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.
We study the effects of uncorrelated quenched disorder to the phase diagram and continuous transitions of three-dimensional lattice Z_{2} gauge Higgs models. For this purpose, we consider two types of quenched disorder, associated with the sites and plaquettes of the cubic lattice. In both cases, for sufficiently weak disorder, the phase diagram remains similar to that of the pure system, showing two different phases (one of them being a topologically ordered phase), separated by two different continuous transition lines. However, the quenched disorder changes the universality classes of the critical behaviors along some of the transition lines. The random-plaquette disorder turns out to be relevant along the topological Z_{2} gauge transition line, so the critical behaviors belong to the different random-plaquette Z_{2} gauge (RPZ_{2}G) universality class with length-scale exponent ν=ν_{rp}≈0.82; on the other hand, it turns out to be irrelevant along the other Ising^{×} transition line (a variant of the Ising transitions with a gauge-dependent order parameter), leaving unchanged its asymptotic critical behaviors with ν=ν_{I}≈0.63. The random-site disorder leads to a substantially different scenario: it destabilizes the Ising^{×} critical behaviors of the pure model, changing them into those of the randomly dilute Ising^{×} (RDI^{×}) universality class with ν=ν_{RDI}≈0.68, while the critical behaviors along the other Z_{2} gauge topological transition line remains stable, with ν=ν_{I}≈0.63.
We have considered the half-filled disordered attractive Hubbard model on a square lattice, in which the on-site attraction is switched off on a fraction
f of sites, while keeping a finite
U on the remaining ones. Through quantum Monte Carlo simulations for several values of
f and
U and for system sizes ranging from $8\ifmmode\times\else\texttimes\fi{}8$ to $16\ifmmode\times\else\texttimes\fi{}16$, we have calculated the configurational averages of the equal-time pair structure factor
Ps and, for a more restricted set of variables, the helicity modulus ${\ensuremath{\rho}}_{s}$, as functions of temperature. Two finite-size scaling Ans\"atze for
Ps have been used: one for zero temperature and the other for finite temperatures. We have found that the system sustains superconductivity in the ground state up to a critical impurity concentration
fc, which increases with
U, at least up to
U=4 (in units of the hopping energy). Also, the normalized zero-temperature gap as a function of
f shows a maximum near $f\ensuremath{\sim}0.07$ for $2\ensuremath{\lesssim}U\ensuremath{\lesssim}6$. Analyses of the helicity modulus and of the pair structure factor led to the determination of the critical temperature as a function of
f for
U=3, 4, and 6: they also show maxima near $f\ensuremath{\sim}0.07$, with the highest
Tc increasing with
U in this range. We argue that, overall, the observed behavior results from both the breakdown of charge-density-wave-superconductivity degeneracy and the fact that free sites tend to ``push'' electrons toward attractive sites; the latter effect being more drastic at weak couplings.
We present in detail a functional renormalization group (FRG) study of a Landau-Ginzburg model of type-II superconductors (generalized to
2N complex fields) in an external magnetic field, both for a pure system and also in the presence of quenched random impurities. If the coupling functions are restricted to the space of functions with nonzero support only at reciprocal lattice vectors corresponding to the Abrikosov lattice, we find a stable FRG fixed point in the presence of disorder for
1<N<4, identical to that of the disordered
O(N) model in $d\ensuremath{-}2$ dimensions. This implies a continuous transition from the vortex crystal to vortex liquid in the presence of disorder, but only for
d>4. The nonzero-temperature transition will disappear in physical dimensions. The pure system has a stable fixed point only for
N>4. Therefore the physical case (
N=2) is likely to have a first-order transition in the absence of quenched disorder. We give a full discussion of both the motivation of the model and the details of the FRG calculation. We also place our results in context with regard to the current experimental scene concerning the high-
Tc compounds. In particular, we discuss the relevance of our results to the recently discovered critical end point in the phase diagram of Bi-Sr-Ca-Cu-O. The main results of this analysis were previously reported in the form of a Letter [M.A. Moore and T.J. Newman, Phys. Rev. Lett. 75, 533 (1995)].
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.
For High-Tc superconductors the fluctuation regime is much wider than conventional superconductors due to the short coherence length and high transition temperature. AC measurements can probe the conductivity at a wide range of frequencies, and therefore provide a test of the scaling properties predicted from theory. The measured critical exponents and width of the fluctuation peak are sample dependent, which suggests that the sample inhomogeneities may play an important role in the critical region. In this thesis we study the effects of disorder on the fluctuation conductivity in the critical regime of zero-field normal-superconducting transition by the time-dependent Ginzburg-Landau theory. We set up a discretized model of the superconductor and calculate the two-dimensional and three-dimensional scaling function without disorder above Tc. The result of the discretized model deviates slightly from previous theoretical studies of the continuous model, which can be explained by a finite short wavelength cutoff in Ginzburg-Landau theory. Our results agree well with other theoretical investigations on cutoff effects from an analytical approach. Disorder in a superconductor is modeled by a distribution of Tc's at the lattice sites. We add random disorder to a two-dimensional lattice and calculate the scaling functions averaged over 1000 disorder configurations. When disorder is weak, the scaling functions are increased when T is close Tc . At strong disorder anomalous behavior may occur at the region close to Tc.