Phase transitions from saddles of the potential energy landscape

Models with symmetry-breaking phase transitions triggered by dumbbell-shaped equipotential surfaces

Recently, a number of sufficiency conditions have been shown for the occurrence of a Z_{2}-symmetry breaking phase transition (Z_{2}-SBPT) starting from geometric-topological concepts of potential energy landscapes. In particular, a Z_{2}-SBPT can be triggered by double-well potentials, or equivalently by dumbbell-shaped equipotential surfaces. In this paper, we introduce two models with a Z_{2}-SBPT that, due to their essential feature, show in the clearest way the generating mechanism of a Z_{2}-SBPT. Although they cannot be considered physical models, they all have the features of such models with the same kind of SBPT. At the end of the paper, the ϕ^{4} model is revisited in light of this approach. In particular, the landscape of one of the models introduced here turned out to be equivalent to that of the mean-field ϕ^{4} model in a simplified version.

Potential energy landscape of the two-dimensional XY model: higher-index stationary points

The application of numerical techniques to the study of energy landscapes of large systems relies on sufficient sampling of the stationary points. Since the number of stationary points is believed to grow exponentially with system size, we can only sample a small fraction. We investigate the interplay between this restricted sample size and the physical features of the potential energy landscape for the two-dimensional XY model in the absence of disorder with up to N = 100 spins. Using an eigenvector-following technique, we numerically compute stationary points with a given Hessian index I for all possible values of I. We investigate the number of stationary points, their energy and index distributions, and other related quantities, with particular focus on the scaling with N. The results are used to test a number of conjectures and approximate analytic results for the general properties of energy landscapes.

Energy-landscape analysis of the two-dimensional nearest-neighbor φ⁴ model

The stationary points of the potential energy function of the φ⁴ model on a two-dimensional square lattice with nearest-neighbor interactions are studied by means of two numerical methods: a numerical homotopy continuation method and a globally convergent Newton-Raphson method. We analyze the properties of the stationary points, in particular with respect to a number of quantities that have been conjectured to display signatures of the thermodynamic phase transition of the model. Although no such signatures are found for the nearest-neighbor φ⁴ model, our study illustrates the strengths and weaknesses of the numerical methods employed.

Computational topology for configuration spaces of hard disks

We explore the topology of configuration spaces of hard disks experimentally and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.

Phase transitions detached from stationary points of the energy landscape

The stationary points of the potential energy function V are studied for the ϕ4 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 ϕ4 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.

Topology, symmetry, phase transitions, and noncollinear spin structures

We use a topological approach to describe the frustration- and field-induced phase transitions exhibited by the infinite-range XY model on the AB2 chain, including noncollinear spin structures. For this purpose, we have computed the Morse number and the Euler characteristic, as well as other topological invariants, which are found to behave similarly as a function of the energy level in the context of Morse theory. In particular, we use a method based on an analogy with statistical mechanics to compute the Euler characteristic, which proves to be quite feasible. We also introduce topological energies which help us to clarify several properties of the transitions, both at zero and finite temperatures. In addition, we establish a nontrivial direct connection between the thermodynamics of the systems, which have been solved exactly under the saddle-point approach, and the topology of their configuration space. This connection allows us to identify the nondegeneracy condition under which the divergence of the density of Jacobian's critical points [jl(E)] at the critical energy of a topology-induced phase transition, proposed by Kastner and Schnetz [Phys. Rev. Lett. 100, 160601 (2008)] as a necessary criterion, is suppressed. Finally, our findings and those available in the literature suggest that the cusplike singularity exhibited both by the Euler characteristic and the topological contribution for the entropy at the critical energy, put together with the divergence of jl(E) , and emerge as necessary and sufficient conditions for the occurrence of the finite-temperature topology-induced phase transitions examined in this work. The general character of this proposal should be subject to a more rigorous scrutiny.

Microcanonical phase diagrams of short-range ferromagnets

A phase diagram is a graph in parameter space showing the phase boundaries of a many-particle system. Commonly, the control parameters are chosen to be those of the (generalized) canonical ensemble, such as temperature and magnetic field. However, depending on the physical situation of interest, the (generalized) microcanonical ensemble may be more appropriate, with the corresponding control parameters being energy and magnetization. We show that the phase diagram on this parameter space looks remarkably different from the canonical one. The general features of such a microcanonical phase diagram are investigated by studying two models of ferromagnets with short-range interactions. The physical consequences of the findings are discussed, including possible applications to nuclear fragmentation, adatoms on surfaces, and cold atoms in optical lattices.

Role of saddles in topologically driven phase transitions: the case of the d -dimensional spherical model

Analyzing the d -dimensional spherical model, we show that underlying saddles, defined through a map in the configuration space, play a central role in driving the phase transition. At the phase transition point the underlying saddle energy reaches its lowest value, corresponding to the trivial boundary topological singularity.

Phase transitions induced by saddle points of vanishing curvature

Based on the study of saddle points of the potential energy landscapes of generic classical many-particle systems, we present a necessary criterion for the occurrence of a thermodynamic phase transition. Remarkably, this criterion imposes conditions on microscopic properties, namely, curvatures at the saddle points of the potential, and links them to the macroscopic phenomenon of a phase transition. We apply our result to two exactly solvable models, corroborating that the criterion derived is not only valid, but also sharp and useful: For both models studied, the criterion excludes the occurrence of a phase transition for all values of the potential energy but the transition energy. This result adds a geometrical ingredient to an established topological condition for the occurrence of a phase transition, thereby providing an answer to the long-standing question of which topology changes in configuration space can induce a phase transition.