Abstract
We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature mu . We are interested in the mean spectral density rho (lambda ) of the Hessian matrix {{mathcal {K}}} at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for rho (lambda ) for a fixed L^d in the N rightarrow infty limit extending d=0 results of our previous work (Fyodorov et al. in Ann Phys 397:1–64, 2018). A particular attention is devoted to analyzing the limit of extended lattice systems by letting Lrightarrow infty . In all cases we show that for a confinement curvature mu exceeding a critical value mu _c, the so-called “Larkin mass”, the system is replica-symmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at mu rightarrow mu _c. For mu <mu _c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (mu _c-mu )^4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called “marginal cases” in d=1,2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning.
Highlights
1.1 The Random Manifold Model and Some Known ResultsNumerous physical systems can be modeled by a collection of points or particles coupled by an elastic energy, usually called an elastic manifold, submitted to a random potential
To d = 0 case treated in detail in [11], it is easy to check that the invariance of the action under rotating matrices P and τ in the replica space implies that the corresponding saddle point solutions must be proportional to the identity matrix: Pαβ = pδαβ and ταβ = τ δαβ
In this paper we have extended our previous work on the spectrum of the Hessian matrix at the global minimum of a high dimensional random potential, to the case of many points coupled by an elastic matrix
Summary
Numerous physical systems can be modeled by a collection of points or particles coupled by an elastic energy, usually called an elastic manifold, submitted to a random potential (see [1,2,3] for reviews). They are often called “disordered elastic systems” and generically exhibit pinning in their statics and depinning transitions and avalanches in their driven dynamics [4,5,6,7,8,9].
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