The Marginally Stable Bethe Lattice Spin Glass Revisited

Abstract

Bethe lattice spins glasses are supposed to be marginally stable, i.e. their equilibrium probability distribution changes discontinuously when we add an external perturbation. So far the problem of a spin glass on a Bethe lattice has been studied only using an approximation where marginal stability is not present, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a marginally stable solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non-perturbative approach to the Bethe lattice spin glass problem using approximations that should be hopefully consistent with marginal stability.

Appendix: The Cavity Equations in the Case of Broken Replica Symmetry

The cavity equations are often a direct way to find the maximum of the free energy.

At this end it is convenient to introduce the cavity equations for the probability over the descriptors i.e. $\mathcal{P}_{}[\mathcal{D}]$. These equations can be derived heuristically by considering a N nodes system to which we add an extra node. A natural requirement is that the properties of the spin the extra node is the same of the others N spins. Alternatively, we could remove one spin from a $N+1$ system forming a cavity, hence the name “cavity” method.

It has been proved [36] that a stochastically stable probability $\mathcal{P}_{}[\mathcal{D}]$ that maximizes the free energy, should also satisfy of some equations that generalize the naive cavity equations (3). They are defined as follows. For each state $\alpha$ we construct a new set of fields and weights using the fields $vh_\alpha$ that are uncorrelated in i and correlated in $\alpha$.

$$\begin{aligned} h'_\alpha =U(\vec {J},\vec {h}_\alpha ) \ \ \ w'_\alpha ={w_\alpha \Delta ^{node}_K(\vec {J},\vec {h}_\alpha )\over Z}\, , \end{aligned}$$

(72)

Z being a normalisation factor that enforces $\sum w'_\alpha =1$. The quantities $h'_\alpha$ and $w'_\alpha$ are the fields and weights corresponding to new spin. These equations can be formally written as

$$\begin{aligned} \mathcal{P}'= \mathcal{C}[\mathcal{P}]. \end{aligned}$$

(73)

According to the heuristic argument we finally impose that the probability distribution of the $\{w'_\alpha ,h'_\alpha \}$ (i.e. $\mathcal{P}'$ is again $\mathcal{P}[\mathcal{D}]$. This constraint is called the cavity equations for $\mathcal{P}_{}[\mathcal{D}]$ [40, 42, 43]: the cavity equations ($\mathcal{P}'= \mathcal{C}[\mathcal{P}]$) are valid in general for stochastically stable distributions, not only in the reproducible case.

The cavity equations can be considered the moral equivalent of

$$\begin{aligned} {\delta \mathcal{F}[\mathcal{P}_{}[\mathcal{D}]]\over \delta \mathcal{P}_{}[\mathcal{D}]}=0 \, . \end{aligned}$$

(74)

These equations are a necessary condition but not a sufficient condition. The cavity equations have many solutions; we need to find out the one that maximizes the free energy $\mathcal{F}[\mathcal{P}_{}[\mathcal{D}]]$.

In the heuristic approach one often requires a further condition of stability. It can formally written as follows. Let us consider a solution ($\mathcal{P}^*[\mathcal{D}]])$ of the cavity equation (74). For small $\delta \mathcal{P}[\mathcal{D}]$ we can write a Taylor expansion

$$\begin{aligned} \mathcal{F}[\mathcal{P}^*[\mathcal{D}]+\delta \mathcal{P}[\mathcal{D}]]=\mathcal{F}[\mathcal{P}^*[\mathcal{D}]]+(\delta \mathcal{P}[\mathcal{D}], \mathbf{\large H}\, \delta \mathcal{P}[\mathcal{D}]) +O(\delta \mathcal{P}[\mathcal{D}]^3), \end{aligned}$$

(75)

where the linear term in $\delta \mathcal{P}[\mathcal{D}]$ is absent as effect of the cavity equation (74).

This equation defines a generalized Hessian (i.e. $\mathbf{\large H}$) that plays a crucial role in the heuristic approach.

Let us call ${\large H}_0$ the largest eigenvalues of $\mathbf{H}$ (the so called replicon eigenvalue). It natural to assume that if $\mathcal{P}^*[\mathcal{D}]$ maximize the free energy only if

$$\begin{aligned} {\large H}_0\le 0, \end{aligned}$$

(76)

or equivalently the matrix $\mathbf{H}$ is non-positive, This condition generalizes the De Almeida–Thouless (DAT) stability condition. It is interesting to note that the DAT stability condition is usually equivalent to the condition:

$$\begin{aligned} \lambda \equiv \lim _{l\rightarrow \infty }\frac{1}{l}\log \left( {\overline{(\langle \sigma _0 \sigma _l\rangle _c)^2 }\over z^l}\right) \le 0, \end{aligned}$$

(77)

where $\langle \sigma _0 \sigma _l\rangle _c$ is the connected correlation function of two spins at distance l on the infinite Bethe Lattice [74].

It is also conjectured that this stability condition in saturated when the replica symmetry is spontaneously broken in a continuous way:

$$\begin{aligned} {\large H}_0=0 \ \ \ \text{ and } \ \ \ \ \lambda =0. \end{aligned}$$

(78)

Indeed in all known cases, the saturation of DAT stability condition is the necessary condition for marginal stability.

It is conjectured that for spin glasses on the Bethe lattice when replica symmetry is spontaneously broken at r steps we have

$$\begin{aligned} {\large H}_0>0. \end{aligned}$$

(79)

This conjecture implies that the stability condition forces us to consider continuous RSB. This is the reason we are interested in finding approximations to spin glasses such that Eq. (78) is satisfied, maybe in some restricted space.