PHYSICAL REVIEW B 87, 064202 (2013) Fragility of the mean-ﬁeld scenario of structural glasses for disordered spin models in ﬁnite dimensions Chiara Cammarota, 1,2,* Giulio Biroli, 1,† Marco Tarzia, 2,‡ and Gilles Tarjus 2,§ 1 IPhT, CEA/DSM-CNRS/URA 2306, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France 2 LPTMC, CNRS-UMR 7600, Universit´ e Pierre et Marie Curie, boˆ ıte 121, 4 Pl. Jussieu, 75252 Paris c´ edex 05, France (Received 11 October 2012; revised manuscript received 18 January 2013; published 21 February 2013) At the mean-ﬁeld level, on fully connected lattices, several disordered spin models have been shown to belong to the universality class of “structural glasses” with a “random ﬁrst-order transition” (RFOT) characterized by a discontinuous jump of the order parameter and no latent heat. However, their behavior in ﬁnite dimensions is often drastically different, displaying either no glassiness at all or a conventional spin-glass transition. We clarify the physical reasons for this phenomenon and stress the unusual fragility of the RFOT to short-range ﬂuctuations, associated, e.g., with the mere existence of a ﬁnite number of neighbors. Accordingly, the solution of fully connected models is only predictive in very high dimension, whereas despite being also mean-ﬁeld in character, the Bethe approximation provides valuable information on the behavior of ﬁnite-dimensional systems. We suggest that before embarking on a full blown account of ﬂuctuations on all scales through computer simulation or renormalization-group approach, models for structural glasses should ﬁrst be tested for the effect of short-range ﬂuctuations and we discuss ways to do it. Our results indicate that disordered spin models do not appear to pass the test and are therefore questionable models for investigating the glass transition in three dimensions. This also highlights how nontrivial is the ﬁrst step of deriving an effective theory for the RFOT phenomenology from a rigorous integration over the short-range ﬂuctuations. DOI: 10.1103/PhysRevB.87.064202 PACS number(s): 64.70.Q−, 75.10.Nr, 64.70.pm I. INTRODUCTION The random ﬁrst-order transition (RFOT) theory 1,2 of the glass transition builds on a mean-ﬁeld scenario in which a complex free-energy landscape with an exponentially large number of metastable states emerges below a critical temper- ature T d and the conﬁgurational entropy associated with these metastable states vanishes at a lower temperature T K . At T K , an RFOT, i.e., a transition with a discontinuous order parameter yet no latent heat, to an ideal glass takes place. This scenario is realized in mean-ﬁeld-like approximations of a variety of glass-forming systems (liquid models, lattice glasses, uniformly frustrated systems) as well as in mean-ﬁeld, fully connected, spin models with quenched disorder 2,3 (e.g., p-spin model, Potts glass). The latter correspond to spin glasses without spin inversion symmetry, and their behavior differs from that of the fully connected Sherrington-Kirkpatrick Ising spin glass, which is characterized by a continuous transition in place of the RFOT. They have been investigated in great detail and have provided most of the clues about the generic behavior of “mean-ﬁeld structural glasses.” To make progress toward a theory of the glass transition, one must, however, go beyond the mean-ﬁeld description and include the effect of ﬂuctuations 4 in ﬁnite-dimensional systems with ﬁnite-range interactions. The main assumption behind the RFOT theory of glass formation is that the mean-ﬁeld scenario with a dynamic and a static critical temperature retains some validity when ﬂuctuations are taken into account. The ergodicity breaking transition at T d is expected to be smeared and the metastable states no longer have an inﬁnite lifetime because of entropically driven nucleation events. This under- lies the picture of a glass-forming liquid as a “mosaic state” with its relaxation to equilibrium dominated by thermally activated rare events involving “entropic droplets.” 1 Yet, the main ingredients associated with the RFOT are assumed to persist, even if renormalized by the effect of the ﬂuctuations. 2 The issue can be understood in the much simpler setting of the liquid-gas transition of simple ﬂuids. The mean-ﬁeld van der Waals approach predicts a liquid-gas transition with a terminal critical point. It is known that this homogeneous mean-ﬁeld picture needs to be modiﬁed, e.g., through the classical nucleation theory and the renormalization group: concepts such as metastability and spinodal are no longer crisply deﬁned, critical exponents as well as nonuniversal quantities are modiﬁed, yet the transition with the two, liquid and gas, free-energy states remains valid, at least when the dimension d is larger than 1. In d = 1, the mean-ﬁeld treatment is plain wrong and predicts a transition that is not present. Fluctuations can therefore have a more or less dramatic inﬂuence on the mean-ﬁeld scenario: this is the key-point concerning the relevance of the RFOT theory to glass-forming liquids. One natural path to follow in order to investigate the effect of the ﬂuctuations on the RFOT and the two-temperature picture is then to consider, both numerically and analytically, the various proposed models of structural glasses in ﬁnite dimensions. The disordered spin models are especially con- venient as they are both well deﬁned at the mean-ﬁeld level and much easier to investigate than more realistic models of structural glass-formers or effective Ginzburg-Landau theories in the replica formalism. In particular, they quite directly lend themselves to computer simulations and to real-space renormalization group (RG) treatments. The major obstacle on this seemingly straightforward route is that so far no traces of the RFOT scenario have been found in such studies on ﬁnite-dimensional, ﬁnite-range models, with either a complete absence of transition to a glass phase 5 or a behavior more compatible with a continuous spin-glass transition 6 than a discontinuous “random ﬁrst-order” one. 064202-1 1098-0121/2013/87(6)/064202(22) ©2013 American Physical Society