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arXiv:1508.03543

Recovering the equivalence of ensembles II: An Ising chain with competing short and long-range interactions

Vera B. Henriques, Silvio R. A. Salinas

Published Aug 14, 2015• 0 citations
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Abstract

In a pioneer work, John Nagle has shown that an Ising chain with competing short and long-range interactions displays second and first-order phase transitions separated by a tricritical point. More recently, it has been claimed that Nagle's model provides an example of the inequivalence between canonical and microcanonical calculations. We then revisit Nagle's original solution, as well as the usual formulation of the problem in a canonical ensemble, which lead to the same results. Also, in contrast to recent claims, we show that an alternative formulation in the microcanonical ensemble, with the adequate choice of the fixed thermodynamic extensive variables, leads to equivalent thermodynamic results.

DOI
10.48550/arxiv.1508.03543

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In a pioneer work, John Nagle has shown that an Ising chain with competing short and long-range interactions displays second and first-order phase transitions separated by a tricritical point. More recently, it has been claimed that Nagle's model provides an example of the inequivalence between canonical and microcanonical calculations. We then revisit Nagle's original solution, as well as the usual formulation of the problem in a canonical ensemble, which lead to the same results. Also, in contrast to recent claims, we show that an alternative formulation in the microcanonical ensemble, with the adequate choice of the fixed thermodynamic extensive variables, leads to equivalent thermodynamic results.
Abstract We study the effect of competing interactions on ensemble inequivalence. We consider a one-dimensional Ising model with ferromagnetic mean-field interactions and short-range nearest-neighbor (NN) and next-NN couplings which can be either ferromagnetic or antiferromagnetic. Despite the relative simplicity of the model, our calculations in the microcanonical ensemble reveal a rich phase diagram. The comparison with the corresponding phase diagram in the canonical ensemble shows the presence of phase transition points and lines which are different in the two ensembles. As an example, in a region of the phase diagram where the canonical ensemble shows a critical point and a critical end point, the microcanonical ensemble has an additional critical point and also a triple point. The regions of ensemble inequivalence typically occur at lower temperatures and at larger absolute values of the competing couplings. The presence of two free parameters in the model allows us to obtain a fourth-order critical point, which can be fully characterized by deriving its Landau normal form.
Abstract In this paper we study an Ising spin chain with short-range competing interactions in the presence of long-range ferromagnetic interactions in the canonical ensemble. The simultaneous presence of the frustration induced by the short-range couplings together with their competition with the long-range term gives rise to a rich thermodynamic-phase diagram. We compare our results with the limit in which one of two local interactions is turned off, which was previously studied in the literature. Eight regions of parameters with qualitatively distinct properties are featured, with different first- and second-order phase transition lines and critical points.
We study the problem of ensemble equivalence in spin systems with short-range interactions under the existence of a first-order phase transition. The spherical model with nonlinear nearest-neighbour interactions is solved exactly for both canonical and microcanonical ensembles. The result reveals apparent ensemble inequivalence at the first-order transition point in the sense that the microcanonical entropy is non-concave as a function of the energy and consequently the specific heat is negative. In order to resolve the paradox, we show that an unconventional saddle point should be chosen in the microcanonical calculation that represents a phase separation. The XY model with nonlinear interactions is also studied by microcanonical Monte Carlo simulations in two dimensions to see how this model behaves in comparison with the spherical model.
Abstract We explore the tricritical points and the critical lines of both Blume–Emery–Grifnths and Ising model within long-range interactions in the microcanonical ensemble. For K = K MTP , the tricritical exponents take the values β = 1/4, 1 = γ − ≠ γ + = 1/2 and 0 = α − ≠ α + = −1/2, which disagree with classical (mean held) values. When K > K MTP , the phase transition becomes second order and the critical exponents have classical values except close to the canonical tricritical parameters (K CTP ), where the values of the critical expoents become β = 1/2, 1 = γ − ≠ γ + = 2 and 0 = α − ≠ α + = 1.
We introduce two variants of the Blume--Emery--Griffiths model with long-range, mean-field-like interactions and quenched disorder. Although the two variants are described by different Hamiltonians, they yield the same free energy. Moreover, both Hamiltonians lead to the same average energy and hence to the same microcanonical entropy. The canonical and microcanonical phase boundaries differ markedly, and so does the multicritical behaviour of the two ensembles. In the microcanonical ensemble, tricritical points appear for any fraction of active spins, whereas in the canonical ensemble they emerge only when this fraction is sufficiently large. At lower fractions, canonical critical endpoints arise instead. We also report the formation of gaps in the entropy as a function of magnetization, another manifestation of ensemble inequivalence.
Models with competing nearest-neighbor and very long-range interactions are solved exactly for several one-dimensional cases, including the usual Ising chain (ICCI), the XY model (XYCI), the transverse Ising model (TICCI), and the spherical model (SCCI). For certain ratios of the competing interaction strengths, ICCI and XYCI display triple points and two critical points in a field. In addition TICCI has an apparently enclosed phase in an H-T phase diagram; however, this phase is really paramagnetic as can be seen when an extended phase diagram is used. The extended phase diagrams for ICCI, XYCI, and TICCI display tricritical points and the tricritical exponents have different values from the usual classical values. In contrast to the rich phase behavior of the preceding models, SCCI shows very simple phase behavior, which is directly related to the ground state. Finally, the introduction of a staggered field to ICCI and a simple transformation allows reinterpretation as a metamagnetic model. Using ICCI as a guide the observability of the tricritical exponents is discussed.