Thermodynamic glass transition in a spin glass without time-reversal symmetry

Abstract

Spin glasses are a longstanding model for the sluggish dynamics that appear at the glass transition. However, spin glasses differ from structural glasses in a crucial feature: they enjoy a time reversal symmetry. This symmetry can be broken by applying an external magnetic field, but embarrassingly little is known about the critical behavior of a spin glass in a field. In this context, the space dimension is crucial. Simulations are easier to interpret in a large number of dimensions, but one must work below the upper critical dimension (i.e., in d < 6) in order for results to have relevance for experiments. Here we show conclusive evidence for the presence of a phase transition in a four-dimensional spin glass in a field. Two ingredients were crucial for this achievement: massive numerical simulations were carried out on the Janus special-purpose computer, and a new and powerful finite-size scaling method. T he glass transition differs from standard phase transitions in that the equilibration time of glass formers (supercooled liquids, polymers, proteins, superconductors, etc.) diverges without dramatic changes in their structural properties (1-3). The reconciliation of the dynamic slowdown with the apparent immutability of glass formers is a major challenge for condensed matter physics.Spin glasses [which are disordered magnetic alloys (4)] enjoy a privileged status in this context, as they provide the simplest model system both for theoretical and experimental studies of a glassy dynamics. On the experimental side, time-dependent magnetic fields provide a wonderful tool to probe the dynamic response, which can be accurately measured with a SQUID (for instance, see ref. 5). On the theoretical side, magnetic systems are notably easier to model and to simulate numerically. In fact, special-purpose computers have been built for the simulation of spin glasses (6-9).Yet, spin glasses differ from most glassy systems in a crucial feature: like all magnetic systems, they enjoy time-reversal symmetry in the absence of an applied magnetic field. In fact, we now know that their glassy dynamics are due to a bona fide phase transition in which the time-reversal symmetry is spontaneously broken (10-12). Yet, in the presence of an applied magnetic field, the experimental spin-glass dynamics is just as glassy, although the field explicitly breaks the symmetry.However, whether spin glasses in a magnetic field undergo a phase transition has been a long-debated and still open question (see refs. 13, 14 for recent, opposed views). In the mean-field approximation, which is valid for large spatial dimension down to the upper critical dimension d u ¼ 6 (15), the de AlmeidaThouless line separates the high-temperature paramagnetic phase from the glassy phase (16)*. Yet, recent numerical simulations in spatial dimensions below d u did not find the transition in a field (18,19). Experimental studies have been conducted as well, with conflicting conclusions (20-23). In spite of these difficulties, it has been argued that the would-be spi...

The three-dimensional Ising spin glass in an external magnetic field: the role of the silent majority

“…If the lattices are large enough, in the presence of a second-order phase transition, the curves are expected to cross at a finite temperature T c (h). The figures on the right show the cumulant R 12 , which in the presence of a magnetic field is a better indicator of a phase transition [29], for the same magnetic fields. At zero field the heights of the crossings (which are universal quantities) are indicated with a point at T c = 1.1019 (29).…”

“…The same strategy has been followed for h > 0, with negative results [27,28]. Yet, this cannot be the whole story: Recent work in D = 4, hence below D u , using a non-standard finitesize scaling method has found clear evidence for a dAT line [29]. Furthermore, one may try to interpolate between D = 3 and D = 4 by tuning long-range interactions in D = 1 chains [30,31].…”

Numerical Construction of the Aizenman-Wehr Metastate

“…RSB theory extends as well to d < d U features found in the mean field solution [25]: many states (infinitely many in the L → ∞ limit), hierarchically organized, contribute to the Gibbs measure, each one with a weight that depends on the disorder realization. Consequences include the existence of the de Almeida-Thouless line [26] (the spin-glass phase transition survives in the presence of a small external magnetic field [27]), or the strong sample-to-sample fluctuations induced by the nonself-averageness of several measurable quantities [28] (these observations [27,28] were, however, obtained by simulating systems of finite size).…”

Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

“…2(c) and 2(d)]. These measurements are necessary for short-range systems because the existence of a spin-glass state in a field remains controversial [28][29][30][31][32][33]. Therefore, under this restriction, we expect to probe an actual (nonequilibrium) spin-glass state.…”