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.

Figure 1

Phase diagram of a simple ferromagnet. The bold line marks the parameter values in the $(T,h)$ plane at which the Gibbs free energy density $g(T,h)$ is nonanalytic.

Figure 2

Microcanonical phase diagram of the spherical model on a three-dimensional cubic lattice. The microcanonical entropy $s$ is defined only within the gray shaded region in the $(\epsilon ,m)$ plane. Within each of the three gray shaded subregions, the entropy is analytic, but not so on their boundaries which are given by Eqs. (5, 6).

Figure 3

(a) Microcanonical specific heat as a function of energy $\epsilon$ for fixed values of the magnetization $m=m_0$. From top to bottom: $m_0=0$, 0.05, 0.1, 0.125, 0.15. The peaks reveal the presence of a phase transition line at relatively high energies. The data shown have been obtained for systems composed of $100\times 100$ spins. (b) Resulting microcanonical phase diagram of the two-dimensional Ising model. The dashed line is the line of spontaneous magnetization separating the ferromagnetic phase at low energies from the paramagnetic phase. The solid line is new and should be compared with the corresponding line in the microcanonical phase diagram of the spherical model in Fig. 2.