Moiré magnetism in a bilayer Ising model

TL;DR

This work investigates moiré magnetism in a bilayer Ising system with moiré-modulated interlayer exchange produced by twist or differential strain. Using large-scale classical Monte Carlo simulations, it shows that the finite-temperature transition remains in the conventional $2$-D Ising universality class, while the low-temperature state can be domain-textured without a separate thermodynamic phase transition. A simple geometric energy balance between bulk interlayer coupling and intralayer domain-wall costs governs a crossover between a uniform ferromagnet and a moiré-domain state, with $J' L_M^2$ competing against $J L_M$ and yielding $J'\sim z/L_M$ (and $J'\sim \tan\phi$ for twists). The results offer a minimal framework for understanding moiré-induced magnetic textures as emergent from geometry rather than thermodynamics, relevant to materials like CrI$_3$ under twist or strain.

Abstract

Moiré patterns in magnetic bilayers generate spatially modulated interlayer exchange interactions that can give rise to nonuniform magnetic textures. We study a minimal classical bilayer Ising model with a moiré-modulated interlayer coupling, generated either by relative twist or differential strain between the layers. Using large-scale classical Monte Carlo simulations, we show that the ordering transition remains in the conventional two-dimensional Ising universality class, even when the low-temperature state is domain-textured. At low temperatures, we find a smooth crossover between a uniform ferromagnet and domain-textured state, in which the spins locally follow the sign of the interlayer exchange. We demonstrate that there is no breaking of layer symmetry for twisted bilayers. The location of the crossover is determined by a simple geometric energy balance between bulk interlayer exchange and intralayer domain-wall costs. Our results provide a minimal framework for understanding how moiré-modulated magnetic textures can emerge from geometric energetics without requiring a thermodynamic phase transition.

Figures (4)

• Figure 1: A map of the interlayer coupling function $\Phi(\mathbf{u}) = \Phi_0 + \sum_a \cos{(\mathbf{b}_a \cdot \mathbf{u})}$, with $\Phi_0 = 0$ for the left figure, and $\Phi_0 = 1/2$ for the right, where $\mathbf{b}_a$ are the reciprocal lattice vectors and $\mathbf{u}$ is the displacement field. The lattice is size $L=72$ with differential strain $a = 24/23$, yielding $N_M=3$ (linear) moiré unit cells. Blue (red) regions correspond to locally ferromagnetic (antiferromagnetic) interlayer exchange. The constant $\Phi_0$ in the interlayer coupling function "softens" the domains. The overall coupling strength is controlled by the parameter $J'$.
• Figure 2: Binder cumulant crossings for the temperature-driven transition from the paramagnetic phase to the ordered, domain-textured state in a strained bilayer. The moiré unit cell size is $L_M = 10$ sites, and $N_M=L/L_M$ denotes the number of moiré unit cells along each linear system dimension. Lines are cubic fits to the Binder cumulant data and are shown as guides to the eye. (Inset) The fitted slopes of $U_2$ at the crossing point (see Sec. \ref{['sec:Temp']}) are used to compute the size-dependent effective exponent $1/\nu^*(L)$, which extrapolates to $1/\nu = 0.97 \pm 0.02$ in the thermodynamic limit, consistent with the conventional 2D Ising universality class.
• Figure 3: No layer symmetry breaking in the twisted bilayer. (Top) Layer polarization order parameter $\langle P^2\rangle$ versus $J'$ for several system sizes, labeled by the number of moiré unit cells $N_M = L/L_M$ ($N_M$ even). $\langle P^2\rangle$ peaks near the crossover $J'_c$, and approaches a size-dependent plateau at large $J'$. (Bottom) The plateau values $\langle P^2(N_M)\rangle_{J'>J'_c}$ decrease with system size and extrapolate to zero in the thermodynamic limit. The largest system sizes ($N_M > 8$) follow the expected scaling $\langle P^2\rangle\propto 1/N_M$.
• Figure 4: Crossover diagram in the interlayer coupling $J'$ and differential strain $a$, which we express as the ratio $(a-1)/a = 1/L_M$, showing the crossover from the ferromagnetic to domain-textured state. This occurs at low temperature; here $T=1\ll T_c$. The color denotes the total magnetization squared of the system. The crossover boundary is fit with the linear function $J' \propto (a-1)/a$ (dashed line), which we show emerges from geometric energy balance of the domains in Sec. \ref{['sec:Cross']}.