title: 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.
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46565, USA.|Department of Chemistry, The University of Cambridge, Cambridge CB2 1EW, United Kingdom.
authors.forename: Dhagash
authors.initials: D
authors.lastname: Mehta
authors.affiliation: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46565, USA.|Simons Institute for the Theory of Computing, University of California, Berkeley, California 94720-2190, USA.
authors.forename: Jonathan D
authors.initials: JD
authors.lastname: Hauenstein
authors.affiliation: Simons Institute for the Theory of Computing, University of California, Berkeley, California 94720-2190, USA.|National Institute of Mathematical Sciences, Daejeon, Korea.
authors.forename: Matthew
authors.initials: M
authors.lastname: Niemerg
authors.affiliation: School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom.
authors.forename: Nicholas J
authors.initials: NJ
authors.lastname: Simm
authors.affiliation: Instituto de Física, Universidade Federal do Rio Grande do Sul and National Institute of Science and Technology for Complex Systems, CP 15051, 91501-970 Porto Alegre, RS, Brazil.
authors.forename: Daniel A
authors.initials: DA
authors.lastname: Stariolo