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arXiv:1004.0303 · Physical Review B

Monte Carlo study of the two-dimensional site-diluted dipolar Ising model

Juan J. Alonso, B. Allés

Published Aug 30, 2010• 4 citations
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Abstract

By tempered Monte Carlo simulations, we study two-dimensional 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 ${x}_{c}<x\ensuremath{\le}1$, where ${x}_{c}=0.79(5)$, we find an antiferromagnetic phase below a temperature which vanishes as $x\ensuremath{\rightarrow}{x}_{c}$ from above. At lower values of $x$, we study (i) distributions of the spin glass (SG) overlap $q$, (ii) their relative mean-square deviation ${\ensuremath{\Delta}}_{q}^{2}$ and kurtosis, and (iii) ${\ensuremath{\xi}}_{L}/L$, where ${\ensuremath{\xi}}_{L}$ 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 $T=0$, $1/\ensuremath{\nu}=0.35(10)$.

DOI
10.1103/physrevb.82.064425

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By tempered Monte Carlo simulations, we study two-dimensional site-diluted dipolar Ising systems. Dipoles are randomly placed on a fraction xxx of all L2{L}^{2}L2 sites in a square lattice and point along a common crystalline axis. For xc<x\ensuremath≤1{x}_{c}<x\ensuremath{\le}1xc​<x\ensuremath≤1, where xc=0.79(5){x}_{c}=0.79(5)xc​=0.79(5), we find an antiferromagnetic phase below a temperature which vanishes as x\ensuremath→xcx\ensuremath{\rightarrow}{x}_{c}x\ensuremath→xc​ from above. At lower values of xxx, we study (i) distributions of the spin glass (SG) overlap qqq, (ii) their relative mean-square deviation \ensuremathΔq2{\ensuremath{\Delta}}_{q}^{2}\ensuremathΔq2​ and kurtosis, and (iii) \ensuremathξL/L{\ensuremath{\xi}}_{L}/L\ensuremathξL​/L, where \ensuremathξL{\ensuremath{\xi}}_{L}\ensuremathξL​ is a SG correlation length. From their variation with temperature and system size, we find that the paramagnetic phase covers the entire T>0T>0T>0 range. Our results enable us to obtain an estimate of the critical exponent associated to the correlation length at T=0T=0T=0, 1/\ensuremathν=0.35(10)1/\ensuremath{\nu}=0.35(10)1/\ensuremathν=0.35(10).
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
We perform high-statistics Monte Carlo simulations of three-dimensional Ising spin glass models on cubic lattices of size LLL: the $\ifmmode\pm\else\textpm\fi{}J$ (Edwards-Anderson) Ising model for two values of the disorder parameter ppp, p=0.5p=0.5p=0.5 and p=0.7p=0.7p=0.7 (up to L=28L=28L=28 and L=20L=20L=20, respectively), and the bond-diluted bimodal model for bond-occupation probability pb=0.45{p}_{b}=0.45pb​=0.45 (up to L=16L=16L=16). The finite-size behavior of the quartic cumulants at the critical point allows us to check very accurately that these models belong to the same universality class. Moreover, it allows us to estimate the scaling-correction exponent $\ensuremath{\omega}$ related to the leading irrelevant operator: $\ensuremath{\omega}=1.0(1)$. Shorter Monte Carlo simulations of the bond-diluted bimodal models at pb=0.7{p}_{b}=0.7pb​=0.7 and pb=0.35{p}_{b}=0.35pb​=0.35 (up to L=10L=10L=10) and of the Ising spin glass model with Gaussian bond distribution (up to L=8L=8L=8) also support the existence of a unique Ising spin glass universality class. A careful finite-size analysis of the Monte Carlo data which takes into account the analytic and the nonanalytic corrections to scaling allows us to obtain precise and reliable estimates of the critical exponents. We obtain $\ensuremath{\nu}=2.45(15)$ and $\ensuremath{\eta}=\ensuremath{-}0.375(10)$.
In this work, we evaluate the probability distribution function of the order parameter for the two-dimensional site-diluted Ising model. Extensive Monte Carlo simulations have been performed for different spin concentrations p (0.70<or=p<or=1) on square lattices with linear sizes L (20<or=L<or=100). The results indicate that the universality class of the diluted Ising model seems to be independent of the amount of dilution. Logarithmic corrections of the finite-size critical temperature behavior of the model can also be inferred even for such small lattices.
Monte Carlo simulations are performed on the square lattice Ising system with both the nearest neighbor and the two-dimensional dipolar interactions in order to investigate the critical property of ferromagnetic phase transitions in this 2-dimensional system. Applying the finite size scaling theory to Monte Carlo data just near the transition point, the critical indices of this dipolar system are calculated and α=0.31±0.04, β =0.20±0.02, γ=1.08±0.20 and ν=0.72±0.10 are obtained. However, in the temperature region considerably far from the transition point, different values of α, β and γ seem to be adequate, which are rather close to those of the rigorous result of the 2-dimensional Ising model.
We study an Ising model with nearest-neighbor antiferromagnetic interactions. It is placed on a triangular lattice, where each site is occupied by a spin with xxx probability. There is no applied magnetic field. Randomness and frustration, two essential ingredients of spin-glasses, are present in this model. We study its critical properties here. The entropy is obtained by a transfer-matrix calculation as a function of xxx at low temperature (T=0.3JkBT=\frac{0.3J}{{k}_{B}}T=kB​0.3J​) for systems on a lattice of 10 \ifmmode\times\else\texttimes\fi{} 20 sites. A fairly shallow minimum appears near $x\ensuremath{\simeq}0.9$, which suggests that this case is the one most likely to show a transition into an ordered state at low temperature. We study the cases x=1,0.9,and0.74x=1, 0.9, \mathrm{and} 0.74x=1,0.9,and0.74, which is about half-way to the critical percolation. We simulate systems on lattices of 50 \ifmmode\times\else\texttimes\fi{} 50 sites and 30 \ifmmode\times\else\texttimes\fi{} 30 sites by the Monte Carlo method. The specific heat has a broad maximum at $T\ensuremath{\simeq}0.9$ for x=0.74and0.9x=0.74 \mathrm{and} 0.9x=0.74and0.9. ${\ensuremath{\chi}}_{\mathrm{SG}}$, defined by ${\ensuremath{\chi}}_{\mathrm{SG}}={N}^{\ensuremath{-}1}\ensuremath{\Sigma}{i,j}^{}{〈{\ensuremath{\sigma}}_{i}{\ensuremath{\sigma}}_{j}〉}^{2}$, and the relaxation time ($\ensuremath{\tau}$) are obtained for $T\ensuremath{\ge}\frac{0.6J}{{k}_{B}}$. Both quantities, $\ensuremath{\tau}$ and ${\ensuremath{\chi}}_{\mathrm{SG}}$, turn out to be proportional to $\mathrm{exp}[\frac{A}{{(T\ensuremath{-}{T}_{0})}^{c}}]$ and $0\ensuremath{\le}{T}_{0}\ensuremath{\lesssim}0.4$; a fit with T0=0{T}_{0}=0T0​=0 yields $c\ensuremath{\approx}1$ for x=1x=1x=1 but $c\ensuremath{\approx}2$ for x=0.74and0.9x=0.74 \mathrm{and} 0.9x=0.74and0.9.
We study the effects of dilution to the critical properties of site-diluted Ising model in two dimensions using Monte Carlo simulations. Quenched disorder from the dilution is incorporated into the Ising model via random empty sites on the square lattice of Ising spins. Thermodynamic quantities such as the magnetization MMM per spin, energy EEE per spin, magnetic susceptibility χχχ per spin, and specific heat CCC per spin are then calculated after the system has equilibrated. At small dilution concentrations d<0.1d<0.1d<0.1, we find that the value of the critical exponent βββ does not deviate from its pure Ising value. At higher dilution concentrations d>0.1d>0.1d>0.1, however, we find βββ to strongly depend on the value of ddd. We are able to locate a critical temperature TcT_cTc​ and a critical dilution concentration dcd_cdc​ where the phase transition occurs. We find TcT_cTc​ to depend linearly on ddd. In the phase diagrams of MMM, EEE, χχχ, and CCC, we find that the phase transition line eventually disappears at high dilutions. Our results suggest that there is a transition from Strong Universality at low dilution to Weak Universality at high dilution. Lastly, we find a wide and nonsingular increase in the magnetic susceptibility χχχ at the low temperature and high dilution region.