Dimensional crossover in the quasi-two-dimensional Ising-O(3) model

We present the results of our Monte Carlo simulation of the Ising-O(3) model on two-dimensional (2D) and quasi-2D lattices. This model is an effective classical model for the stacked square-lattice $J_1$-$J_2$ Heisenberg model, where the nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_2$) couplings are frustrated and we assume that $J_2$ is dominant. We find an Ising ordered phase in which the O(3) spins remain disordered in a moderate quasi-2D region. There is a single first-order transition for a sufficiently large 3D coupling, in agreement with a renormalization group treatment. The subtle region in which the single transition splits into two transitions is also discussed and compared against recent measurements of two very close transitions in BaFe${}_2$As${}_2$. Our results can provide a qualitative explanation of the experiments on ferropnictides, namely the observed sequence and orders of the structural and magnetic transitions, in terms of the ratio between the interlayer and intralayer coupling.

Figure 1

Schematic pictures of the sequence of the transition(s) observed in the parent compounds of the iron-based superconductors. Shown are the disordered state with tetragonal lattice symmetry at high $T$'s (left), the Ising ordered state with broken lattice symmetry and short-range magnetic ordering at intermediate $T$ (center), and the lowest-$T$ state with broken lattice symmetry and long-range stripelike magnetic ordering (right). The symmetry also allows for the other lattice distortion obtained by rotating the presented lattice by $\pi /2$.

Figure 2

Phase diagram of the quasi-2D Ising-O(3) model. The Ising ordered phase ($\langle \mathbf{S}\rangle =0$, $\langle \sigma \rangle \ne 0$) appears in the quasi-2D region $J_z\lesssim 0.01$. For large enough interlayer coupling, a single first-order transition separates the paramagnetic state ($\langle \mathbf{S}\rangle =0$, $\langle \sigma \rangle =0$) from the lowest-$T$ phase that simultaneously breaks the $Z_2$ and O(3) symmetries ($\langle \mathbf{S}\rangle \ne 0$, $\langle \sigma \rangle \ne 0$). The detailed structure for $0.01\lesssim J_z\lesssim 0.0204$ remains to be clarified; see the text. The line representing a phase boundary between the lowest-$T$ phase and the Ising ordered phase is a schematic one. The other lines are guides to the eye.

Figure 3

(a) ${\xi }_{\sigma }/L$ and (b) $U_{\sigma }$ of the 2D Ising-O(3) model. The lines are guides to the eye. The horizontal lines marked as “2D Ising (CFT)' indicate the universal critical values of the square-lattice Ising model given in Ref. 38. The inset shows their FSS plots for $L\ge 48$, where $\nu =1$ and $T_{c1}^{2\text{D}}=1.0514\left(3\right)$.

Figure 4

FSS of the Ising correlation function $G_{\left(L/2,L/2\right)}^{\sigma }$ at the largest distance in a given 2D system defined by Eq. (6), where $\eta =0.25$, $\nu =1$, and $T_{c1}^{2\text{D}}=1.0514\left(3\right)$.

Figure 5

The inverse susceptibility of the O(3) order parameter of the 2D Ising-O(3) model ($J_z=0$). The vertical line indicates the 2D Ising transition temperature $T_{c1}^{2\text{D}}$.

Figure 6

Two separate transitions for $J_z=0.01$: (a) specific heat, (b) the Binder parameters, (c) the normalized correlation lengths in the $x$ direction, and (d) the normalized correlation lengths in the $z$ direction. The vertical lines show the estimated critical temperatures based on the FSS analysis. The lines are guides to the eye.

Figure 7

FSS plots for the higher-$T$ Ising transition for $J_z=0.01$ [$t_1\equiv \left(T-T_{c1}\right)/T_{c1}$]. The upper panel (a) shows the FSS of $U_{\sigma }$. The inset shows the relation to the real temperature $T$; the thick line indicates the lower-$T$ transition close to which a severe finite-size effect appears. The middle panel (b) shows the FSS of ${\xi }_{\sigma }^x/L_x$, with the inset showing the FSS of ${\xi }_{\sigma }^z/L_z$. The lower panel (c) shows the FSS of $G_{L_x/2}^{\sigma }\equiv G^{\sigma }(L_x/2,0,0)$ and $G_{L_z/2}^{\sigma }\equiv G^{\sigma }(0,0,L_z/2)$ (inset) [see Eq. (7)].

Figure 8

FSS plots for the lower-$T$ O(3) transition for $J_z=0.01$ [$t_2\equiv \left(T-T_{c2}\right)/T_{c2}$]. The upper panel (a) shows the FSS of $U_m$. The inset shows the relation to the real temperature $T$; the thick line indicates the higher-$T$ transition close to which a severe finite-size effect appears. The middle panel (b) shows the FSS of ${\xi }_m^x/L_x$, with the inset showing the FSS of ${\xi }_m^z/L_z$. The lower panel (c) shows the FSS of $G_{L_x/2}^m\equiv G^m(L_x/2,0,0)$ and $G_{L_z/2}^m\equiv G^m(0,0,L_z/2)$ (inset) [see Eq. (7)].

Figure 9

Bimodal internal energy density distribution at the first-order phase transition. The inset shows the $J_z$ dependence of the peak-to-peak distance $\Delta E$ of the distributions.

Figure 10

Binder parameters $U_m$ and $U_{\sigma }$ near the first-order transition for (a) $J_z=0.0625$, (b) $J_z=0.0278$, and (c) $J_z=0.0204$.

Figure 11

Two possible scenarios that may explain our observations in the intervening region. (a) The two transitions continue to be of second order until they collapse into the direct first-order transition. (b) Only the lower-$T$ O(3) transition becomes of first order before the transitions merge. The arrows schematically indicate the possible ways in which the system is cooled down for $0.01\lesssim J_z\lesssim 0.0204$.