Existence and nature of the de Almeida–Thouless transition in the three-dimensional Edwards–Anderson spin glass in a magnetic field

In 1990, Caracciolo, Parisi, Patarnello and Sourlas simulated the three-dimensional Ising spin glass in a uniform magnetic field and found behavior strikingly similar to mean-field theory — a non-trivial, non-self-averaging overlap distribution, a spin-glass susceptibility growing as $L^\omega$ with $\omega = 1.8 \pm 0.25$, and a coupled-replica response steepening as the system grew — but they could not control finite-size effects and could not decide whether a genuine thermodynamic spin-glass transition in a field exists. This is the finite-dimensional version of the de Almeida–Thouless (dAT) question: does replica symmetry breaking survive in a magnetic field in three dimensions, or does droplet theory apply, with no transition at any nonzero field? Thirty-six years later the question remains open and actively debated: correlation-length finite-size-scaling studies in three dimensions find no transition, newer conditioned-overlap and tail-robust susceptibility analyses find evidence for one, field theory provides no perturbative fixed point below six dimensions while zero-temperature loop expansions and exact-form-factor arguments support a finite-dimensional transition, and simulations in four, five and six dimensions increasingly find the transition. The problem is to settle, for the thermodynamic limit of the three-dimensional model, whether the dAT line exists, and if so to determine its critical line, order and universality class, and the scaling description of the low-temperature phase; if not, to supply the crossover theory that explains the finite-size mean-field-like signals.

Background

A spin glass is a magnet with quenched (frozen) random interactions whose low-temperature state is frozen but lacks conventional magnetic order. The standard lattice model is the Edwards–Anderson (EA) Ising spin glass: spins $\sigma_i = \pm 1$ on a cubic lattice with random nearest-neighbor couplings $J_{ij}$ (typically Gaussian with zero mean, or bimodal $\pm J$), governed by $H = -\sum_{\langle ij \rangle} J_{ij}\sigma_i\sigma_j - h\sum_i \sigma_i$, where $h$ is a uniform magnetic field. Because the magnetization is not the order parameter, one studies the overlap $q = N^{-1}\sum_i \sigma_i\tau_i$ between two independent equilibrium replicas (two copies of the system sharing the same disorder realization), with probability distribution $P(q)$. In zero field the three-dimensional model has a well-established transition at $T_c(h=0)$ below which $P(q)$ broadens. In the infinite-range Sherrington–Kirkpatrick mean-field model, the exact Parisi replica-symmetry-breaking (RSB) solution describes the glassy phase: the replica-symmetric solution becomes unstable below the de Almeida–Thouless (AT) line $h_{\mathrm{AT}}(T)$, and below that line $P(q)$ is non-trivial even at $h>0$, with an ultrametric (hierarchical-tree) organization of pure states. For short-range finite-dimensional models this picture is contested. Droplet theory (Fisher and Huse) instead predicts only two pure states related by a global spin flip; a magnetic field then acts as a random field on the order parameter and destroys the transition at any $h>0$ in any finite dimension, so no AT line should exist. The field-theoretic treatment of the field-induced transition has upper critical dimension $d=6$ (the dimension above which mean-field critical behavior becomes exact), and the classic one-loop analysis of Bray and Roberts found no stable renormalization-group fixed point below six dimensions, so perturbation theory cannot decide the issue in low dimensions. The 1990 paper of Caracciolo, Parisi, Patarnello and Sourlas attacked the three-dimensional problem in a field with two diagnostics that still define the modern agenda: (i) the field-dependent overlap distribution $P(q,h)$, the link-overlap distribution (the analogue of $P(q)$ built from nearest-neighbor bond variables rather than site spins), and the spin-glass susceptibility $\chi_{\mathrm{SG}}$ (the fluctuations of the overlap), which in an ordered phase grows with linear size $L$ as $L^{\omega}$; and (ii) two real replicas coupled by a small perturbation $\varepsilon\sum_i \sigma_i\tau_i$, whose response $Q(\varepsilon)$ becomes singular (discontinuous) at $\varepsilon = 0$ in the RSB scenario but stays smooth in a two-state (droplet) phase. They found mean-field-like signals at the sizes then available but judged the finite-size effects uncontrolled, leaving open whether the AT line exists in three dimensions, whether the apparent singularities are genuine, whether the exponent $\omega$ is asymptotic or effective, and how to formulate a controlled finite-size-scaling description of the low-temperature phase in a field.

Problem Statement

For the three-dimensional Edwards–Anderson Ising spin glass with quenched random couplings in a uniform magnetic field $h > 0$: does a thermodynamic spin-glass phase transition exist below the zero-field critical temperature $T_c(h=0)$ — that is, is there a nonempty de Almeida–Thouless critical line $h_{\mathrm{dAT}}(T) > 0$ in the field–temperature plane at which replica symmetry is broken — or is the mean-field-like behavior observed at finite sizes a long crossover with no transition at any nonzero field, as droplet theory predicts? If the transition exists, determine its critical line $h_{\mathrm{dAT}}(T)$, its order (continuous versus first-order-like), and its universality class, including the finite-size-scaling forms of the overlap distribution $P(q,h,L)$, the coupled-replica response $Q(\varepsilon,h,L)$ (in particular whether $Q(\varepsilon)$ becomes singular at $\varepsilon = 0$ in the thermodynamic limit), the growth $\chi_{\mathrm{SG}} \sim L^{\omega}$ with its asymptotic exponent $\omega$, and the organization of the low-temperature phase (ultrametric, many-state RSB structure versus a two-state droplet structure). If no transition exists, provide the scaling theory that accounts for the observed finite-size mean-field-like signals — the broadening of $P(q,h)$, the steepening of $Q(\varepsilon)$, and the apparent growth of $\chi_{\mathrm{SG}}$ — as crossover phenomena. A deciding answer must establish the existence claim for the thermodynamic limit of the three-dimensional short-range model itself; evidence restricted to long-range one-dimensional surrogates, to spatial dimensions $d \geq 4$, or to finite sizes without a controlled extrapolation does not by itself close the question.

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