Frustrated Ising model on the honeycomb lattice: Metastability and universality

Abstract

We study the Ising model with competing ferromagnetic nearest- and antiferromagnetic next-nearest-neighbor interactions of strengths $J_1 > 0$ and $J_2 < 0$, respectively, on the honeycomb lattice. For $J_2 > - J_1 / 4$ it has a ferromagnetic ground state, and previous work has shown that at least for $J_2 \gtrsim -0.2 J_1$ the transition is in the Ising universality class. For even lower $J_2$ some indicators pointing towards a first-order transition were reported. By utilizing population annealing Monte Carlo simulations together with a rejection-free and adaptive update, we can equilibrate systems with $J_2$ as low as $-0.23 J_1$. By means of a finite-size scaling analysis we show that the system undergoes a second-order phase transition within the Ising universality class at least down to $J_2 =-0.23 J_1$ and, most likely, for all $J_2 > - J_1 / 4$. As we show here, there exist very long-lived metastable states in this system explaining the first-order like behavior seen in only partially equilibrated systems.

Figures (14)

• Figure 1: Ising model on a honeycomb lattice. Black circles represent spin sites, solid (respectively dashed) lines nearest-neighbor (respectively next-nearest-neighbor) interactions.
• Figure 2: Phase diagram in the $J_2-T$ plane. The square symbols correspond to the pseudocritical temperature obtained from the peak locations of the specific heat for $L=48$ systems. Triangles denote the exactly known values for $J_2 = 0$ and $J_2=-1/4$. The solid colored lines show EFT approximations for various cluster sizes $n$ from Ref. Bobak2016, with the end points indicating the predicted tricritical locations. The black solid line shows exact pseudocritical points for an $L=4$ system obtained by exact enumeration. Blue circles show previous MC results from Ref. Zukovic2021. Green diamonds denote FSS extrapolations for $T_c$ from our previous work in Ref. Gessert2024 using MC data. The dashed line is merely a guide to the eye. Note that the EFT lines for $n > 1$ do not exist below a certain value of $J_2$, where they (erroneously) predict a tricritical point.
• Figure 3: Equilibrium snapshots at the inverse temperature $\beta=0.6 \beta_c$ for $L=64$, and (a) $J_2=0$, (b) $J_2=-0.1$, (c) $J_2=-0.2$, and (d) $J_2=-0.23$. In (c) and (d), for $J_2$ close to $-1/4$, the structures appear much more stable, consistent with the observed metastability and slow relaxation at these values of $J_2$.
• Figure 4: Results from PA simulations using Metropolis updates with $\theta = 2000$ (respectively $\theta=500$ for $J_2=0$) and $L=48$. Top panel: Magnetization per spin $m$ for various coupling strengths. For $J_2 \gtrsim -0.2$ we obtain good agreement with the rejection-free data (not shown here). For $J_2 < -0.22$ the simulations fail to reach the fully ordered state at low temperatures. Bottom panel: Replica-averaged family size $\rho_t$. As expected Wang2015$\rho_t$ monotonously increases during the annealing process with decreasing temperature. The failure to equilibrate for $J_2 < -0.2$ is reflected in a sharp increase in $\rho_t$ near the corresponding critical temperatures $T_c$ for $J_2 \leq -0.2$.
• Figure 5: Overview over the thermodynamic quantities measured as functions of temperature for $J_2 \in \{-0.2, -0.21, -0.22, -0.23\}$ for a system of linear size $L=48$ from PA simulations using $n$-fold way updates. The shaded areas indicate $2\sigma$ error environments of the data for $J_2 \in \{-0.21, -0.22, -0.23\}$ (but note that in many cases they are hardly visible). Top left: Energy per spin (dashed lines correspond to the ground-state energy), Top right: Magnetization per spin. Bottom left: Specific heat. Bottom right: Magnetic susceptibility.