Kastner (2009) computed the microcanonical entropy s(ε,m)s(\varepsilon,m) of the dd-dimensional spherical model with nearest-neighbour ferromagnetic interactions exactly in the thermodynamic limit, finding a concave entropy — and hence equivalence of the microcanonical and canonical ensembles despite the effectively long-range spherical constraint — together with two transition lines that, at fixed magnetization mm, separate a paramagnetic phase from ferromagnetic and antiferromagnetic ones. The paper left open three directions: whether this two-transition structure is generic for short-range ferromagnets, how the strict fixed-energy/fixed-magnetization constraints could be realized experimentally or numerically, and its stated expectation that only properly long-range interactions would make the entropy develop nonconcave regions and ensemble nonequivalence. This problem asks for the microcanonical entropy of the spherical model with genuinely long-range (e.g. power-law) but non-mean-field couplings, interpolating between the solved nearest-neighbour case (concave, ensembles equivalent) and the solved mean-field case (partial nonequivalence); for the boundary and transitional scenarios at which nonconcavity first appears; for a determination of whether the two-transition-per-fixed-mm structure is generic for short-range ferromagnets; and, as an auxiliary component flagged by the source, for concrete protocols realizing fixed-(ε,m)(\varepsilon,m) constraints to probe the predicted transitions.

Background

In the microcanonical ensemble an isolated many-body system is described by the entropy density s(ε,m)s(\varepsilon,m), the logarithm of the phase-space volume of states with energy ε\varepsilon and magnetization mm per spin. The canonical ensemble instead describes a system coupled to a heat bath. The two descriptions agree in the thermodynamic limit for suitable short-range interactions, but for long-range interactions they can disagree (ensemble nonequivalence): by the large-deviation results of Ellis, Haven and Turkington, microcanonical–canonical equivalence holds precisely where ss is a concave function of its arguments, and nonconcave regions of ss correspond to negative specific heat and to energy ranges the canonical ensemble cannot reach. The spherical model of Berlin and Kac is a classical lattice spin model whose spins are continuous variables subject to a global constraint ∑isi2=N\sum_i s_i^2=N, which is effectively long-range; the model is exactly solvable while showing a ferromagnetic phase transition, which makes it a natural laboratory for ensemble-equivalence questions. Behringer (2005) computed part of the microcanonical entropy of the nearest-neighbour spherical model, but only outside the phase-coexistence region where nonequivalence could occur. Kastner (2009) completed the calculation for all accessible (ε,m)(\varepsilon,m): the entropy is concave (albeit not strictly concave), so the ensembles are equivalent — provided the canonical ensemble is allowed negative temperatures, which generate the antiferromagnetic branch — and the microcanonical phase diagram in the (ε,m)(\varepsilon,m) plane shows two transition lines at fixed magnetization, ε+(m)\varepsilon_+(m) (paramagnet–ferromagnet) and ε−(m)\varepsilon_-(m) (antiferromagnet–paramagnet). The companion work of Kastner and Pleimling (2009) showed that the Ising model has a microcanonical phase diagram of the same qualitative shape, and proposed physical settings with fixed energy and fixed density (nuclear fragmentation, adatoms on surfaces, cold atoms in optical lattices after switching off cooling) where such diagrams could matter. Two interaction-range endpoints of the spherical model are already solved microcanonically: the nearest-neighbour case (concave, equivalent; Kastner 2009) and the mean-field case, where Kastner and Schnetz (2006) found partial ensemble nonequivalence in the thermodynamic limit. On the canonical side, Joyce (1966) solved the spherical model with ferromagnetic power-law couplings J(r)∝r−(d+σ)J(r)\propto r^{-(d+\sigma)} for all σ\sigma, providing the natural interpolation family between the two endpoints. What is missing is the microcanonical analysis at intermediate range, the location of the onset of nonconcavity as the range is increased, and any general statement of whether the two-transition structure is generic for short-range ferromagnets.

Problem Statement

For the dd-dimensional spherical model with ferromagnetic interactions of tunable range — specifically Joyce-type power-law couplings J(r)∝r−(d+σ)J(r)\propto r^{-(d+\sigma)}, with σ\sigma the tunable range parameter, which interpolate between the mean-field limit σ→0\sigma\to 0 and the nearest-neighbour limit σ→∞\sigma\to\infty — compute the microcanonical entropy s(ε,m)s(\varepsilon,m) in the thermodynamic limit over the full accessible (ε,m)(\varepsilon,m) domain, and determine the conditions on interaction range and strength under which ss develops nonconcave regions, with the associated partial or complete nonequivalence of the microcanonical and canonical ensembles. In particular, characterize the boundary between the concave, ensemble-equivalent nearest-neighbour regime (Kastner 2009) and the partially nonequivalent mean-field regime (Kastner and Schnetz 2006), including the manner (e.g. continuous growth of a nonconcave region versus first appearance at a threshold range) in which nonconcavity sets in, and the transitional scenarios at the boundary. As a linked component, establish whether the two-transition-per-fixed-mm structure of the microcanonical phase diagram found for the nearest-neighbour spherical model and the Ising model is generic for short-range ferromagnets, identifying the model conditions under which it holds or fails. As an auxiliary component flagged by the source, identify concrete experimental or simulational protocols that realize the strict fixed-energy, fixed-magnetization constraints well enough to probe the predicted two transitions and coexistence phenomena. A complete resolution requires both (i) the entropy and onset analysis — the thermodynamic-limit s(ε,m)s(\varepsilon,m) of this tunable-range spherical model, the conditions on interaction range and strength for nonconcavity with the associated ensemble nonequivalence, and the characterization of the onset boundary and transitional scenarios — and (ii) the genericity statement about the two-transition-per-fixed-mm structure for short-range ferromagnets; resolving only one of the two is a partial resolution. The protocols component is auxiliary and is not required for a complete resolution.

References

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