Phase Transitions from Saddles of the Potential Energy Landscape

Abstract

The relation between saddle points of the potential of a classical many-particle system and the analyticity properties of its thermodynamic functions is studied. For finite systems, each saddle point is found to cause a nonanalyticity in the Boltzmann entropy, and the functional form of this nonanalytic term is derived. For large systems, the order of the nonanalytic term increases unboundedly, leading to an increasing differentiability of the entropy. Analyzing the contribution of the saddle points to the density of states in the thermodynamic limit, our results provide an explanation of how, and under which circumstances, saddle points of the potential energy landscape may (or may not) be at the origin of a phase transition in the thermodynamic limit. As an application, the puzzling observations by Risau-Gusman et al. on topological signatures of the spherical model are elucidated. Phase transitions, like the boiling and evaporating of water at a certain temperature and pressure, are common phenomena both in everyday life and in almost any branch of physics. Loosely speaking, a phase transition brings about a sudden change of the macroscopic properties of a many-particle system while smoothly varying a parameter (the temperature or the pressure in the above example). The mathematical description of phase transitions is conventionally based on (grand)canonical thermodynamic functions, relating their loss of analyticity to the occurrence of a phase transition. Such a nonanalytic behavior in a (grand)canonical thermodynamic function can occur only in the thermodynamic limit in which the number of degrees of freedom N of the system goes to infinity [1].Many researchers took it for granted that the same were true also for microcanonical thermodynamic functions. Recently, however, it was observed that the microcanonical entropy, or Boltzmann entropy, of a finite system is not necessarily analytic, and nonanalytic entropy functions of finite systems have been reported for certain classical models [2,3,4].In light of their conceptual similarity to the definition of a phase transition, it is tempting to regard finitesystem nonanalyticities of the entropy as phase transitionlike phenomena. This point of view is advocated in [4], and the authors of that reference argue that such nonanalyticities should also be measurable experimentally, at least in very small systems. However, such an interpretation is complicated by the fact that, as discussed in [3], for typical models the number of nonanalytic points of the entropy increases unboundedly with the number of degrees of freedom N .Because of their typically large number, one might assume that nonanalyticities of the finite-system entropy were unrelated to the occurrence of a phase transition in the thermodynamic limit. A theorem by Franzosi and Pettini [5], however, indicates the existence of a relation between finite-system and infinite-system nonanalyticities (and we will come back to that theorem later). The purpose of the present Letter is to clarify and quantify the ...

Quantum quench influenced by an excited-state phase transition

“…III B that the ground-state QPTs are located at the critical points λ (n) c0 from Eqs. (25) and (27). It turns out that for λ > λ (n) c0 the singularity propagates into the excited spectrum.…”

“…Such nonanalyticities most commonly follow from the presence of the Hamiltonian stationary points [40,41]. The relation between such points and thermodynamic phase transitions was recently investigated [27] for systems with asymptotically increasing numbers of degrees of freedom. The situation is similar also in the present type of system, described by finite algebraic models, in which the number of degrees of freedom is fixed (independent of the increasing size parameter).…”

“…At the critical point (27), the determinant of the Hessian matrix (composed from second derivatives of V (3) with respect to both variables) evaluated at the minimum becomes negative, which means that (x,y) = (0,0) becomes a saddle point of the potential. Starting at this point, two degenerate minima deviate symmetrically to the quadrants with xy < 0 (for λ > 0).…”

“…The ESQPTs can be viewed as close analogs of thermal phase transitions formulated in the microcanonical language [27]. An important step is the scaling of energy and other observables by a suitably defined size parameter ℵ.…”

Phase Transitions Detached from Stationary Points of the Energy Landscape

“…At a stationary point, we have ∇V = 0, the integrand on the righthand side of (1) diverges, and we may expect the stationary point to give an important contribution to the integral. Indeed, it has been shown that, for finite N , every stationary point q s of V induces nonanalytic behavior in s N (v) precisely at the potential energy of the stationary point, v = V (q s )/N [5]. Nonanalyticities of thermodynamic functions are hallmarks of phase transitions.…”