PACS. 05.10.Cc – Renormalization group methods. PACS. 64.60.-i – General studies of phase transitions. PACS. 75.10.Hk – Classical spin models. Abstract. – A study of the influence of different types of disorder on systems in the Ising universality class by employing both a dynamical field theory approach and extensive Monte Carlo simulations is presented. We reproduce some well-known results for the case of quenched disorder (random temperature and random field), and analyze the effect of four different types of time-dependent disorder scarcely studied so far in the literature. Some of them are of obvious experimental and theoretical relevance (as, for example, globally fluctuating temperatures or random fields). All the predictions coming from our field-theoretical analysis are fully confirmed by extensive simulations in two and three dimensions, and novel qualitatively different, non-Ising transitions are reported. Possible experimental setups designed to explore the described phenomenologies are also briefly discussed. The Ising model has played a central role in the development of modern statistical mechanics. Its universality class includes a broad variety of models and real materials, as magnetic
A study of the influence of different types of disorder on systems in the Ising universality class by employing both a dynamical field theory approach and extensive Monte Carlo simulations is presented. We reproduce some well-known results for the case of quenched disorder (random temperature and random field), and analyze the effect of four different types of time-dependent disorder scarcely studied so far in the literature. Some of them are of obvious experimental and theoretical relevance (as, for example, globally fluctuating temperatures or random fields). All the predictions coming from our field-theoretical analysis are fully confirmed by extensive simulations in two and three dimensions, and novel qualitatively different, non-Ising transitions are reported. Possible experimental setups designed to explore the described phenomenologies are also briefly discussed.
We present an extensive study of the effects of quenched disorder on the dynamic phase transitions of kinetic spin models in two dimensions. We undertake a numerical experiment performing Monte Carlo simulations of the square-lattice random-bond Ising and Blume-Capel models under a periodically oscillating magnetic field. For the case of the Blume-Capel model we analyze the universality principles of the dynamic disordered-induced continuous transition at the low-temperature regime of the phase diagram. A detailed finite-size scaling analysis indicates that both nonequilibrium phase transitions belong to the universality class of the corresponding equilibrium random Ising model.
By performing a high-statistics simulation of the D=4 random-field Ising model at zero temperature for different shapes of the random-field distribution, we show that the model is ruled by a single universality class. We compute to a high accuracy the complete set of critical exponents for this class, including the correction-to-scaling exponent. Our results indicate that in four dimensions (i) dimensional reduction as predicted by the perturbative renormalization group does not hold and (ii) three independent critical exponents are needed to describe the transition.
Disorder is an inevitable part of any condensed matter system and therefore its study has always been of great importance. The effect of quenched randomness on a system that exhibits a continuous phase transition in the absence of any impurity has been studied in the past and the results are relatively well understood. However, the effect of quenched randomness on \\emph{first-order} phase transitions is still not well understood. In this dissertation, we study the effect of quenched bond-randomness on the classical and quantum first-order phase transitions.In Chapter 2, we study the effect of the disordered three-color Ashkin-Teller model, whose pure version undergoes a first-order phase transition. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as a critical point. We conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model. Furthermore, we find that the exponents
beta and
nu vary with disorder and the four-spin coupling strength.In Chapter 3, we extend our study of the disordered three-color Ashkin-Teller model. Utilizing extensive cluster Monte Carlo simulations on large lattice sizes of up to
128times128 spins, each of which is represented by three colors taking values
pm1, we show that the rounding of the first-order phase transition is an emergent criticality.We find that the critical exponents,
nu and
beta, change as the strength of disorder or the four-spin coupling varies, and we show that the correlation length critical exponent violates the lower bound
2/Dlenu, where
D is the dimension of the system.In Chapter 4, we study the \\emph{quantum} three-color Ashkin--Teller model and show that the quantum critical point in
(1+1) dimension is an unusual one, with activated scaling at the critical point and Griffiths-McCoy phase away from it. We find that the behavior is similar to the transverse random field Ising model, even though the pure system has a first-order transition in this case.
For random systems with quenched disorders distributed randomly in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions and perfectly correlated in the remaining ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ dimensions, the renormalization-group theory has been performed in the past in two different ways. In the single-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as finite and expands the perturbation terms only for \ensuremath{\epsilon}=
dc-d, where
dc is the upper critical dimension. In the double-expansion method, one treats ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$ as infinitesimal and performs an additional expansion for ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$. The former predicts smeared phase transitions, while the latter predicts sharp second-order phase transitions. We wish to find out which of the two predictions holds for a three-dimensional Ising model for which ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{C}}$=2 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$=1. We argue, using the replica method, that the spin correlation is isotropic for large distances as long as the fluctuation among the random bonds (in ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{R}}$ dimensions) is sufficiently smaller than the thermal energy
kBT. With parameters chosen to satisfy this condition in the model, a Monte Carlo simulation has been performed. The results favor a sharp second-order phase transition.
We review and compare different approaches for studying the influence of quenched, random disorder in three-dimensional Ising and Potts models for ferromagnets subject to impurities.From a theoretical view point, field theoretic renormalization group studies provide quite accurate results.Experiments carried out on crystalline mixtures of compounds lead to measurements of criticial exponents as accurate as three digits.Numerically, extensive Monte Carlo simulations are shown to be of comparable accuracy.Finally, we also discuss recently generated high-temperature series expansions for the free energy and susceptibility.Within this approach, using the star-graph expansion technique, quenched disorder averages can be calculated exactly while keeping the disorder strength p as well as the dimension d as symbolic parameters.