Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization

Abstract

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite size $2$-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

9 pages, 17 figures

References in corpus (7)

• Phase transitions and configuration space topology
• Topological and Dynamical Complexity of Random Neural Networks
• Enumerating Gribov copies on the lattice
• Exploring the energy landscape of XY models
• On a microcanonical relation between continuous and discrete spin models
• Gauge-fixing on the Lattice via Orbifolding
• Experiments on the zeros of harmonic polynomials using certified counting

Cited by in corpus (2)

• Topology trivialization transition in random non-gradient autonomous ODE's on a sphere
• Replica-symmetry breaking transitions in the large deviations of the ground-state of a spherical spin-glass