Eigenspace Stability Conditions in Mean-Field Replica Theories

Abstract

I consider the eigenspace form of the saddle point conditions that apply for mean-field replica models that are characterized by a single overlap matrix order parameter. These allow for a characterization of the full set of saddle point solutions. Further, they motivate consideration of a particular class of solutions, which I call the principally-commuting set. These are the solutions that simultaneously satisfy the replica-space rotational stability constraints for all physical models. I show that this set satisfies the replica equivalence symmetry, as well as other more stringent conditions, and that it also contains the set of Parisi-type ultrametric solutions as a subset.

Appendices

Appendix 1: Parisi’s ultrametric (UM) overlaps

The Parisi-type UM matrices are hierarchical in structure—an example is shown in Fig. 4. In the notation of the main text, they are characterized by principal matrices that take the form

$$ \begin{aligned} P^{(i)}_{\alpha \beta } = {\left\{ \begin{array}{ll} 1, &{}\quad \text {if } \lfloor \frac{ \alpha }{m_i} \rfloor \not = \lfloor \frac{\beta }{m_i}\rfloor \& \lfloor \frac{\alpha }{m_{i+1}}\rfloor = \lfloor \frac{\beta }{m_{i+1}}\rfloor \\ 0, &{}\quad \text {otherwise,} \end{array}\right. } \end{aligned}$$

(26)

where the $\{m_i\}$ are some set of specified integers ($m_i \equiv \sum _{j<i} n_{Q_j}$, with $m_1 \equiv 1$ and $m_{\vert \mathcal {C}_Q \vert +1} \equiv n$ in the notation of this text) [20]. It is a simple matter to prove that the UM matrices are RES.

Fig. 4

A $\vert \mathcal {C}_Q \vert =3$ UM overlap with block sizes $m_1 = 1$, $m_2=2$, $m_3 = 4$, and $m_4 \equiv n = 12$. These correspond to $n_{Q_i}$ values ($Q_i$ per row) of $n_{Q_1} = 1$, $n_{Q_2}= 2$, and $n_{Q_3} = 8$

For UM $Q$, it is convenient to define

$$\begin{aligned} Q(x) \equiv Q_i, \text { for } m_i < x < m_i+1. \end{aligned}$$

(27)

This notation allows one to express the Hamiltonian (1) in an integral form. For example, one can write

$$\begin{aligned} \frac{1}{n} \sum _{\alpha \not = \beta } f(Q_{\alpha \beta }) = \sum _{i>0} n_{Q_i} f(Q_i) = \int _1^n dx f(Q(x)). \end{aligned}$$

(28)

Varying the integral form of (1) with respect to $Q(x)$ then allows one to easily seek the optimal $Q$ within the full UM subspace.

Appendix 2: Lautrup’s Simply Extensible (SE) Overlaps

In [17], Lautrup considered many of the issues that I discuss here, but from a slightly different perspective. His results are highly complementary to mine, and I review some of them here: Lautrup began by arguing that, because the overlap matrices must by symmetric, the symmetry group $\mathcal {A}(\mathcal {G}_Q)$ should enforce transposition symmetry. The necessary and sufficient condition for the full centralizer ring of $\mathcal {A}(\mathcal {G}_Q)$ (i.e., the set of overlaps invariant under $\mathcal {A}(\mathcal {G}_Q)$; these form a ring under matrix multiplication and addition) to be symmetric is the RES condition.⁷ Once the RES condition was imposed, Lautrup demonstrated that the principal matrices $P^{(i)}$ ($T_i$ in his notation) form a closed algebra, satisfying (24). From this condition, he derived various eigenvalue identities for $Q$, some of which I also present in Sect. 4, albeit, starting from weaker assumptions.

Writing $m_i= \sum _{j<i} n_{Q_j}$, with $m_1 \equiv 1$ and $m_{\vert \mathcal {C}_Q \vert +1} \equiv n$, Lautrup next noted that the Parisi $Q(x)$ integral notation (cf. Appendix 1) can be applied to any RT overlap. In order to allow for variability of $n$, he argued that it should be possible to extend $Q(x)$ to an $n^{\prime } > n$ without altering its value on $1<x<n$. A natural way to incorporate this condition is to require the principal matrix algebra to be simply extensible (SE), which Lautrup defines as one for which the subsets $\{P^{(1)}, P^{(2)}, \ldots P^{(k)} \}$ individually form a closed algebra for each $k \le \vert \mathcal {C}_{Q} \vert$. This allows one to tack on additional principal matrices to the algebra without altering the prior structure. For SE graphs, the algebra simplifies to [17]

$$\begin{aligned} P^{(i)} P^{(j)}&= n_{Q_i} P^{(j)}, \text { for } i<j \nonumber \\ P^{(i)}P^{(i)}&= (n_{Q_i} - m_i) P^{(i)} + n_{Q_i} \sum _{j<i} P^{(j)}. \end{aligned}$$

(29)

Using (29) and the fact that the $P^{(i)}$ are 0–1 matrices, Lautrup then showed that the $V^{(i)} \equiv \sum _{j<i} P^{(j)}$ must be block-diagonal, for some suitable labeling of the replica indices. In other words, the SE symmetry implies the Parisi ansatz.