Nematic phase without Heisenberg physics in FeAs planes

We use Monte Carlo simulations and analytical arguments to analyze a frustrated Ising model with nearest-neighbor antiferromagnetic coupling $J_1$ and next-nearest-neighbor coupling $J_2$. The model is inspired on the physics of pnictide superconductors and to some extent we argue that it can be more representative of this physics than the Heisenberg counterpart. Parameters are chosen such that the ground state is a columnar or striped state, as observed experimentally, but is close to the transition to the simple Néel ordered antiferromagnetic state $R=J_2/\left|J_1\right|\gtrsim 0.5$. We find that a nematic phase is induced close to $R=0.5$ by finite-size effects and argue that this explains experiments in imperfect samples that find a more robust nematic state as the quality of the sample decreases [A. Jesche et al., Phys. Rev. B 81, 134525 (2010)]. Including the effect of a weak coupling with the lattice, we find that a structural transition occurs associated with a nematic phase, with a magnetic transition occurring at a lower temperature. These two transitions merge into a single structural and magnetic transition with a stronger first-order character for larger spin-lattice couplings. These two situations are in agreement with the different phenomenologies found in different families of pnictides.

Figure 1

Phase diagram of the system. Each point represents the critical temperature for a given value of $R$, and the types of points depends on the different methods used to compute them. Empty circles (green online) have been obtained with MC, empty triangles (blue online) have been obtained with real space renormalization group, crosses (purple online) have been obtained with MC renormalization group, and the solid (yellow online) circles have been obtained with series expansion. The open square (light blue online) marks the tricritical point at $R=1.144$.[22] The solid squares (red online) mark the critical temperature values obtained with our MC simulations. The solid lines are guides to the eyes. Notice that Ref.22 considers an equivalent model with ferromagnetic nn interactions so that our Néel-AF ordered phase corresponds to their ferromagnetic phase.

Figure 2

Illustration of a spin domain (gray arrows, red online) in which the staggered magnetization in $x$ direction has opposite sign with respect to the embedding cluster (black arrows).

Figure 3

(a) Example of an MC simulation on a $50\times 50$ lattice for $R=0.51$ and $k_{B}T/J_1=0.52$. Statistically independent points where binned according to the value of the 3D magnetization $(m_{(\pi ,0)},m_{(0,\pi )},m_{\phi })$ in cells of size $0.1\times 0.1\times 0.1$. We show an isosurface with 500 points per bin. Since the highest density of points occurs inside the two regions around $m_{\phi }\approx \pm 1$ and $m_{(\pi ,0)}\approx 0$, $m_{(0,\pi )}\approx 0$, this phase is identified as nematic. (b) Same as in (a), but at a lower temperature $k_{B}T/J_1=0.46$. Also here $m_{\phi }\approx \pm 1$, but the highest density of points occur for $m_{(\pi ,0)}\approx \pm 1$ and $m_{(0,\pi )}\approx \pm 1$, thereby signaling that the system is in the smectic (i.e., magnetized) phase.

Figure 4

Temperature dependence of $U_4\left[m_{\phi }\right]$ and $U_4\left[m_{\text{stagg}}\right]$, calculated for different ($50\times 50$, $24\times 24$, $10\times 10$) system sizes.

Figure 5

${\chi }_{(0,\pi )}$ and ${\chi }_{\phi }$ scaling functions for lattice size $10\times 10$ (triangles, red and magenta online), $24\times 24$ (circles, green and ligth blue), and $50\times 50$ (squares, blue and black). The temperature step is $5\times {10}^{-4}$. We have fixed $\gamma =7/6$, $\nu =2/3$, and $T_{\mathrm{AF}}=0.9715$ for the magnetic transition, and $\gamma =7.00/4.15$, $\nu =0.975$, and $T_{\text{nem}}=0.9720$ for the nematic transition.

Figure 6

Level curves of the bin counts in the 2D space $[m_{(\pi ,0)},m_{\phi }]$ resulting from the simulations of a $50\times 50$ lattice, with $R=0.51$. The results for $T=0.53$, 0.52, $0.51$, and $0.50$ are shown. The higher is the brightness of the bins, the higher is the value of the counts. The lowest and the highest level curves correspond to 300 and 2400 bin counts, respectively. Adjacent level curves differ by 300 bin counts. Above 2400 bin counts, the contour plot intensity saturates and produces the bright white regions.

Figure 7

Magnetic configurations of a $50\times 50$ system with $T=0.52$; (a) real spin configuration, where the red squares represent spin up, while the blue squares spin down, and (b) staggered configuration along $y$ axis, where the squares are red if ${\sigma }_i{(-1)}^{y_i}=+1$, while they are blue if ${\sigma }_i{(-1)}^{y_i}=-1$.

Figure 8

Level curves of the bin counts in the $[m_{(\pi ,0)},m_{\phi }]$ 2D space resulting from the simulations of a $24\times 24$ lattice, with $R=0.51$. The results for $T=0.54$ and $0.49$ are reported. The higher is the brightness of the bins, the higher is the value of the counts. The lowest and the highest level curves correspond to 300 and 2400 bin counts, respectively. Adjacent level curves differ by 300 bin counts. Above 2400 bin counts, the contour plot intensity saturates and produces the bright white regions.

Figure 9

Phase diagram of the system with $R=0.51$ in the $\lambda$ vs $T$ space. We report the numerical results for the structural (or nematic) critical temperatures (open circles, blue online) and the magnetic critical temperatures (open squares, green online), and the structural critical temperatures in the stability limit (open circles, red online, joined by the solid line). The shaded (magenta online) region represents the nematic range in a $50\times 50$ lattice. The inset displays an expansion of the nematic region of the phase diagram. The nematic range in a $24\times 24$ system (darker shaded region, yellow online) is also shown.