Metastable states in the $J_1\text{−}{J}_2$ Ising model

We study the $J_1\text{−}{J}_2$ Ising model on the square lattice using the random local field approximation (RLFA) and Monte Carlo (MC) simulations for various values of the ratio $p=J_2/\left|J_1\right|$ with antiferromagnetic coupling $J_2$, ensuring spin frustration. RLFA predicts metastable states with zero order parameter (polarization) at low temperature for $p\in (0,1)$. This is supported by our MC simulations, in which the system relaxes into metastable states with not only zero, but also with arbitrary polarization, depending on its initial value, external field, and temperature. We support our findings by calculating the energy barriers of these states at the level of individual spin flips relevant to the MC calculation. We discuss experimental conditions and compounds appropriate for experimental verification of our predictions.

Figure 1

The $J_1\text{−}{J}_2$ Ising model scheme. (a) Square lattice of Ising spins (white: up; blue: down) with the interaction constant $J_1$ between nearest neighbors along horizontal and vertical (solid) lines and $J_2$ between next-nearest neighbors along (dashed) diagonals. (b) Random (very high-temperature) spin configuration; the number of spins along each side of the square sample is $L=100$.

Figure 2

Polarization as a function of temperature obtained using the RLFA analytical approach and MC simulations. (a) Solution of the RLFA equation (lines) and MC results (markers) for uniform polarization in a uniform field at $p<p_0$, and (b) for striped polarization in a striped field at $p>p_0$. The magnitude of the field is $h=0.001$ (purple dashed line and triangles) and $h=0$ (blue solid line and circles). Each data point is derived from a single MC run with a random initial spin configuration at each temperature. The magenta diamonds in the bottom left panel of (a) indicate the temperatures $T_0<T_1<T_c$.

Figure 3

Critical temperatures and energies in the $J_1\text{−}{J}_2$ Ising model as functions of $p=J_2/\left|J_1\right|$. (a) Phase diagram obtained using RLFA, standard MFA, and MC. Solid dark blue curves correspond to MFA; open blue circles, magenta up-triangles, and purple down-triangles are $T_c$, $T_0$, and $T_1$, respectively, obtained within RLFA [Fig. 2(a)]. The dependence $T_c\left(p\right)$ shows the logarithmic scaling in the striped phase as $p\to p_0$. The red square is the exact Onsager's solution for the 2D Ising model. Dark blue filled circles are calculated using the MC method. (b) Energy per dipole obtained using MC simulation at zero temperature in the absence of a field (red squares), at a temperature $T={10}^{-9}$ and a field $h=0$ (blue circles), and $T=0$, $h={10}^{-9}$ (cyan up-triangles for uniform and magenta down-triangles for striped field). Each data point is averaged over 100 samples of size $L=100$. Standard deviations of the energy distribution histogram over 100 samples for each data point are smaller than the markers. Blue solid and dashed lines correspond to the energy of the FM and AFM states, respectively.

Figure 4

Spin configurations of final absorbing states after energy relaxation at zero temperature, starting from a random spin configuration, in the absence or in a very small uniform or striped external field $h$ for the values of $p=J_2/\left|J_1\right|$ equal to 0, 0.01, 0.99, and 1.

Figure 5

Relaxation during MC simulation. (a) Energy relaxation starting from a random spin configuration at each temperature at $p=0.1$, $L=100$. In some cases, the system is stuck in a state with an energy of about $E_1=-1.77$, which is higher than the FM ground state with $E_0=-1.8$ by the energy of two domain walls. These two-stripe states are metastable with an activation energy of $4|J_1|$ and disappear in an external field, as discussed for the 2D Ising model in [26, 27]. Large red dots at energy $E_0$ correspond to the relaxation time. Inset: the relaxation time fitting to the activation law $Aexp(0.4/T)$ vs $1/T$. (b) Polarization relaxation at $p=0.3$ and zero temperature for 100 samples, starting from a random spin configuration with polarization $m_{\text{init}}=0.2$.