Competing ferromagnetic and antiferromagnetic phases on the frustrated Ising honeycomb lattice

TL;DR

The paper addresses phase transitions in the frustrated honeycomb Ising model with ferromagnetic $J_1$ and $J_2$ and antiferromagnetic $J_3$. It adopts cluster mean-field theory (CMF) with $n_s=6$ and $n_s=18$ to resolve ferromagnetic, antiferromagnetic, and paramagnetic phases and to locate tricritical, critical-endpoint, and bicritical points by analyzing the cluster free-energy derivatives $\\eta_2^{\alpha}$ and $\\eta_4^{\alpha}$. Key findings include that FE–PM transitions are always second order, while AF–PM transitions exhibit first-order behavior within a finite window of $g_3$ that shrinks as $g_2$ increases, with tricritical points moving toward $g_3\approx -1$ and a bicritical point emerging at large $g_2$; order-by-disorder effects also appear near $g_3=-1$, and two successive AF–FE and FE–PM transitions can occur near the AF–FE boundary. These results delineate a rich multicritical landscape with implications for two-dimensional van der Waals magnets and guide future numerical and experimental investigations.

Abstract

We investigate the frustrated $J_1$-$J_2$-$J_3$ Ising model on the honeycomb lattice, featuring first- and second-neighbor ferromagnetic couplings ($J_1>0$ and $J_2>0$) and third-neighbor antiferromagnetic interactions ($J_3<0$). Using the cluster mean-field method, we analyze the phase transitions in the regime $1/2 < J_2/J_1 \le 1$, where ferromagnetic and antiferromagnetic phases compete. Our results reveal that near the strongly frustrated limit $J_3/J_1 = -1$, the system exhibits order-by-disorder state selection, tricritical and bicritical behavior, critical endpoints, and two successive phase transitions. The ferromagnetic-paramagnetic transition remains second order across the entire interaction range, whereas the antiferromagnetic-paramagnetic boundary shows a richer behavior, including both first- and second-order transitions as well as tricriticality. Increasing the second-neighbor coupling $J_2/J_1$ narrows the range of $J_3/J_1$ where first-order antiferromagnetic-paramagnetic transitions occur; beyond a certain threshold, only second-order order-disorder transitions persist. Consequently, the tricritical point shifts toward $J_3/J_1 \approx -1$ as $J_2/J_1$ increases, culminating in a bicritical point where the antiferromagnetic, ferromagnetic, and paramagnetic phases meet.

Figures (7)

• Figure 1: (a) Schematic representation of the first-, second-, and third-neighbor interactions in the honeycomb lattice. (b) Portion of the ground-state phase diagram in the $(g_2, g_3)$ plane, where $g_2 = J_2/J_1$ and $g_3 = J_3/J_1$. Panels (c)--(e) illustrate the magnetic configurations of the ferromagnetic (FE), Néel antiferromagnetic (AF), and zigzag (ZZ) antiferromagnetic phases, respectively. Red and black circles denote spins with opposite orientations.
• Figure 2: Schematic representation of the CMF method applied to the honeycomb lattice using clusters of (a) 6 and (b) 18 sites. Second- and third-neighbor interactions are omitted for clarity. Arrows indicate the influence of the mean fields on the central cluster (hatched in red). Red and black filled circles denote spins with opposite directions.
• Figure 3: Phase transitions for $g_2 = 0.6$ within the 6-site CMF approximation. (a) Phase diagram of temperature versus third-neighbor interaction $g_3$, where solid and dashed lines represent second- and first-order phase transitions, respectively. The circle marks a tricritical point, and the square denotes a critical endpoint. Panels (b) and (c) show the free-energy difference $\Delta f = f_{\textrm{AF}} - f_{\textrm{PM}}$ for values of $g_3$ near the tricritical point. The red, black, and blue curves in panel (b) correspond to $T/J_1 = 3.365500$, $3.365443$, and $3.365400$, respectively, while panel (c) shows results for $T/J_1 = 3.339000$ (red), $3.338915$ (black), and $3.338000$ (blue). Panel (d) presents the loci of $\eta^{\textrm{AF}}_2 = 0$ (blue) and $\eta^{\textrm{AF}}_4 = 0$ (red) in the $(g_3, T)$ plane; the light-red and white regions correspond to $\eta^{\textrm{AF}}_4 < 0$ and $\eta^{\textrm{AF}}_4 > 0$, respectively. (e) Discontinuity of the antiferromagnetic order parameter as a function of $g_3$ at the AF–PM transition. The dotted lines are guides to the eye.
• Figure 4: Phase transitions for $g_2 = 0.75$. (a) Temperature–coupling phase diagrams obtained within the CMF approach using clusters of 6 (dark) and 18 (red) sites. (b) Curves satisfying $\eta_2^{\textrm{AF}} = 0$ and $\eta_4^{\textrm{AF}} = 0$ in the temperature–$g_3$ plane for $n_s=6$. The regions with $\eta_4^{\textrm{AF}} < 0$ and $\eta_4^{\textrm{AF}} > 0$ are indicated in light red and white, respectively. (c) Discontinuity of the antiferromagnetic order parameter for $n_{\rm s} = 6$ as a function of $g_3$. Panels (d) and (e) show the temperature dependence of the antiferromagnetic order parameter obtained from the CMF approximation with $n_{\rm s} = 6$ and $n_{\rm s} = 18$, respectively.
• Figure 5: (a) Difference between the free energies per spin of the ferromagnetic and antiferromagnetic phases, $\delta f = (f_{\textrm{FE}} - f_{\textrm{AF}})/n_{\rm s}$, as a function of temperature for $g_2 = 0.75$ and $g_3 = -1.0$, obtained using the 6-site ($n_{\rm s} = 6$, dark) and 18-site ($n_{\rm s} = 18$, red) CMF approximations. Panels (b) and (c) show the temperature dependence of the entropy per spin and the order parameters of the antiferromagnetic and ferromagnetic phases for $g_2 = 0.75$ and $g_3 = -1.01$ within the 6-site and 18-site CMF approaches, respectively.