Block diagonalizing ultrametric matrices

Abstract

The problem of diagonalizing a class of complicated matrices, to be called ultrametric matrices, is investigated. These matrices appear at various stages in the description of disordered systems with many equilibrium phases by the technique of replica symmetry breaking. The residual symmetry, remaining after the breaking of permutation symmetry between replicas, allows us to bring all ultrametric matrices to a block diagonal form by a common similarity transformation. A large number of these blocks are, in fact, of size 1 × 1, i.e. in a vast sector the transformation actually diagonalizes the matrix. In the other sectors we end up with blocks of size (R + 1) × (R + 1) where R is the number of replica symmetry breaking steps. These blocks cannot be further reduced without giving more information, in addition to ultrametric symmetry, about the matrix. Similar results for the inverse of a generic ultrametric matrix are also derived. PACS classification numbers: 75.10.Nr, 05.50.+q. * E-mail:temtam@hal9000.elte.hu IntroductionLow temperature disordered systems often possess many equilibrium phases. The technique of replica symmetry breaking (RSB) provides a theoretical framework in which these systems can be described analytically , starting from a microscopic basis. Discovered and developed in the theory of spin glasses [1], RSB has recently penetrated into a number of other problems, including the theory of random manifolds [2][3][4], random field problems [5,6], protein folding [7][8][9], vortex pinning [10], etc. In each of these theories randomness is handled via the replica trick, and the multitude of equilibrium phases is captured by breaking the permutation symmetry between the replicas. As always, symmetry breaking means that the low temperature solutions realize a particular subgroup of the underlying symmetry group of the theory, here of the permutation group. The proper choice of the subgroup proved to be a highly nontrivial task in the case of RSB. The succesful Ansatz for the symmetry breaking pattern, proposed by Parisi originally in the context of spin glasses, turned out to embody a particular, hierarchical organization of the equilibrium phases, usually referred to as ultrametricity [1].The corresponding subgroup determines the structure not only of the order parameter, but also of all other quantities in the theory, like self-energies, propagators, etc. The structure imposed by this residual symmetry on quantities depending on two replica indices is by now widely known. The algebra of such quantities has been worked out by Parisi [11] with further results, most notably on the inversion problem, added by Mézard and Parisi [2]. At a certain stage of the development of RSB theories, however, one has to face also more complicated objects, depending on three or four replica indices. The structure of these is much harder to grasp and their algebra is much more involved than that of the two-index quantities. Our purpose here is to analyse and exploit the structure imposed by ultametricity on fou...

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