Tests of conformal invariance in randomness-induced second-order phase transitions

Abstract

The conformal covariance of correlation functions is checked in the second-order transition induced by random bonds in the two-dimensional eight-state Potts model. The decay of correlations is obtained via transfer matrix calculations in a cylinder geometry, and large-scale Monte Carlo simulations provide access to the correlations and the profiles inside a square with free or fixed boundary conditions. In both geometries, conformal transformations constrain the form of the spatial dependence, leading to accurate determinations of the order parameter scaling index, in good agreement with previous independent determinations obtained through standard techniques. The energy density exponent is also computed.PACS numbers: 64.60. Cn, 05.50.+q,05.70.Jk, 64.60.Fr It is well known that quenched randomness can deeply affect the critical properties at second-order phase transitions [1], and is liable to smooth first-order transitions, eventually leading to continuous transitions [2,3].In random systems, owing to strong inhomogeneities inherent in disorder, the usual symmetry properties required by conformal invariance (CI) [4] do not hold. However, by averaging over disorder realizations (denoted by [. . .] av ), translation and rotation invariance are restored, and it is generally believed that conformal invariance techniques should apply in principle. Recent results based on this assumption have recently been obtained at randomness-induced second-order transitions [5,6], but clear evidence of the validity of conformal invariance is still missing. The question is of both fundamental and practical interest. From the fundamental point of view, situations are known where a diverging correlation length does not guarantee the validity of CI. A few years ago, lattice animals were indeed found to be not conformally invariant although they display isotropic critical behavior with correlation lengths satisfying the usual scaling ξ ∼ L with the system size [7]. If conformal invariance works, on the other hand, its powerful techniques might be applied with no restriction to investigate the critical behavior of 2D random systems [8].In the 2D random-bond Ising Model, both analytic [9] and numerical results [10] are available. Transfer matrix (TM) calculations were also used to study the correlation function decay along strips and to compute the conformal anomaly (defined as a universal amplitude in finite-size corrections to the free energy) [11] and, since disorder is marginally irrelevant in the 2D Ising model, conformal invariance techniques were indeed efficient. At randomness-induced second-order phase transitions, a direct comparison between the results deduced from conformal invariance, and standard techniques, such as finitesize scaling (FSS) Monte Carlo (MC) simulations, have nevertheless not yet been made. After the pioneering work of Imry and Wortis, the first large-scale MC simulations devoted to the influence of quenched randomness, in a system whose pure version undergoes a strong firstorder phase transition, were a...

Magnetic critical behavior of two-dimensional random-bond Potts ferromagnets in confined geometries

“…In the two-dimensional RBIM, randomness being a marginally irrelevant perturbation, many results have been obtained via these techniques: Conformal anomaly, correlation decay, gap-exponent relation for long strips [34][35][36]. At randomness-induced second-order phase transitions, conformal techniques have also been used already [17,18,37] and numerical evidences for the validity of the conformal covariance assumption for correlation functions and density profiles were recently reported [38]. It is well known that in disordered spin systems, the strong fluctuations of couplings from sample to sample require careful averaging procedures [39][40][41].…”

Monte Carlo study of the triangular Blume-Capel model under bond randomness

“…This outcome is in contradiction to relevant results concerning the emerging under disorder continuous transitions of the (q > 4) Potts model in d = 2. In fact, it is now well established in the current literature [7,8,36,60,61,66,67,[71][72][73][74][75][76][77] that the universality class of the random Potts model is never that of the pure (or disordered) Ising model, but changes as a function of the number of states of the model. Thus, one may suspect that this should also be the case for the current BC model, i.e., the appearance of second-order phase transitions that belong to different universality classes, depending continuously on the value of the crystal-field coupling.…”

“…1), the following issue should be treated with caution: Two characteristic values of the crystal-field coupling should be chosen and at the same time a value of the disorder strength r, under which the originally first-order phase transition of the system undoubtedly switches to second order. In the current model this is a nontrivial issue, compared to the case of the Potts model, for which Picco [60] and Chatelain and Berche [61] have proposed techniques which allow one to find an optimal value for the ratio of couplings to secure the minimum scaling corrections to the critical behavior.…”

“…Although the first work on the random q = 8 Potts model by Chen et al [33] suggested that the emerging, under disorder, continuous transition belongs to the universality class of the Ising model, this result is known now to be false. In particular, many works have been devoted to this issue since then [7,8,36,60,61,66,67,[71][72][73][74][75][76][77] and the values of the critical exponents are pretty well known for various values of q (below and above q = 4). So, the situation is now completely clarified, with the universality class of the random Potts model changing continuously with the values of q.…”

Universality, frustration, and conformal invariance in two-dimensional random Ising magnets

“…21,23 The validity of conformal invariance ideas for ͑unfrustrated͒ disordered q-state Potts models has also been verified. 24,25 This paper is organized as follows. In Sec.…”