Phase transitions, order by disorder, and finite entropy in the Ising antiferromagnetic bilayer honeycomb lattice

We present an analytical and numerical study of the Ising model on a bilayer honeycomb lattice including interlayer frustration and coupling with an external magnetic field. First, we discuss the exact $T=0$ phase diagram, where we find finite entropy phases for different magnetizations. Then, we study the magnetic properties of the system at finite temperature using complementary analytical techniques (Bethe lattice) and two types of Monte Carlo algorithms (Metropolis and Wang-Landau). We characterize the phase transitions and discuss the phase diagrams. The system presents a rich phenomenology: There are first- and second-order transitions, low-temperature phases with extensive degeneracy, and order-by-disorder state selection.

Figure 1

Bilayer honeycomb lattice. The (antiferromagnetic) couplings are indicated by different colors. $A,B,C,D$ label spins in a unit cell. The interlayer coupling $J_p$ is indicated by (brown) vertical lines (joining sites $A\text{−}C$ and $B\text{−}D$), the intralayer coupling $J_1$ by (blue) horizontal ones (joining sites $A\text{−}B$ and $C\text{−}D$), and intralayer frustrating coupling $J_x$ is drawn as (violet) dashed lines (joining sites $A\text{−}D$ and $B\text{−}C$).

Figure 2

$T=0$ phase diagrams. (a) $J_x$ vs. $J_1$ in units of $J_p$ for $h=0$. $h/J_p$ vs. $J_1/J_p$ phase diagrams for (b) $J_1=J_x$, (c) $J_x<J_1$ ($\alpha =J_x/J_1=1/2$), and (d) $J_x>J_1$ ($\alpha =J_x/J_1=2$).

Figure 3

The $q=3$ bilayer Bethe lattice that reproduces the ground-state configurations of the bilayer honeycomb lattice.

Figure 4

(a) $J_x/J_p$ vs. $T/J_p$ phase diagram for $J_1=J_p$ for Bethe lattice calculations [full (blue) circles] and Metropolis simulations [full (red) squares] for $L=48$. The transition lines between the paramagnetic and the ordered phases are of second order as it is confirmed by the Binder cumulant (b) and scaled order parameter susceptibility (c) for $(J_x/J_p=0.2,J_1/J_p=1)$.

Figure 5

Phase diagram obtained by BL and MC-M methods for $h=0$ and $J_1=J_x$. The empty (blue) triangles joined by a finely dashed line represent a second-order phase transition and the empty (red) circles joined by a dashed line represent first-order transitions. The $+$ (green) symbol is a tricritical point and the $\times$ (green) symbol corresponds to the exact $T=0$ first-order point. The empty (black) squares correspond to the maximum of the specific heat for $L=48$ MC-M simulations.

Figure 6

Curves of ${\mathrm{AF}}_1$ order parameter, ${\mathcal{C}}_{J_p}$ correlator, specific heat, and entropy (in units of $ln2/N$) as a function of $T/J_p$ for three values of the couplings (indicated in bounded box over each set of curves) along the $J_1=J_x$ line: $J_x=0.2J_p<J_p/3$, $J_p/3<J_x=0.4J_p<J^{*}$, and $J^{*}<J_x=0.6J_p$ obtained by comparison of Metropolis simulations (MC-M) (full triangles), Bethe lattice approximation (BL) (full line), and Wang-Landau (MC-WL) (empty circles). In each region a different behavior is seen. For the larger values of $J_x/J_p$ the system orders in the ${\mathrm{AF}}_1$ phase through a second-order transition from the paramagnetic phase. For a smaller range of $J_1=J_x>J_p/3$, the system orders in the ${\mathrm{AF}}_1$ at low temperatures through a first-order transition, clearly seen as a jump in the order parameter and an off-scale peak in the specific heat. For $J_1=J_x<J_p/3$, the system does not order at low temperature remaining in a macroscopically degenerate state. In the latter case, the specific heat shows a broad peak, as seen, for example, in spin ice systems. The entropy in this region does not vanish as the temperature is lowered towards $T=0$ but tends to the residual value $s=1/2$.

Figure 7

[(a)–(c)] MC-WL simulation results for free energy vs. order parameter ${\varphi }_{{\text{AF}}_1}$ around the critical temperature $T_c$. $T>T_c$ is indicated by circles and $T<T_c$ by squares for different system sizes $L$. (a) For $J_1=J_x=0.4J_p$, where there is a first-order transition; (b) $J_1=J_x=0.6J_p$, where there is a second-order transition. (c) Landau free energy at the critical temperature for different sizes for a first-order phase transition. As the size of the system is increased, the Landau free energy is flatter. The dotted flat line is the behavior in the thermodynamic limit. (d) The difference of the free energy of the ${\mathrm{AF}}_1$ and the ${\mathrm{U}}_2$ states as a function of the $J_x$ parameter along the degenerate line $J_1=J_p/3;J_x>J_p/3$ for two different temperatures obtained from the Bethe lattice approximation.

Figure 8

Magnetization $m$ and entropy per spin [in units of $ln\left(2\right)]$ $s$ as a function of the magnetic field $h/J_p$ obtained with Wang-Landau simulations for the highly frustrated point $J_1=J_x=J_p/3$ (left) and for an arbitrary value of the couplings $J_1=0.2J_p,J_x=0.5J_p$ (right). The top panel shows the magnetization curves at different temperatures. The middle and bottom panels are the density plots corresponding to $h/J_p$ vs. $T/J_p$ phase diagrams for $m$ and $s$, respectively.