addition to describing dynamics in numerous intractable long-range interacting lattice models, our results can be experimentally verified in a variety of ultracold-atomic and solid-state systems.