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Register a new userMonte-Carlo analysis of critical properties of the two-dimensional randomly site-diluted Ising model via Wang-Landau algorithm
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Ioannis Hatziagapiou A
Publication date
2008
fields
Physics
and research's language is
English
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The influence of random site dilution on the critical properties of the two-dimensional Ising model on a square lattice was explored by Monte Carlo simulations with the Wang-Landau sampling. The lattice linear size was $L = 20-120$ and the concentration of diluted sites $q=0.1, 0.2, 0.3$. Its pure version displays a second-order phase transition with a vanishing specific heat critical exponent $alpha$, thus, the Harris criterion is inconclusive, in that disorder is a relevant or irrelevant perturbation for the critical behavior of the pure system. The main effort was focused on the specific heat and magnetic susceptibility. We have also looked at the probability distribution of susceptibility, pseudocritical temperatures and specific heat for assessing self-averaging. The study was carried out in appropriate restricted but dominant energy subspaces. By applying the finite-size scaling analysis, the correlation length exponent $ u$ was found to be greater than one, whereas the ratio of the critical exponents ($alpha / u$) is negative and ($gamma / u$) retains its pure Ising model value supporting weak universality.
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By tempered Monte Carlo simulations, we study 2D site-diluted dipolar Ising systems. Dipoles are randomly placed on a fraction x of all L^2 sites in a square lattice, and point along a common crystalline axis. For x_c< x<=1, where x_c = 0.79(5), we find an antiferromagnetic phase below a temperature which vanishes as x approaches x_c from above. At lower values of x, we study (i) distributions of the spin--glass (SG) overlap q, (ii) their relative mean square deviation Delta_q^2 and kurtosis and (iii) xi_L/L, where xi_L is a SG correlation length. From their variation with temperature and system size, we find that the paramagnetic phase covers the entire T>0 range. Our results enable us to obtain an estimate of the critical exponent associated to the correlation length at T=0, 1/nu=0.35(10).
We report results of a Wang-Landau study of the random bond square Ising model with nearest- ($J_{nn}$) and next-nearest-neighbor ($J_{nnn}$) antiferromagnetic interactions. We consider the case $R=J_{nn}/J_{nnn}=1$ for which the competitive nature of interactions produces a sublattice ordering known as superantiferromagnetism and the pure system undergoes a second-order transition with a positive specific heat exponent $alpha$. For a particular disorder strength we study the effects of bond randomness and we find that, while the critical exponents of the correlation length $ u$, magnetization $beta$, and magnetic susceptibility $gamma$ increase when compared to the pure model, the ratios $beta/ u$ and $gamma/ u$ remain unchanged. Thus, the disordered system obeys weak universality and hyperscaling similarly to other two-dimensional disordered systems. However, the specific heat exhibits an unusually strong saturating behavior which distinguishes the present case of competing interactions from other two-dimensional random bond systems studied previously.
Statistical mechanical models with local interactions in $d>1$ dimension can be regarded as $d=1$ dimensional models with regular long range interactions. In this paper we study the critical properties of Ising models having $V$ sites, each having $z$ randomly chosen neighbors. For $z=2$ the model reduces to the $d=1$ Ising model. For $z= infty$ we get a mean field model. We find that for finite $z > 2$ the system has a second order phase transition characterized by a length scale $L={rm ln}V$ and mean field critical exponents that are independent of $z$.
We study the hopping transport of a quantum particle through finite, randomly diluted percolation clusters in two dimensions. We investigate how the transmission coefficient T behaves as a function of the energy E of the particle, the occupation concentration p of the disordered cluster, the size of the underlying lattice, and the type of connection chosen between the cluster and the input and output leads. We investigate both the point-to-point contacts and the busbar type of connection. For highly diluted clusters we find the behavior of the transmission to be independent of the type of connection. As the amount of dilution is decreased we find sharp variations in transmission. These variations are the remnants of the resonances at the ordered, zero-dilution, limit. For particles with energies within 0.25 <= E <= 1.75 (relative to the hopping integral) and with underlying square lattices of size 20x20, the configurations begin transmitting near p_a = 0.60 with T against p curves following a common pattern as the amount of dilution is decreased. Near p_b = 0.90 this pattern is broken and the transmission begins to vary with the energy. In the asymptotic limit of very large clusters we find the systems to be totally reflecting except when the amount of dilution is very low and when the particle has energy close to a resonance value at the ordered limit or when the particle has energy at the middle of the band.
We implement a two-stage approach of the Wang-Landau algorithm to investigate the critical properties of the 3D Ising model with quenched bond randomness. In particular, we consider the case where disorder couples to the nearest-neighbor ferromagnetic interaction, in terms of a bimodal distribution of strong versus weak bonds. Our simulations are carried out for large ensembles of disorder realizations and lattices with linear sizes $L$ in the range $L=8-64$. We apply well-established finite-size scaling techniques and concepts from the scaling theory of disordered systems to describe the nature of the phase transition of the disordered model, departing gradually from the fixed point of the pure system. Our analysis (based on the determination of the critical exponents) shows that the 3D random-bond Ising model belongs to the same universality class with the site- and bond-dilution models, providing a single universality class for the 3D Ising model with these three types of quenched uncorrelated disorder.
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