One-step replica-symmetry-breaking phase below the de Almeida-Thouless line in low-dimensional spin glasses

Evidence of a second-order phase transition in the six-dimensional Ising spin glass in a field

The very existence of a phase transition for spin glasses in an external magnetic field is controversial, even in high dimensions. We carry out massive simulations of the Ising spin-glass in a field, in six dimensions (which, according to classical-but not generally accepted-field-theoretical studies, is the upper critical dimension). We obtain results compatible with a second-order phase transition and estimate its critical exponents for the simulated lattice sizes. The detailed analysis performed by other authors of the replica symmetric Hamiltonian, under the hypothesis of critical behavior, predicts that the ratio of the renormalized coupling constants remain bounded as the correlation length grows. Our numerical results are in agreement with this expectation.

Erratum: Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field [Phys. Rev. E 105, 054106 (2022)]

This corrects the article DOI: 10.1103/PhysRevE.105.054106.

Replica symmetry broken states of some glass models

We have studied in detail the M-p balanced spin-glass model, especially the case p=4. These types of model have relevance to structural glasses. The models possess two kinds of broken replica states; those with one-step replica symmetry breaking (1RSB) and those with full replica symmetry breaking (FRSB). To determine which arises requires studying the Landau expansion to quintic order. There are nine quintic-order coefficients, and five quartic-order coefficients, whose values we determine for this model. We show that it is only for 2≤M<2.4714⋯ that the transition at mean-field level is to a state with FRSB, while for larger M values there is either a continuous transition to a state with 1RSB (when M≤3) or a discontinuous transition for M>3. The Gardner transition from a 1RSB state at low temperatures to a state with FRSB also requires the Landau expansion to be taken to quintic order. Our result for the form of FRSB in the Gardner phase is similar to that found when 2≤M<2.4714⋯, but differs from that given in the early paper of Gross et al. [Phys. Rev. Lett. 55, 304 (1985)0031-900710.1103/PhysRevLett.55.304]. Finally we discuss the effects of fluctuations on our mean-field solutions using the scheme of Höller and Read [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] and argue that such fluctuations will remove both the continuous 1RSB transition and discontinuous 1RSB transitions when 8>d≥6 leaving just the FRSB continuous transition. We suggest values for M and p which might be used in simulations to confirm whether fluctuation corrections do indeed remove the 1RSB transitions.

Theory and experiments for disordered elastic manifolds, depinning, avalanches, and sandpiles

Domain walls in magnets, vortex lattices in superconductors, contact lines at depinning, and many other systems can be modeled as an elastic system subject to quenched disorder. The ensuing field theory possesses a well-controlled perturbative expansion around its upper critical dimension. Contrary to standard field theory, the renormalization group (RG) flow involves a function, the disorder correlator Δ(w), and is therefore termed the functional RG. Δ(w) is a physical observable, the auto-correlation function of the center of mass of the elastic manifold. In this review, we give a pedagogical introduction into its phenomenology and techniques. This allows us to treat both equilibrium (statics), and depinning (dynamics). Building on these techniques, avalanche observables are accessible: distributions of size, duration, and velocity, as well as the spatial and temporal shape. Various equivalences between disordered elastic manifolds, and sandpile models exist: an elastic string driven at a point and the Oslo model; disordered elastic manifolds and Manna sandpiles; charge density waves and Abelian sandpiles or loop-erased random walks. Each of the mappings between these systems requires specific techniques, which we develop, including modeling of discrete stochastic systems via coarse-grained stochastic equations of motion, super-symmetry techniques, and cellular automata. Stronger than quadratic nearest-neighbor interactions lead to directed percolation, and non-linear surface growth with additional Kardar-Parisi-Zhang (KPZ) terms. On the other hand, KPZ without disorder can be mapped back to disordered elastic manifolds, either on the directed polymer for its steady state, or a single particle for its decay. Other topics covered are the relation between functional RG and replica symmetry breaking, and random-field magnets. Emphasis is given to numerical and experimental tests of the theory.

Numerical test of the replica-symmetric Hamiltonian for correlations of the critical state of spin glasses in a field

A growing body of evidence indicates that the sluggish low-temperature dynamics of glass formers (e.g., supercooled liquids, colloids, or spin glasses) is due to a growing correlation length. Which is the effective field theory that describes these correlations? The natural field theory was drastically simplified by Bray and Roberts in 1980. More than 40 years later, we confirm the tenets of Bray and Roberts's theory by studying the Ising spin glass in an externally applied magnetic field, both in four spatial dimensions (data obtained from the Janus collaboration) and on the Bethe lattice.

Complexity as information in spin-glass Gibbs states and metastates: Upper bounds at nonzero temperature and long-range models

In classical finite-range spin systems, especially those with disorder such as spin glasses, a low-temperature Gibbs state may be a mixture of a number of pure or ordered states; the complexity of the Gibbs state has been defined in the past roughly as the logarithm of this number, assuming the question is meaningful in a finite system. As nontrivial pure-state structure is lost in finite size, in a recent paper [Phys. Rev. E 101, 042114 (2020)2470-004510.1103/PhysRevE.101.042114] Höller and the author introduced a definition of the complexity of an infinite-size Gibbs state as the mutual information between the pure state and the spin configuration in a finite region, and applied this also within a metastate construction. (A metastate is a probability distribution on Gibbs states.) They found an upper bound on the complexity for models of Ising spins in which each spin interacts with only a finite number of others, in terms of the surface area of the region, for all T≥0. In the present paper, the complexity of a metastate is defined likewise in terms of the mutual information between the Gibbs state and the spin configuration. Upper bounds are found for each of these complexities for general finite-range (i.e., short- or long-range, in a sense we define) mixed p-spin interactions of discrete or continuous spins (such as m-vector models), but only for T>0. For short-range models, the bound reduces to the surface area. For long-range interactions, the definition of a Gibbs state has to be modified, and for these models we also prove that the states obtained within the metastate constructions are Gibbs states under the modified definition. All results are valid for a large class of disorder distributions.

Unexpected Upper Critical Dimension for Spin Glass Models in a Field Predicted by the Loop Expansion around the Bethe Solution at Zero Temperature

The spin-glass transition in a field in finite dimension is analyzed directly at zero temperature using a perturbative loop expansion around the Bethe lattice solution. The loop expansion is generated by the M-layer construction whose first diagrams are evaluated numerically and analytically. The generalized Ginzburg criterion reveals that the upper critical dimension below which mean-field theory fails is D_{U}≥8, at variance with the classical result D_{U}=6 yielded by finite-temperature replica field theory. Our expansion around the Bethe lattice has two crucial differences with respect to the classical one. The finite connectivity z of the lattice is directly included from the beginning in the Bethe lattice, while in the classical computation the finite connectivity is obtained through an expansion in 1/z. Moreover, if one is interested in the zero temperature (T=0) transition, one can directly expand around the T=0 Bethe transition. The expansion directly at T=0 is not possible in the classical framework because the fully connected spin glass does not have a transition at T=0, being in the broken phase for any value of the external field.

Spin Glasses in a Field Show a Phase Transition Varying the Distance among Real Replicas (And How to Exploit It to Find the Critical Line in a Field)

We discuss a phase transition in spin glass models that have been rarely considered in the past, namely, the phase transition that may take place when two real replicas are forced to be at a larger distance (i.e., at a smaller overlap) than the typical one. In the first part of the work, by solving analytically the Sherrington-Kirkpatrick model in a field close to its critical point, we show that, even in a paramagnetic phase, the forcing of two real replicas to an overlap small enough leads the model to a phase transition where the symmetry between replicas is spontaneously broken. More importantly, this phase transition is related to the de Almeida-Thouless (dAT) critical line. In the second part of the work, we exploit the phase transition in the overlap between two real replicas to identify the critical line in a field in finite dimensional spin glasses. This is a notoriously difficult computational problem, because of considerable finite size corrections. We introduce a new method of analysis of Monte Carlo data for disordered systems, where the overlap between two real replicas is used as a conditioning variate. We apply this analysis to equilibrium measurements collected in the paramagnetic phase in a field, h > 0 and T c ( h ) < T < T c ( h = 0 ) , of the d = 1 spin glass model with long range interactions decaying fast enough to be outside the regime of validity of the mean field theory. We thus provide very reliable estimates for the thermodynamic critical temperature in a field.