Complexity of metastable states and the magnetization-driven response of the short-range random-field Ising model: does the mean-field picture survive?

The zero-temperature, fully connected soft-spin random-field Ising model (RFIM) admits an exact static solution: the quenched complexity, the disorder-averaged logarithmic count of metastable states, can be computed analytically, the saturation hysteresis loop is the curve where this complexity vanishes, and at low disorder the reentrant zero-complexity curve coincides with the response obtained when the magnetization rather than the field is externally controlled. Rosinberg and Munakata (2009), who obtained this solution, asked how much of the picture survives short-range interactions in finite dimensions. The subsequent literature gives only a partial answer: the replica-based extension to three dimensions works above the nonequilibrium critical disorder but breaks down inside the hysteresis loop and at criticality; Bethe-lattice calculations suggest the complexity need not vanish along the whole boundary of the metastable-state domain; and the magnetization-driven response, with its strong system-size-independent fluctuations of the induced field and its conjectured interface/depinning interpretation, is not yet captured by a static complexity-based description in finite dimensions. The problem posed here is to establish, analytically or numerically, the structure of the metastable-state complexity of the short-range RFIM and its relation to both the field-driven and the magnetization-driven response. The companion conjecture that equilibrium and out-of-equilibrium RFIM critical behaviors coincide was resolved negatively below d ≈ 5.1 by nonperturbative functional renormalization group in 2018 and is therefore not part of the open core.

Background

The random-field Ising model (RFIM) places Ising spins on a lattice with ferromagnetic couplings and quenched (frozen-in) random local fields $h_i$; the competition between coupling and disorder generates its physics. At zero temperature, when the uniform external field $H$ is swept adiabatically, the system sticks to a metastable configuration until it loses local stability and relaxes through an avalanche, a burst of coordinated spin flips. Sethna, Dahmen, Kartha, Krumhansl, Roberts, and Shore (1993) showed that this driven RFIM has a disorder-induced critical point: in three dimensions with Gaussian random fields of width $\Delta$, the saturation hysteresis loop (the limiting loop traced once the field has been swept over its full range) carries a macroscopic jump for $\Delta<\Delta_c^{\mathrm{hyst}}\approx 2.16$ and becomes smooth with scale-free avalanche statistics at $\Delta_c^{\mathrm{hyst}}$, whereas the equilibrium (ground-state) transition occurs at the strictly larger disorder $\Delta_c^{\mathrm{GS}}\approx 2.27$ (values compiled by Liu and Dahmen). A perturbative renormalization group for the nonequilibrium critical point was built by Dahmen and Sethna (1996) as an expansion around mean-field theory within the Martin-Siggia-Rose dynamical formalism (a field-theoretic representation of the classical dynamics of a driven system), but doubts about how the history dependence is implemented in that expansion have persisted.

Two structural facts organize the theory of this athermal dynamics. First, the no-passing rule (the property that as the external field is monotonically swept, a configuration reached earlier evolves into one reached later, so that different preparation histories cannot cross), proved by Middleton (1992) for charge-density-wave models (modulated electron-density phases of a solid driven through pinning disorder) and valid for ferromagnetic RFIM dynamics, makes the adiabatic field-driven evolution history-independent and implies that no one-spin-flip-stable metastable state exists outside the saturation loop. Second, inside the loop the metastable states, configurations stable against flipping any single spin, are exponentially numerous in the system size (Barrat, Kurchan, Loreto, and Sellitto, 2001), which invites a thermodynamic description: the complexity $\Sigma(m,H)$, the disorder-averaged logarithm of the number of metastable states with magnetization $m$ at field $H$, is an entropy density of metastability.

Rosinberg and Munakata (2009) computed this object exactly for the mean-field (fully connected) soft-spin version of the model, in which the discrete spins are replaced by continuous variables in a double-well potential,

$\mathcal{H}=-\sum_i(H+h_i)s_i+\frac{J}{2N}\sum_{i\neq j}s_i s_j+\sum_i V(s_i)$

with ferromagnetic coupling $J$, Gaussian random fields of width $\Delta$, and a piecewise-quadratic double well $V$ stiff enough ($k>J$) to keep the magnetization finite. Their results: at large disorder the domain $D(m,H)$ of the field-magnetization plane where the complexity is positive is convex, the complexity is strictly positive inside the saturation loop and vanishes along it (the loop is the envelope of the metastable states), and the system responds identically whether the field or the magnetization is controlled; at low disorder $D(m,H)$ is nonconvex and disconnected, the zero-complexity curve has a reentrant part, and this reentrant part coincides with the loop obtained under the magnetization-driven (M-driven) protocol, in which the magnetization is imposed and the field becomes an induced variable $H(m)$, the constraint acting like an infinite-range antiferromagnetic demagnetizing field. The message is that the zero-temperature nonequilibrium response can be defined unambiguously from a purely static description of the metastable states, and the paper closes by asking how much of this survives when the interaction is short-ranged.

Three strands bound what is known beyond mean field. On the Bethe lattice, a loopless graph of fixed connectivity $z$ (finite connectivity but no geometry), Dhar, Shukla, and Sethna (1997) solved the zero-temperature hysteresis loop, and Rosinberg, Tarjus, and Pérez-Reche (2008, 2009) computed the complexity: a smooth loop still coincides with the zero-complexity envelope, a jumping loop comes with a gap in the magnetization of the metastable states and a reentrant envelope, the unstable branch of the naive recursions is an artifact of mean field, and, putatively from the order-$1/z$ expansion, the complexity may not vanish along the whole boundary of $D(m,H)$. Pérez-Reche, Rosinberg, and Tarjus (2008) computed the annealed and quenched complexities (the disorder average of the state count itself, and of its logarithm, respectively) numerically for random regular graphs (graphs in which every vertex has the same fixed number of neighbors) and for the cubic lattice, confirming that at low disorder a region inside the loop contains no typical metastable states (a nonconvex hole). Rosinberg and Tarjus (2010) then constructed the analytical replica formalism (introducing independent copies of the system, the standard device for performing the disorder average of a logarithm) for the three-dimensional soft-spin model; it describes the loop, the correlation functions, and the second moment of the avalanche-size distribution above the nonequilibrium critical disorder, but breaks down inside the loop, where avalanches generate a singular contribution that a tractable local approximation cannot cancel, and near criticality, where a nonperturbative renormalization-group treatment would be needed. On the M-driven side, Illa, Rosinberg, and Vives (2006) and Illa, Rosinberg, Shukla, and Vives (2006) compared field- and magnetization-driven protocols in local mean-field theory and showed that the M-driven dynamics at zero temperature violates the no-passing rule, so the state reached at $m+\delta m$ depends on the step $\delta m$; Pérez-Reche, Truskinovsky, and Zanzotto (2008) showed in a related setting that controlling the conjugate variable can turn classical criticality into self-organized criticality (critical behavior reached by the dynamics itself, without tuning a control parameter); and simulations on the cubic lattice at low disorder show the induced field $H(m)$ fluctuating strongly around a magnetization-independent value $H_d$, with fluctuations reminiscent of a critical depinning transition (the loss of stability of an interface pushed through a disordered medium), suggesting that single-interface configurations dominate this regime.

Problem Statement

For the zero-temperature random-field Ising model with short-range (nearest-neighbor) ferromagnetic interactions and Gaussian random fields on finite-dimensional lattices (the physically relevant cases being the square and cubic lattices), determine which features of the exact mean-field complexity picture of rate-independent hysteresis survive, by establishing a static, complexity-based (or equivalent) statistical-mechanical characterization of the metastable states and its relation to the two driving protocols. The overarching objective is a self-contained static description of the short-range model that plays the role the exact mean-field solution plays for the fully connected one; its load-bearing deliverables are (1) and (2), with (3) entering insofar as it bears on the definition of the magnetization-driven response.

(1) Structure and boundary of the metastable-state domain. Compute, analytically or through controlled numerics with thermodynamic-limit extrapolation, the quenched complexity $\Sigma(m,H)$ of the one-spin-flip-stable states (configurations stable against flipping any single spin; $\Sigma$ is the disorder-averaged logarithm of their number at given magnetization $m$ and field $H$) in the field-magnetization plane: the shape of the domain $D(m,H)$ where $\Sigma>0$ as a function of the disorder (convex at large disorder; nonconvex with a magnetization gap at low disorder), and whether $\Sigma$ vanishes along the entire boundary of $D(m,H)$ or remains positive along parts of it, which is equivalently the question whether the saturation hysteresis loop still coincides with the zero-complexity envelope in all disorder regimes (order-$1/z$ Bethe-lattice calculations suggest it may not).

(2) Mapping to the magnetization-driven response. Decide whether the mean-field correspondence survives in finite dimensions: at low disorder the reentrant zero-complexity curve coincides with the M-driven hysteresis loop, while at large disorder the field-driven and magnetization-driven responses are identical. This includes characterizing the strong, system-size-independent fluctuations of the induced field $H(m)$ around the disorder-dependent, magnetization-independent value $H_d$ observed at low disorder: are they the signature of single-interface (domain-wall) configurations, and is their statistics that of a critical depinning transition, as conjectured?

(3) Protocol independence under magnetization control. Because the M-driven dynamics violates the no-passing rule (the monotone-evolution property that makes the field-driven response history-independent), the metastable state reached when going from $m$ to $m+\delta m$ depends on the step $\delta m$; determine whether a protocol-independent M-driven response exists in the thermodynamic limit and how the selection among the exponentially many metastable states is resolved.

Progress could take the form of an analytical theory (for example, an extension of the replica complexity formalism beyond the local self-energy approximation, or a nonperturbative functional renormalization-group treatment), a controlled numerical determination, or a combination of both.

References

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