It is shown that (i) the N-component Ashkin-Teller model in d=2 has continuous O(N) symmetry, (ii) its critical properties near the decoupled Ising transition are determined by the massive Gross-Neveu [(\ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\psi}
)2] model in general and the (integrable) massless version along a line, (iii) the known results for the massless Gross-Neveu model imply that the first-order transition found as N\ensuremath{\rightarrow}\ensuremath{\infty} persists down to N=3 along this line, (iv) the supersymmetry of the N=3 model implies that the leading singularity of the free energy is zero along the first-order line, and (v) the N=4 model along this line decouples into two identical N=2 models up to subleading corrections.
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 critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large-distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent \ensuremath{\nu} of the correlation length for the three- and four-state random-bond Potts models is also calculated in a three-loop approximation. We show that this exponent is given by an apparently convergent series in \ensuremath{\epsilon}=c-1/2 (with c the central charge of the Potts model) and that the numerical values of \ensuremath{\nu} are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the three- and four-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries. \textcopyright{} 1996 The American Physical Society.
The critical behavior of 2D anisotropic systems with weak quenched disorder described by the so{called generalized Ashkin{Teller model (GATM), including the Ising model with random bonds, the dilute Baxter model, the impure N{color Ashkin{Teller model, and minimal conformal eld theory models (MCFTM) with c < 1 (c is the central charge) perturbed by randomness is discussed.All these models except MCFTM were found to belong to the Ising model universality class.Critical exponents of disordered MCFTM are calculated within perturbative expansions in the powers of = c 1=2. RG ows exhibit the rounding of uctuation{driven rst{order phase transitions by disorder.
Abstract The N -color Ashkin–Teller model corresponds to N Ising models coupled by four-spin interactions. We consider the two-dimensional case in presence of quenched disorder and use scale invariant scattering theory to determine all the solutions of the exact renormalization group fixed points equations. The weak disorder sector is characterized by a solution that, for any fixed N larger than 1, is a line of fixed points with Ising thermal exponents and continuously varying magnetic exponents. The number of fixed point solutions allowed by the symmetries of the model increases at strong disorder illustrating the growing dependence on the distributions of the two random couplings. The presence of some critical exponents which do not depend on the symmetry parameter N confirms this type of superuniversality as a peculiar feature of random criticality.
An intriguing result of statistical mechanics is that a first-order phase transition can be rounded by disorder coupled to energylike variables. In fact, even more intriguing is that the rounding may manifest itself as a critical point, quantum or classical. In general, it is not known, however, what universality classes, if any, such criticalities belong to. In order to shed light on this question we examine in detail the disordered three-color Ashkin-Teller model by Monte Carlo methods. Extensive analyses indicate that the critical exponents define a new universality class. We show that the rounding of the first-order transition of the pure model due to the impurities is manifested as criticality. However, the magnetization critical exponent, β, and the correlation length critical exponent, ν, are found to vary with disorder and the four-spin coupling strength, and we conclusively rule out that the model belongs to the universality class of the two-dimensional Ising model.