Phase transitions detached from stationary points of the energy landscape

Abstract

The stationary points of the potential energy function V are studied for the ϕ^4 model on a two-dimensional square lattice with nearest-neighbor interactions. On the basis of analytical and numerical results, we explore the relation of stationary points to the occurrence of thermodynamic phase transitions. We find that the phase transition potential energy of the ϕ^4 model does in general not coincide with the potential energy of any of the stationary points of V. This disproves earlier, allegedly rigorous, claims in the literature on necessary conditions for the existence of phase transitions. Moreover, we find evidence that the indices of stationary points scale extensively with the system size, and therefore the index density can be used to characterize features of the energy landscape in the infinite-system limit. We conclude that the finite-system stationary points provide one possible mechanism of how a phase transition can arise, but not the only one.

5 pages, 3 figures

References in corpus (6)

• Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method
• Phase transitions induced by saddle points of vanishing curvature
• Nonanalyticities of the entropy induced by saddle points of the potential energy landscape
• Topological approach to phase transitions and inequivalence of statistical ensembles
• Stationary point approach to the phase transition of the classical XY chain with power-law interactions
• On a microcanonical relation between continuous and discrete spin models

Cited by in corpus (42)

• On the completeness of solutions of Bethe's equations
• Persistent Homology analysis of Phase Transitions
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Finding all flux vacua in an explicit example
• Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Enumerating Gribov copies on the lattice
• Numerical Polynomial Homotopy Continuation Method and String Vacua
• Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model
• Exploring the energy landscape of XY models
• Communication: Newton homotopies for sampling stationary points of potential energy landscapes
• A Collection of Challenging Optimization Problems in Science, Engineering and Economics
• The Loss Surface of XOR Artificial Neural Networks
• Topological Approach to Microcanonical Thermodynamics and Phase Transition of Interacting Classical Spins
• Numerical Analyses on Moduli Space of Vacua
• Certifying the Potential Energy Landscape
• Topological Theory of Phase Transitions
• Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions
• Potential Energy Landscapes for the 2D XY Model: Minima, Transition States and Pathways
• The Geometric Theory of Phase Transitions
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization
• An Inversion-Relaxation Approach for Sampling Stationary Points of Spin Model Hamiltonians
• Certification and the Potential Energy Landscape
• Topology of configuration space of the mean-field phi^4 model by Morse theory
• Statistics of Stationary Points of Random Finite Polynomial Potentials
• Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces
• Configuration spaces of hard spheres
• The topological hypothesis for discrete spin models
• On the apparent failure of the topological theory of phase transitions
• Gauge-fixing on the Lattice via Orbifolding
• Experiments on the zeros of harmonic polynomials using certified counting
• Quotient Maps and Configuration Spaces of Hard Disks
• An index-resolved fixed-point homotopy and potential energy landscapes
• Certified counting of roots of random univariate polynomials
• Numerical optimization using flow equations
• Exact Topology of Dynamic Probability Surface of an Activated Process by Persistent Homology
• Toward a refining of the topological theory of phase transitions
• A geometric conjecture about phase transitions
• Simplified energy landscape of the ${\varphi }^4$ model and the phase transition
• On the origin of Phase Transitions in the absence of Symmetry-Breaking
• Exploring the Potential Energy Landscape Over a Large Parameter-Space