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.
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Notes
An example is provided by the Sherrington-Kirkpatrick (SK) Hamiltonian, which describes the mean-field Ising spin-glass, \( H_{SK}/N = a Q_{\alpha \beta }^2 + b Q_{\alpha \beta } Q_{\beta \gamma } Q_{\gamma \alpha } + c Q_{\alpha \beta }^4 + \cdots \), where repeated indices are summed from \(1\) to \(n\) [28].
This is because \(\Pi \vert \lambda _i - \lambda _j \vert \) does not scale as \(N\) when brought into the exponent of the Boltzmann factor. In the spherical case, the eigenvalues all sit at the same saddle location—even in this extreme case, the interactions are irrelevant in the thermodynamic limit [16].
That is, I require the variable graphs to satisfy the separability condition of [26]. However, the \(Q_i\) are free to take on common values at particular overlap locations.
A graph automorphism \(\pi \) is an invertible mapping of the vertex set onto itself that satisfies \(E_{\pi (v_i) \pi (v_j)} = E_{v_i v_j}\) for each pair of vertices \(v_i\) and \(v_j\). The automorphism group for a graph is the set of all automorphisms for that graph. It forms a group under composition.
If some \(H\) is encountered that does not already satisfy this condition, one can formally symmetrize it by writing \(H \rightarrow \frac{1}{2} ( H \vert _Q + H \vert _{Q^T})\). The only effect is to change the behavior at nonphysical, non-symmetric \(Q\).
Requiring RES removes from consideration all symmetric overlaps \(Q\) that have a centralizer ring—associated with \(\mathcal {A}(\mathcal {G}_Q)\)—containing non-symmetric matrices.
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Acknowledgments
I thank J. Rudnick, P. Pincus, D. Chandler, J. Bergknoff, and an anonymous referee for support and/or helpful comments and the USA NSF for support through grant Nos. DMR-1101900 and CHE-1265664.
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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
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.
For UM \(Q\), it is convenient to define
This notation allows one to express the Hamiltonian (1) in an integral form. For example, one can write
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.Footnote 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]
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.
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Landy, J. Eigenspace Stability Conditions in Mean-Field Replica Theories. J Stat Phys 159, 62–74 (2015). https://doi.org/10.1007/s10955-014-1172-y
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DOI: https://doi.org/10.1007/s10955-014-1172-y