Complete analysis of ensemble inequivalence in the Blume-Emery-Griffiths model

We study inequivalence of canonical and microcanonical ensembles in the mean-field Blume-Emery-Griffiths model. This generalizes previous results obtained for the Blume-Capel model. The phase diagram strongly depends on the value of the biquadratic exchange interaction $K$, the additional feature present in the Blume-Emery-Griffiths model. At small values of $K$, as for the Blume-Capel model, lines of first- and second-order phase transitions between a ferromagnetic and a paramagnetic phase are present, separated by a tricritical point whose location is different in the two ensembles. At higher values of $K$ the phase diagram changes substantially, with the appearance of a triple point in the canonical ensemble, which does not find any correspondence in the microcanonical ensemble. Moreover, one of the first-order lines that starts from the triple point ends in a critical point, whose position in the phase diagram is different in the two ensembles. This line separates two paramagnetic phases characterized by a different value of the quadrupole moment. These features were not previously studied for other models and substantially enrich the landscape of ensemble inequivalence, identifying new aspects that had been discussed in a classification of phase transitions based on singularity theory. Finally, we discuss ergodicity breaking, which is highlighted by the presence of gaps in the accessible values of magnetization at low energies: it also displays new interesting patterns that are not present in the Blume-Capel model.

Figure 1

$(\mathrm{\Delta },T)$ phase diagram of the BEG model in the canonical ensemble for two different values of the biquadratic interaction parameter $K$. Solid and dashed lines are second- and first-order phase transition lines, respectively, while the red dot denotes the canonical tricritical point (CTP). (a) $K=0$, i.e., the BC model, ${\mathrm{\Delta }}_{\text{CTP}}\backsimeq 0.4621$, $T_{\text{CTP}}=1/3$; (b) $K=1$, ${\mathrm{\Delta }}_{\text{CTP}}\backsimeq 0.7726$, $T_{\text{CTP}}=0.6$.

Figure 2

Entropy ${\tilde{s}}_{-}(\epsilon ,m)$ vs. magnetization $m$ for $K=1$ and for two values of the single-spin energy parameter $\mathrm{\Delta }$, plotted for several different values of the energy. (a) $\mathrm{\Delta }=0.8$: second-order phase transition. (b) $\mathrm{\Delta }=0.93$: first-order phase transition; at the transition energy the maxima at $m=0$ and at $m\ne 0$ are at the same height.

Figure 3

(a) $(\mathrm{\Delta },T)$ phase diagram of the BEG model in the microcanonical ensemble at $K=1$. Solid and dashed lines are the second- and first-order phase transition lines, respectively. The second-order transition line ends at the microcanonical tricritical point (MTP), located at ${\mathrm{\Delta }}_{\text{MTP}}\backsimeq 0.8439$, $T_{\text{MTP}}\backsimeq 0.5078$. (b) Zoom of the region of the first-order transitions, showing also the canonical tricritical point (CTP), located at ${\mathrm{\Delta }}_{\text{CTP}}\backsimeq 0.7726$, $T_{\text{CTP}}=0.6$, and the canonical first-order transition line (red dashed line). The coordinates of the tricritical points in the phase diagram can be calculated from Eqs. (15), (16), (B3), and (B5). The first-order lines reach the $T=0$ line at $\mathrm{\Delta }=(K+1)/2=1$.

Figure 4

Energy dependence of the temperature (caloric curve) for $K=1$ and different values of the single-spin energy parameter $\mathrm{\Delta }$. The blue lines are the microcanonical curves; they coincide with the canonical ones in the energy ranges where ensembles are equivalent. Where instead the ensembles are inequivalent, the canonical curves are given by the horizontal red lines (Maxwell construction). The energies within the range covered by these lines are forbidden in the canonical ensemble.

Figure 5

The most interesting region of the $(\mathrm{\Delta },\epsilon )$ phase diagram for the canonical and microcanonical ensembles at $K=1$. The solid blue line denotes second-order transitions and ends at the CTP and the MTP for the canonical and microcanonical ensembles, respectively. Red (blue) dashed lines are the first-order transition lines for the canonical (microcanonical) ensemble. The coordinates of the tricritical points are ${\mathrm{\Delta }}_{\text{CTP}}\backsimeq 0.7726$, ${\epsilon }_{\text{CTP}}\backsimeq 0.2836$ and ${\mathrm{\Delta }}_{\text{MTP}}\backsimeq 0.8439$, ${\epsilon }_{\text{MTP}}\backsimeq 0.2996$. The region inside the two red dashed lines is forbidden in the canonical ensemble. The first-order lines meet at the point with coordinates $\mathrm{\Delta }=1$ and $\epsilon =0$ (the minimum allowed energy for $\mathrm{\Delta }=1$).

Figure 6

$(\mathrm{\Delta },T)$ phase diagram of the canonical and microcanonical ensembles at $K=2.85$: panel (a) shows the most interesting region, which is further zoomed in panel (b). The solid blue line denotes the line of second-order transitions, ending at the CTP and the MTP for the canonical and microcanonical ensembles, respectively. Red (blue) dashed lines are the first-order transition lines for the canonical (microcanonical) ensemble. The first-order canonical line branches at the canonical triple point (TP): one branch reaches $T=0$ where it meets the two microcanonical first-order lines at $\mathrm{\Delta }=(K+1)/2=1.925$; the other branch ends at the canonical critical point (CCP). The coordinates of the relevant points, CTP, MTP, TP, and CCP, are given in the text.

Figure 7

Caloric curves for $K=2.85$ and different values of the single-spin energy parameter $\mathrm{\Delta }$. The meaning of the lines is the same as in Fig. 4.

Figure 8

The most interesting region of the $(\mathrm{\Delta },\epsilon )$ phase diagram for the canonical and microcanonical ensembles at $K=2.85$. The blue solid line denotes the line of second-order transitions, ending at CTP and MTP for the canonical and microcanonical ensembles, respectively. Red (blue) dashed lines are the first-order transition lines for the canonical (microcanonical) ensemble. Between the CTP and the MTP the second-order microcanonical line is extremely close (from below) to one of the first-order canonical lines, so that in the plot they are practically indistinguishable. The plot shows the three energies of the canonical triple point (TP) and the canonical critical point (CCP). The first-order lines meet at the point with coordinates $\mathrm{\Delta }=1.925$ and $\epsilon =0$ (the minimum allowed energy for $\mathrm{\Delta }=1.925$). The region inside the red dashed lines is forbidden in the canonical ensemble.

Figure 9

The most interesting region of the phase diagrams of the canonical and microcanonical ensembles at $K=3.6$. The solid blue line denotes the line of second-order transitions, ending at the canonical tricritical point (CTP) and at a microcanonical branching point (MBP) for the two ensembles, respectively. Red (blue) dashed lines are the first-order transition lines for the canonical (microcanonical) ensemble. (a) $(\mathrm{\Delta },T)$ phase diagram. Between the CTP and the triple point (TP) the first-order canonical line is very close to the second-order microcanonical line; the inset shows a zoom of the region near the TP. One canonical first-order branch and the two microcanonical first-order branches meet at $T=0$ at $\mathrm{\Delta }=(K+1)/2=2.3$. (b) $(\mathrm{\Delta },\epsilon )$ phase diagram. The three energies of the canonical triple point (TP) are indicated. The first-order lines meet at the point with coordinates $\mathrm{\Delta }=2.3$ and $\epsilon =0$ (the minimum allowed energy for $\mathrm{\Delta }=2.3$). The coordinates of the relevant points, CTP, MBP, MCP, TP, and CCP, are given in the text.

Figure 10

Caloric curves for $K=3.6$ and different values of the single-spin energy parameter $\mathrm{\Delta }$. The meaning of the lines is the same as in Fig. 4.

Figure 11

The most interesting region of the phase diagrams of the canonical and microcanonical ensembles at $K=5$. The solid blue line denotes the line of second-order transitions, ending at the canonical branching point (CBP) and at a microcanonical branching point (MBP) for the two ensembles, respectively. Red (blue) dashed lines are the first-order transition lines for the canonical (microcanonical) ensemble. (a) $(\mathrm{\Delta },T)$ phase diagram. One canonical first-order branch and the two microcanonical first-order branches meet at $T=0$ at $\mathrm{\Delta }=(K+1)/2=3$. (b) $(\mathrm{\Delta },\epsilon )$ phase diagram. The first-order lines meet at the point with coordinates $\mathrm{\Delta }=3$ and $\epsilon =0$ (the minimum allowed energy for $\mathrm{\Delta }=3$). The coordinates of the relevant points, CBP, MBP, MCP, and CCP, are given in the text.

Figure 12

Caloric curves for $K=5$ and different values of the single-spin energy parameter $\mathrm{\Delta }$. The meaning of the lines is the same as in Fig. 4.

Figure 13

A schematic representation of the accessible region in the $m\text{−}\epsilon$ plane for fixed values of $\mathrm{\Delta }$ and $\mathrm{\Delta }/K$.