Using 2-loop renormalization group calculations, we study a system of N two-dimensional Potts models with random bonds coupled together by their local energy density. This model can be seen as a generalization of the random Ashkin-Teller model. We found that, depending on the sign of the coupling term, the universality class of the system in the presence of randomness is different. Under particular consideration, this model presents an example of a first-order phase transition rounded by randomness.
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.
The effects of bond randomness on the universality aspects of a two-dimensional (d = 2) Blume-Capel model embedded in the triangular lattice are discussed. The system is studied numerically in both its first- and second-order phase-transition regimes by a comprehensive finite-size scaling analysis for a particularly suitable value of the disorder strength. We find that our data for the second-order phase transition, emerging under random bonds from the second-order regime of the pure model, are compatible with the universality class of the two-dimensional (2D) random Ising model. Furthermore, we find evidence that, the second-order transition emerging under bond randomness from the first-order regime of the pure model, belongs again to the same universality class. Although the first finding reinforces the scenario of strong universality in the 2D Ising model with quenched disorder, the second is in difference from the critical behavior, emerging under randomness, in the cases of the ex-first-order transitions of the Potts model. Finally, our results verify previous renormalization-group calculations on the Blume-Capel model with disorder in the crystal-field coupling.
We analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of N coupled two-dimensional Ising models. For N> 2 the pure model exhibits a fluctuation-driven first order transition, characterised by runaway renormalisation group behaviour. We show that the addition of weak randomness acts to stabilise these flows, in such a way that the trajectories ultimately flow back towards the pure decoupled Ising fixed point, with the usual critical exponents α = 0, ν = 1, apart from logarithmic corrections. We also show by examples that, in higher dimensions, such transitions may either become continuous or remain first order in the presence of randomness. 1 The effect of quenched randomness coupling to the local energy density of a system which, in its absence, undergoes a continuous phase transition is well understood from the point of view of the renormalisation group version
We investigate the effects of weak disorder on different coupled q -state Potts models with q ⩽ 4 using two-loop renormalisation group equations. This study presents new examples of first-order transitions driven by randomness. We found that weak disorder makes the models decouple. Therefore, it appears that no relation emerges, at a perturbation level, between the disordered ( q 1 × q 2 )-state Potts model and the two disordered q 1 , q 2 -state Potts models ( q 1 ≠ q 2 ), though recent numerical investigations have shown that their central charges are approximately similar.
Motivated by recent experiments on phase behavior of systems confined in porous media, we have studied the effect of randomness on the nature of the phase transition in the two-dimensional Potts model. To model the effects of the porous matrix we introduce a random distribution of couplings P(J(ij))=pdelta(J(ij)-J1)+(1-p)delta(J(ij)-J2) in the q state Potts Hamiltonian. An extensive Monte Carlo study is made on this system for q=5. We studied two different cases of disorder (a) J(1)/J(2)-->infinity and p=0.8 and (b) J(1)/J(2)=10 and p=0.5. We observed, in both cases, that the weak first order transition that appears in the pure case, changes to a second-order transition. A finite size scaling analysis shows that the correlation length exponent nu is close to 1 and the best fit to the dependence of the specific heat on system size is logarithmic. This suggests that both cases belong to the universality class of the Ising model. In contrast, the magnetic exponents point to a different universality class.