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arXiv:cond-mat/9801215 · Journal of Physics A Mathematical and General

Crossovers in the two-dimensional Ising spin glass with ferromagnetic next-nearest-neighbour interactions

Giorgio Parisi A, Juan Jesus Ruiz-Lorenzo, Daniel A. Stariolo

Published May 22, 1998• 6 citations
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Abstract

By means of extensive computer simulations we analyse in detail the two-dimensional Ising spin glass with ferromagnetic next-nearest-neighbour interactions. We found a crossover from ferromagnetic to `spin glass'-like order both from numerical simulations and analytical arguments. We also present proof of a second crossover from the `spin glass' behaviour to a paramagnetic phase for the largest volume studied.

DOI
10.1088/0305-4470/31/20/007

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By means of extensive computer simulations we analyse in detail the two-dimensional Ising spin glass with ferromagnetic next-nearest-neighbour interactions. We found a crossover from ferromagnetic to spin glass'-like order both from numerical simulations and analytical arguments. We also present proof of a second crossover from the spin glass' behaviour to a paramagnetic phase for the largest volume studied.
The search for problems where quantum adiabatic optimization might excel over classical optimization techniques has sparked a recent interest in inducing a finite-temperature spin-glass transition in quasiplanar topologies. We have performed large-scale finite-temperature Monte Carlo simulations of a two-dimensional square-lattice bimodal spin glass with next-nearest ferromagnetic interactions claimed to exhibit a finite-temperature spin-glass state for a particular relative strength of the next-nearest to nearest interactions [Phys. Rev. Lett. 76, 4616 (1996)]. Our results show that the system is in a paramagnetic state in the thermodynamic limit, despite zero-temperature simulations [Phys. Rev. B 63, 094423 (2001)] suggesting the existence of a finite-temperature spin-glass transition. Therefore, deducing the finite-temperature behavior from zero-temperature simulations can be dangerous when corrections to scaling are large.
The authors consider the one-dimensional Ising model which besides the interaction of nearest neighbours includes a long-range term. The latter is ferromagnetic while the nearest-neighbour interaction can have any sign. The crossover between short- and long-range interactions when varying the Hamiltonian parameters is analysed for such a model. For some values of the antiferromagnetic short-range interaction the transition from a ferro- to paramagnetic state is of first order. At the tricritical point where the transition order changes from second to first they obtain classical tricritical exponents. For a ferromagnetic short-range interaction the Curie temperature surpasses the transition temperature of the long-range model.
The critical behaviour of an Ising model with competing first- and third-nearest-neighbour interactions ('biaxial next-nearest-neighbour Ising' or BNNNI model) on the square lattice is investigated by finite lattice methods. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.