Site-decorated model for unconventional frustrated magnets: Ultranarrow phase crossover and two-dimensional spin reversal transition

Abstract

The site-decorated Ising model is introduced to advance the understanding and experimental realization of the recently discovered one-dimensional (1D) finite-temperature ultranarrow phase crossover in an external magnetic field, while mitigating the geometric complexities of traditional bond-decorated models. The unconventional frustration and physics are clarified by exactly mapping the 1D site-decorated Ising model in a magnetic field onto a zero-field bond-decorated $J_1$-$J_2$ Ising model with conventional geometrical frustration. Furthermore, although higher-dimensional Ising models in an external field remain unsolved exactly, an exact solution for a spin-reversal transition -- driven by an exotic, hidden half-ice, half-fire state induced by site decoration -- is derived. This transition, triggered by a slight variation in temperature or magnetic field -- without changing its direction -- even in the weak-field limit, offers a promising route toward energy-efficient applications such as data storage and processing. The results suggest that site decoration offers an avenue for materials and device design, particularly in systems such as mixed $d$-$f$ compounds, optical lattices, and neural networks, calling for further studies with site-decorated Heisenberg models. In addition, the site-decorated model offers a rigorous test ground for artificial intelligence (AI) in science, as the analytic derivation of the present results was not only validated but also improved by a general-purpose large language model, inspiring the use of AI as scientific discoverer.

Figures (8)

• Figure 1: Schematics of minimally decorated Ising models in 1D and 2D: (a) and (c) bond decoration and (b) and (d) site decoration. Green balls depict the backbone, type-a spins coupled by the FM interaction $J$ (red and orange bonds); gray balls depict the decorating, type-b spins coupled to the backbone by the AFM interaction $J_{ab}$ (gray bonds). The magnetic moments of the spins are $\mu_a$ and $\mu_b$, respectively. While the conventional geometric frustration is absent, frustration comes from the competition between an external field and the AFM coupling $J_{ab}$. (e) The zero-field $J_1$-$J_2$ Ising model. (f) The zero-field bond-decorated $J_1$-$J_2$ Ising model, where the decorating spins are coupled to two nearest-neighbor backbone spins by interactions $J_b$ and $J_{ab}$, yielding geometric frustration for $J_1 J_b J_{ab}<0$. Model (b) can be exactly mapped onto model (f) when $J_1=h\mu_a$, $J_b=h\mu_b$, and $J_2=J$.
• Figure 2: (a) Density plot of $h_\mathrm{eff}$ in the $h$-$T$ plane. The dashed line is the $h_\mathrm{eff}=0$ contour line, i.e., the $T_0$ curve given by Eq. (\ref{['eq:T0']}). (b) OP $\langle \sigma_i\rangle$ and the definition of the crossover width $2\delta T = 2\left(\partial \langle \sigma_i \rangle/\partial T\right)^{-1}_{T=T_0}$. Here, $J_{ab}=-2$, $\mu_a=1$, and $\mu_b=4/3$, resulting in $h_c\equiv|J_{ab}/\mu_a|=2$ and $h_f\equiv|J_{ab}/\mu_b|=1.5$.
• Figure 3: The density plots of $\langle\sigma_i\rangle$, $\langle b_i\rangle$, and entropy $S/\ln2$ in the $h$-$T$ plane. Top panels: (a)--(c) for $J=0$. Bottom panels: (d)--(f) for $J=20$. Here $J_{ab}=-2$, $\mu_a=1$, and $\mu_b=4/3$, resulting in $h_c=2$ (solid circle) and $h_f=1.5$ (open circle). The dashed lines in (a),(d),(e) denote the contour of zero value: In (a), the $\langle\sigma_i\rangle=0$ line defines $T_0$ the phase boundary in (d) by Eq. (\ref{['eq:T0']}). In (c), the black arrow points to the zero-temperature critical point at $h=h_c$ hosting the half-fire, half-ice state. In (e), the vertical line $\langle b_i\rangle=0$ at $h=h_f$ and above the phase boundary line hosts the opposite half-ice, half-fire state---shown in (f)---originates from the hidden frustration, and divides the region above $T$=$T_0$ in half where the decorated spins flip for $h<h_f$.
• Figure 4: Approaching the zero field. (a) $T_0$ as a function of $\mu_a/\mu_b$ for $h\to0$. The temperature dependence of (b) $\langle\sigma_i\rangle$, (c) $\langle b_i\rangle$, and (d) the total magnetization $m=\mu_a\langle\sigma_i\rangle+\mu_b\langle b_i\rangle$ in weak fields. $J=20$, $J_{ab}=-2$, $\mu_a=1$, and $\mu_b=4/3$ unless specified.
• Figure 5: The magnetic field dependence of (a) $\langle \sigma_i\rangle$, (b) $\langle b_i \rangle$, and (c) the total magnetization $m=\mu_a\langle \sigma_i \rangle + \mu_b\langle b_i \rangle$ at $T_0$ as $h_\mathrm{eff}\to 0^\pm$ in the site-decorated square-lattice Ising model for $J=1$ with $T_c^*\approx 2.269/k_\mathrm{B} > T_0^\mathrm{max}$ (blue solid and dashed lines) and $J=0.88$ with $T_c^*\approx 2/k_\mathrm{B} < T_0^\mathrm{max}$ (red solid and dashed lines). Here $J_{ab}=-2$, $\mu_a=1$, $\mu_b=4/3$ yielding $T_0^\mathrm{max}\approx 2.056/k_\mathrm{B}$.