Microcanonical Analysis of the Curie–Weiss Anisotropic Quantum Heisenberg Model in a Magnetic Field

Abstract

The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy $s(e,\varvec{m})$ is obtained as a function of the energy $e$ and the magnetization vector ${\varvec{m}}$ in the thermodynamic limit. Since, for this model, $e$ is uniquely determined by ${\varvec{m}}$, the same information can be encoded either in $s(\varvec{m})$ or $s(e,m_1,m_2)$. Although these two entropies correspond to the same physical setting of fixed $e$ and ${\varvec{m}}$, their concavity properties differ. The entropy $s_{{\varvec{h}}}(u)$, describing the model at fixed total energy $u$ and in a homogeneous external magnetic field ${\varvec{h}}$ of arbitrary direction, is obtained by reduction from the nonconcave entropy $s(e,m_1,m_2)$. In doing so, concavity, and therefore equivalence of ensembles, is restored. $s_{{\varvec{h}}}(u)$ has nonanalyticities on surfaces of co-dimension 1 in the $(u,\varvec{h})$-space. Projecting these surfaces into lower-dimensional phase diagrams, we observe that the resulting phase transition lines are situated in the positive-temperature region for some parameter values, and in the negative-temperature region for others. In the canonical setting of a system coupled to a heat bath of positive temperatures, the nonanalyticities in the microcanonical negative-temperature region cannot be observed, and this leads to a situation of effective nonequivalence even when formal equivalence holds.

Appendix: Evaluation of \(\fancyscript{F}\) at the Stationary Points

It is shown how to evaluate $\fancyscript{F}$ as given in (20) at a stationary point determined by equations (18a)–(18c).

We start by writing (18b) in the form

$$\begin{aligned} -\frac{m_\alpha r}{\tanh r} = t_\alpha -x_\alpha \sqrt{\lambda _\alpha }. \end{aligned}$$

(39)

Upon squaring, summing over $\alpha$, and then taking the square root on both sides of this equation, we obtain

$$\begin{aligned} \frac{|\varvec{m}|r}{\tanh r}=\sqrt{\sum _{\alpha =1}^3\left( t_\alpha -x_\alpha \sqrt{\lambda _\alpha }\right) ^2}=r, \end{aligned}$$

(40)

where the second equality sign is due to definition (19). We therefore have $\tanh r=|\varvec{m}|$ and can write

$$\begin{aligned} \fancyscript{F}(s,\varvec{t},\varvec{x})=2es+\varvec{m}\cdot \varvec{t}-\frac{1}{2}\ln \left( 1-\varvec{m}^2\right) . \end{aligned}$$

(41)

The first two terms on the right hand side of (41) can be written in the form

$$\begin{aligned} \varvec{m}\cdot \varvec{t}+2es =\sum _{\alpha =1}^3\left( m_\alpha t_\alpha - \frac{1}{s}x_\alpha x_\alpha \right) =\sum _{\alpha =1}^3 m_\alpha \left( t_\alpha - x_\alpha \sqrt{\lambda _\alpha }\right) {,} \end{aligned}$$

(42)

where first (18a) and then (18c) have been used. With (39) and (40), this expression simplifies to

$$\begin{aligned} \varvec{m}\cdot \varvec{t}+2es =-\frac{\varvec{m}^2 r}{\tanh r} = -|\varvec{m}|r = -|\varvec{m}|\mathrm {\,arctanh\,}|\varvec{m}|. \end{aligned}$$

(43)

Inserting this into (41), the derivation of (21) is complete.