The nature of the low-temperature phase of the Edwards–Anderson (EA) Hamiltonian [1] in finite dimensions H J = − ∑ < x , y > J x y σ x σ y (1) remains unresolved. Here, σ x = ± 1 is the Ising spin at site x , and ⟨ x , y ⟩ denotes a nearest-neighbor edge in the edge set E d of the d -dimensional cubic lattice Z d . The couplings J x y are taken to be independent, identically distributed continuous random variables chosen from a distribution ν ( d J x y ) , with random variable J x y assigned to the edge ⟨ x , y ⟩ . Our requirements on ν are that it be supported on the entire real line, distributed symmetrically about 0, and has finite variance; e.g., a Gaussian with mean 0 and variance 1. We denote a particular realization of the couplings by J .

There are, at present, four scenarios for the spin glass phase that are consistent both with numerical results and, as far as is currently known, mathematically consistent: replica symmetry breaking (RSB) [2–12], droplet-scaling [13–17], trivial–non-trivial spin overlap (TNT) [18, 19], and chaotic pairs [10, 20–22]. One of the central open questions in spin glass theory is which (if any) of these scenarios is correct and for which dimensions and temperatures.

The differences among the four scenarios at positive temperature are described elsewhere [12, 23, 24]; here, we are concerned with their different predictions at zero temperature, i.e., for the ground-state structure of the EA Hamiltonian. Of the four, two (RSB and chaotic pairs) predict the existence of many ground states, and the other two (droplet-scaling and TNT) predict the existence of only a single pair [17, 25, 26]. Although important, these differences are less fundamental than the nature of the interfaces that separate their ground states from their lowest-lying long-wavelength excitations. The presence or absence of multiplicity of ground states follows as a consequence of the nature of these excitations.

In this paper, we focus on the nature of low-energy long-wavelength excitations above the ground state and how they relate to ground state stability, with a view toward distinguishing different predictions of the four scenarios. Aside from elucidating the different (and potentially testable) predictions of these scenarios, determining the stability properties of the ground state is crucial in determining the low-temperature properties of the spin glass phase, including central questions such as multiplicity of pure states and the presence or absence of an Almeida-Thouless (AT) line [27]. We begin by defining the parameters of the study.

A finite volume Λ L was chosen corresponding to a cube of side L centered at the origin. A finite-volume ground state σ L is the lowest-energy spin configuration in Λ L , which is subject to a specified boundary condition. An infinite-volume ground state σ is a spin configuration on all of Z d , which is defined by the condition that its energy cannot be lowered by flipping any finite subset of spins. ( σ is always defined with respect to a specific J , but we suppress its dependence for notational convenience.) The condition for σ to be a ground state is then E S = ∑ ⟨ x , y ⟩ ∈ S J x y σ x σ y > 0 , (2) where S is any closed surface (or contour in two dimensions) in the dual lattice. The surface S encloses a connected set of spins (a “droplet”), and ⟨ x , y ⟩ ∈ S is the set of edges connecting spins in the interior of S to spins outside S . The inequality in Equation 2 is strict because, by the continuity of ν ( d J x y ) , there is zero probability of any closed surface having exactly zero energy in σ . The condition in Equation 2 must also hold for finite-volume ground states for any closed surface completely inside Λ L . It is then not hard to show that an alternative (and equivalent) definition, which we also use sometimes, is that an infinite-volume ground state is any convergent limit of an infinite sequence of finite-volume ground states. Given the spin-flip symmetry of the Hamiltonian, a ground state, whether of finite or infinite volume, generated by a spin-symmetric boundary condition, such as free or periodic, will appear as one part of a globally spin-reversed pair; we therefore refer generally to ground state pairs (GSPs) rather than individual ground states.

An interface between two infinite-volume spin configurations α and β comprises the set of edges whose associated couplings are satisfied in α and unsatisfied in β , or vice versa; they separate regions in which the spins in α agree with those in β from regions in which their spins disagree. An interface may consist of a single connected component or multiple disjoint ones, but (again using the continuity of the coupling distribution) if α and β are ground states, any such connected component must be infinite in extent.

Interfaces can be characterized by their geometry and energy. They can either be “space-filling,” meaning they comprise a positive density of all edges in E d , or zero-density, in which the dimensionality of the interface is strictly less than the dimension d . Ground states are called incongruent if they differ by a space-filling interface [28, 29].

Interfaces can also differ by how their energies scale with volume. The energy might diverge (though not monotonically) as one examines interfaces contained within increasingly larger volumes, or it might remain O ( 1 ) , independent of the volume considered. We will denote the former as a “high-energy interface” and the latter as a “low-energy interface.”

An excitation above the ground state is any spin configuration obtained by overturning one or more spins in the ground state (while leaving an infinite subset of spins in the original ground state intact); therefore, an interface is the boundary of an excitation. We are primarily interested in excitations consisting of overturning droplets of large, or possibly infinite, size; because an interface is the boundary of such an excitation, the energy of the excitation is simply twice the interface energy. An excitation above a ground state may itself be a new ground state (this would require the excitation to involve overturning an infinite number of spins such that Equation 2 remains satisfied). Indeed, as proven elsewhere [9], an excitation having a space-filling interface with the original ground state may generate a new ground state entirely.

With this in mind, we present the four low-temperature spin glass scenarios in Table 1, which illustrates their various relationships (and clarifies why we consider these four scenarios together).

As shown elsewhere [9], the existence of space-filling interfaces in the first row scenarios (RSB and chaotic pairs) implies the presence of multiple GSPs, whereas droplet-scaling and TNT both predict a single GSP [9, 25, 26, 28, 29].

Remarks on Table 1. The droplet-scaling scenario predicts a broad distribution of (free) energies for a minimal energy compact droplet of diameter O ( L ) , with a characteristic energy growing as L θ with θ > 0 in dimensions where a low-temperature spin glass phase is present. The distribution is such that there exist droplets of O ( 1 ) energy on large length scales, but these appear with a probability falling off as L − θ as L → ∞ . In contrast, both the RSB and TNT scenarios require droplets with O ( 1 ) energy to appear with positive probability bounded away from 0 on all length scales. Thus, the droplet-scaling scenario is shown in the second column of Table 1.

We now focus on the concepts of flexibility and critical droplets, which were introduced by Newman et al. [30, 31] and whose properties were described extensively in [24] (see also [26, 32]). Here, we only summarize their main features. We first provide some definitions (all with respect to some fixed coupling realization J ):

Definition 2.1

(Newman et al. [24]) Consider the GSP σ L for the EA Hamiltonian (Equation 1) on a finite-volume Λ L with boundary conditions chosen independently of J (for specificity, we always use periodic boundary conditions (PBCs) in this paper). Choose an edge b x y = ⟨ x , y ⟩ with x , y ∈ Λ L , and consider all closed surfaces in the dual-edge lattice E L * , which includes the dual edge b x y * . From Equation 2 and the continuity of the couplings, these all have distinct positive energies. There then exists a closed-surface ∂ D ( b x y , σ L ) , passing through b x y * , with least energy in σ L . We call ∂ D ( b x y , σ L ) the critical droplet boundary of b x y in σ L and the set of spins D ( b x y , σ L ) enclosed by ∂ D ( b x y , σ L ) the critical droplet of b x y in σ L .

Remarks. Critical droplets are defined with respect to edges rather than associated couplings to avoid confusion, given that we often vary the coupling value associated with specific edges, while the edges themselves are fixed, geometric objects.

We define the energy E D ( b x y , σ L ) of the critical droplet of b x y in σ L to be the energy of its boundary as given by Equation 2: E D b x y , σ L = ∑ < x , y > ∈ ∂ D b x y , σ L J x y σ x σ y . (3)

Definition 2.2

(Newman et al. [24]) The critical value of the coupling J x y associated with b x y in σ L is the value of J x y , where E D ( b x y , σ L ) = 0 , while all other couplings in J are held fixed.

We next define the flexibility f ( J x y , σ L ) :

Definition 2.3

(Newman et al. [24]) Let J x y be the value of the coupling assigned to the edge b x y in coupling realization J and J c ( b x y , σ L ) be the critical value of b x y in σ L . We define the flexibility f ( b x y , σ L ) of b x y in σ L to be f ( b x y , σ L ) = | J x y − J c ( b x y ) | .

Remarks. The critical value J c of an edge b x y with coupling value J x y is determined by all couplings in J , except J x y . Because couplings are chosen independently from ν ( d J x y ) , it follows that the value J x y is independent of J c . Therefore, given the continuity of ν ( d J x y ) , there is zero probability in a ground state that any coupling has exactly zero flexibility.

It follows from the definitions above and Equation 3 that f b x y , σ L = E D b x y , σ L .

Therefore, couplings which share the same critical droplet have the same (strictly positive) flexibility.

A rigorous definition of critical droplets and flexibilities within infinite-volume ground states requires use of the excitation metastate, whose definition and properties are presented in [26, 30, 31, 33]. Here, we simply note that finite-volume critical droplets and their associated flexibilities converge with their properties preserved in the infinite-volume limit, for reasons presented in [24]. This result would be trivial if all critical droplets in infinite-volume ground states were finite. However, it could also be that critical droplets can be infinite in extent in one or more directions, in which case metastates can be used to define such unbounded critical droplets which enclose an infinite subset of spins: they are the infinite-volume limits of critical droplets in finite-volume ground states.

In [24], critical droplets in infinite-volume ground states were classified according to the size of their boundary ∂ D ( b x y , σ ) , which is the relevant factor in associating the presence of a given type of critical droplet with one of the scenarios in Table 1. We simplify the nomenclature used in that paper by focusing on three different kinds of critical droplets. Let | ∂ D ( b x y , σ ) | denote the number of edges in the critical droplet boundary. A finite critical droplet is one in which | ∂ D ( b x y , σ ) | < ∞ ; in two and more dimensions, this implies that the critical droplet D ( b x y , σ ) itself consists of a finite set of spins and thus can be completely contained within some finite volume. (A 1 D chain is an exception: here, the critical droplet boundary of any edge consists of that edge alone, but the associated critical droplet consists of a semi-infinite chain of spins.) If these are the only type of critical droplets present, then the distribution of their sizes becomes important in answering fundamental questions involving edge disorder chaos and ground-state structure [26]. It is not hard to show that in any dimension, an EA ground state must contain at least a positive density of edges with finite critical droplets (whereas in 1 D , this is the case for all edges).

There are two kinds of critical droplets with | ∂ D ( b x y , σ ) | = ∞ . The first class includes those with infinite boundary ∂ D ( b x y , σ ) having a lower dimensionality than the space dimension d ; that is, the critical droplet boundary is infinite but zero-density in E d . We refer to these as zero - density\/ critical droplets (ZDCDs). (a finite critical droplet boundary also has zero density in E d , but we reserve the term “ZDCD” to apply only to critical droplets with an infinite boundary.)

Finally, there is the possibility that there exist infinite number of critical droplets whose boundary has dimension d , i.e., ∂ D ( b x y , σ ) comprises a positive density of edges in E d . We refer to these as space-filling critical droplets (SFCDs). These critical droplets have boundaries that pass within a distance O ( 1 ) of any site in Z d ; i.e., the closest distance from any site in Z d to ∂ D ( b x y , σ ) is essentially independent of the location of the site.

Because our ground states are chosen from the zero-temperature PBC metastate (denoted κ J ), we can adapt a result from [25, 34, 35], which is described below:

Theorem 3.1

Let σ denote an infinite-volume spin configuration. Then, for almost every ( J , σ ) pair at zero temperature (which restricts the set of σ s to ground states corresponding to particular coupling realizations J ), and for any type of critical droplet (finite, zero-density, or positive-density), either a positive density of edges in σ has a critical droplet of that type or else no edges do.

The method of proof of this theorem is essentially identical to that used in [25, 35] and so will be omitted here. The conclusion is that there is zero probability that a ground state σ chosen from κ J has a (finite or infinite) set of edges with zero density in E d and has SFCDs (or finite critical droplets or ZDCDs).

In [24], it was shown that there is a close connection between critical droplets and the four scenarios shown in Table 1. However, the results obtained were incomplete for the most prominent of the four scenarios, namely, replica symmetry breaking. In particular, it was proven there that the existence of SFCDs was a sufficient condition for some pairs of incongruent ground states to be separated by space-filling low-energy interfaces, hereafter referred to simply as “RSB interfaces” in accordance with Table 1. However, they were not shown to be necessary. This paper aimed to complete the correspondence between critical droplets and spin glass scenarios by demonstrating that the presence of SFCDs is not only sufficient but also a necessary condition for RSB interfaces to be present.

Replica symmetry breaking predicts that there exist space-filling, low-energy interfaces between ground states in three and higher dimensions. We have shown that this prediction is equivalent to the presence of SFCDs for a positive density of edges in E d in a typical ground state; that is, the presence of SFCDs is both a necessary and sufficient condition for the appearance of RSB interfaces. A stronger conclusion can be drawn in two dimensions, where ground state multiplicity relies on SFCDs: if they are absent, the zero-temperature PBC metastate κ J is supported on a single pair of spin-reversed ground states.

Where does this leave the other three scenarios appearing in Table 1? Like RSB, the chaotic pair scenario also predicts the appearance of multiple incongruent ground states separated by space-filling interfaces, but unlike RSB, the interface energy in chaotic pairs scales with L . To address this scenario, we require the following quantities, introduced in [24]. Let K * ( b , σ ) denote the number of edges in E d whose critical droplet boundaries in ground-state σ pass through the edge b . Then, for k = 1,2,3 , … , P ( k , σ ) is defined to be the fraction of edges b ∈ E d such that K * ( b , σ ) = k , and let E σ K * = ∑ k = 1 ∞ k P k , σ . (6) That is, E σ [ K * ] is the average number of edges whose critical droplet boundaries a typical edge belongs to in the GSP σ . Using results from this paper, Equation 6, and [24], we conclude that if SFCDs are absent and (a positive fraction of) ground states in κ J are characterized by E σ [ K * ] = ∞ , then the chaotic pair scenario should hold.

It follows that neither RSB nor chaotic pairs will hold if E σ [ K * ] < ∞ , which follows if P ( k , σ ) falls off faster than k − ( 2 + ε ) for any ε > 0 as k → ∞ . If this is the case, then κ J is supported on a single pair of spin-reversed ground states and either droplet-scaling or TNT should hold.