Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

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 µ. We are interested in the mean spectral density ρ(λ) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for ρ(λ) for a fixed L d in the N → ∞ limit extending d = 0 results of our previous work [11]. A particular attention is devoted to analyzing the limit of extended lattice systems by letting L → ∞. In all cases we show that for a confinement curvature µ exceeding a critical value µ c , the so-called "Larkin mass", the system is replicasymmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at µ → µ c . For µ < µ 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 (µ c − µ) 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. arXiv:1903.07159v1 [cond-mat.dis-nn]

Low-frequency vibrational spectrum of mean-field disordered systems

“…The vibrational spectrum of structural glasses displays a series of universal features in different frequency ranges, which are responsible for important material properties, such as wave attenuation, heat transport, and plasticity [1][2][3]. Motivated by these observations, several authors have constructed simple models of the nonphononic vibrational density of states of structurally disordered systems, D(ω) [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Meanfield models typically display a quadratic spectrum, D(ω) ∼ ω 2 [10,12], of delocalized and featureless modes [21]; this delocalization is inherently different from the one associated with phononic excitations in solids (which are absent in the mean-field limit), and it is a manifestation of the marginal stability associated to replica symmetry breaking [22,23].…”

Landscape complexity beyond invariance and the elastic manifold

“…Remark Here we take Δ to be the lattice Laplacian, which is the classic choice in the elastic manifold, but as suggested in [32] one would also want to replace Δ everywhere with another $$L^d \times L^d$$ matrix M . For example, this would allow for interactions beyond pairwise.…”

“…[38] and [39] for a review of disordered elastic media in general and to ref. [32] for a review of this specific Hamiltonian, which we summarize briefly here.…”

Hessian Eigenspectra of More Realistic Nonlinear Models

“…Hessian eigenspectra: a statistical physics perspective. Statistical physicists are interested in the energy "landscape" of random functions, by means of studying the Hessian of such functions [35][36][37][38]98]. This is somewhat different from the Hessian of the loss function in optimization, and it is mainly in the context of spin glass theory.…”