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Physica A Statistical Mechanics and its Applications

Critical dynamics: A consequence of a random, stepwise growth of clusters?

Z. Alexandrowicz

Published Aug 1, 1990• 22 citations
Publisher
DOI
10.1016/0378-4371(90)90117-b

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The exact dynamics of the entanglement between two harmonic modes generated by an angular momentum coupling is examined. Such a system arises when considering a particle in a rotating anisotropic harmonic trap or a charged particle in a fixed harmonic potential in a magnetic field, and it exhibits a rich dynamical structure, with stable, unstable, and critical regimes according to the values of the rotational frequency or field and trap parameters. Consequently, it is shown that the entanglement generated from an initially separable Gaussian state can exhibit quite distinct evolutions, ranging from quasiperiodic behavior in stable sectors to different types of unbounded increase in critical and unstable regions. The latter lead, respectively, to logarithmic and linear growth of the entanglement entropy with time. It is also shown that entanglement can be controlled by tuning the frequency, such that it can be increased, kept constant, or returned to a vanishing value with just stepwise frequency variations. Exact asymptotic expressions for the entanglement entropy in the different dynamical regimes are provided.
We formulate the Wolff algorithm as a site-bond percolation problem, apply it to the ferromagnetic and antiferromagnetic planar triangular spin models, and study the percolation critical behavior using finite-size scaling. In the former case the Wolff algorithm is successful as an accelerating algorithm, whereas in the latter case it is not. We found the percolation temperatures and the cluster exponents for both models. In the antiferromagnetic model, the percolation temperature is higher than the critical temperature of the spin system. The cluster exponents are found to be the same as the random two-dimensional (2D) percolation. In the ferromagnetic model, the percolation temperature agrees with the critical temperature, and the cluster exponents are different from the random 2D percolation, meaning that they are in different universal classes. For the ferromagnetic model we discuss the mechanism of the cluster growth in the regime of the Kosterlitz-Thouless transition. We also note a relation between the dynamic exponent and the percolation exponents.