Interplay between magnetic and vestigial nematic orders in the layered $J_1-J_2$ classical Heisenberg model

We study the layered $J_1-J_2$ classical Heisenberg model on the square lattice using a self-consistent bond theory. We derive the phase diagram for fixed $J_1$ as a function of temperature $T, J_2$, and interplane coupling $J_z$. Broad regions of (anti)ferromagnetic and stripe order are found, and are separated by a first-order transition near $J_2\approx 0.5$ (in units of $|J_1|$). Within the stripe phase the magnetic and vestigial nematic transitions occur simultaneously in first-order fashion for strong $J_z$. For weaker $J_z$, there is in addition, for $J_2^{*}<J_2<J_2^{**}$, an intermediate regime of split transitions implying a finite temperature region with nematic order but no long-range stripe magnetic order. In this split regime, the order of the transitions depends sensitively on the deviation from $J_2^{*}$ and $J_2^{**}$, with split second-order transitions predominating for $J_2^{*}\ll J_2\ll J_2^{**}$. We find that the value of $J_2^{*}$ depends weakly on the interplane coupling and is just slightly larger than 0.5 for $|J_z|\lesssim 0.01$. In contrast, the value of $J_2^{**}$ increases quickly from $J_2^{*}$ at $|J_z|\lesssim 0.01$ as the interplane coupling is further reduced. In addition, the magnetic correlation length is shown to directly depend on the nematic order parameter and thus exhibits a sharp increase (or jump) upon entering the nematic phase. Our results are broadly consistent with the predictions based on itinerant electron models of the iron-based superconductors in the normal state and, thus, help substantiate a classical spin framework for providing a phenomenological description of their magnetic properties.

Figure 1

(a) Schematic phase diagram of Eq. (1) for a fixed value of $J_z>0$. The phase diagram is invariant under $J_1\to -J_1$ and antiferromagnetic order (AFM) $\to$ ferromagnetic order (FM). For $J_2/J_1>1/2$, the system enters the stripe phase via a simultaneous (magnetic and nematic) first-order transition when $J_2<J_2^{*}$, depicted in (b). For $J_2>J_2^{*}$, the magnetic and nematic order parameters, $M$ and $I$, develop at different temperatures, $T_M$ and $T_I$, respectively. Near the branching point, $J_2\sim J_2^{*}$, one (or both) of the split transitions may remain first-order, while for $J_2\gtrsim J_2^{*}$ the transitions are predominantly second-order, as shown in (d). In the case where the magnetic transition is first-order, as shown in (c), one has a metanematic transition whereby the nematic order parameter jumps from a finite value to a higher one.

Figure 2

(a) The $n=3$ term coming from expanding $S_{\lambda }$. (b) The $m=3$ diagram in the expansion of the integrand for the momentum dependent susceptibility.

Figure 3

Dyson equation for the renormalized spin propagator.

Figure 4

Dyson equation for the effective constraint-field propagator.

Figure 5

Self-consistent equations for the self-energy and the polarization. Note that the bold lines on the right-hand sides also include the self-energy and the polarization.

Figure 6

Magnetic order parameter squared for the unfrustrated 3D FM: $J_z=-1.0, J_2=0$. Both panels show finite-size curves with $L=100–400$ where darker curves indicate larger $L$. (b) shows a zoom-in in (a) near the phase transition point. A vertical red line indicating a jump in the order parameter is drawn for the largest $L$ curve.

Figure 7

Order parameters for $J_z=-1$, and (a) $J_2=0.51$, (b) $J_2=2.0$ for various system sizes $L=100–400$. Darker curves indicate larger sizes. The stripe magnetization squared is shown as positive values, while the nematic order parameter $I$ is shown as negative values. The red vertical lines indicate the temperature where the magnetization curves of the largest system size turns back. We take such an overshoot to indicate a first-order phase transition and the red vertical line to mark the associated $T_c$.

Figure 8

Order parameters for $J_z=-1.0$ and $J_2=0.495$. The nematic order parameter $I$ is shown as negative values. The dominant magnetization is shown as positive values. An intermediate regime is clearly visible with nonzero nematic order parameter $I$.

Figure 9

Phase diagrams for (a) $J_z=-1.0$ and (b) $J_z=-0.1$. Disorder-FM (solid black), disorder-stripe (dashed red), and FM-stripe (dotted blue) phase boundaries.

Figure 10

Order parameters for $J_z=-0.0001$. (a) $J_2=0.8$, (b) 0.513, (c) 0.506, (d) 0.5054, (e) 0.503, and (f) 0.5005. The different curves are for different system sizes, the darkest being the largest system size. The nematic order parameter $I$ is shown as negative values while the stripe magnetic order parameter squared is shown as positive values. The orange thick lines indicate temperature intervals of very slow convergence in solving the nematic bond equations.

Figure 11

Phase diagram for $J_z=-0.0001$. The inset shows a blow-up of the region where all lines meet. The black line is the disorder-FM phase transition. The orange line is the disorder-nematic phase transition where the nematic phase has no long-range stripe magnetic order. The green line is where the magnetic order sets in. The dashed red line is the simultaneous Id/Md phase transition. The dotted blue line is the phase boundary between the ordered FM and the ordered stripe magnetic phases.

Figure 12

The nematic $T_c$ vs $J_2$ for the two dimensional model ($|J_z|=0$). The inset shows a zoom in on the region $J_2\approx 0.5$.

Figure 13

The magnetic $T_c$ vs. $|J_z|$ for $J_2=1$. The values of $T_c\mathrm{s}$ are here obtained by the Kouvel-Fisher analysis locating the temperature where the spin-spin correlation length diverges as described in conjunction with Fig. 20 in Sec. 6.

Figure 14

Order parameter curves for $J_z=-0.01$, (a) $J_2=0.505$, (b) 0.52, and (c) 0.55 for various system sizes.

Figure 15

Order parameters for $J_z=-0.002$, (a) $J_2=0.505$, (b) 0.51, (c) 0.55, (d) 0.9, (e) 1.1, and (f) $J_2=1.2$ for various system sizes.

Figure 16

$J_2^{*}$ and $J_2^{**}$ as functions of interplane coupling $|J_z|$.

Figure 17

${\chi }_{\vec{q}}$ in the $(q_x,q_y)$ plane for $J_2=0.6, J_z=-0.0001$ at a temperature $T=0.31$ just below the nematic phase transition.

Figure 18

Inverse correlation lengths vs $T$ for $J_z=-0.0001$ and $J_2=1$ for a lattice with $2400\times 2400\times 48$ sites. Shown are $0.01{\xi }_z^{-1}$ (green dashed), ${\xi }_2^{-1}$ (red dot-dashed), ${\xi }_1^{-1}$ (black solid), and the nematic order parameter I (light blue solid).

Figure 19

Scaled spin correlation lengths and susceptibility peak heights vs. $I$ close to the nematic phase transition for $J_z=-0.0001$ and $J_2=1$. The quantities specified in the legends are plotted as the y values. The dashed lines have slopes 4.46 and 8.79. For $J_z=-0.0001$ and $J_2=1$, Eq. (30) gives $\alpha =8.55$.

Figure 20

Kouvel-Fisher plot showing $K\left(T\right)\equiv -{\left(\frac{\partial ln{\xi }_1}{\partial T}\right)}^{-1}$ vs $T$. $J_z=-0.0001$ and $J_2=1$. The different curves are for different system sizes and aspect ratios with grey scale coding such that the darkest curves correspond to the largest system sizes. Two families of system sizes are used $L\times L\times L/50$ with $L=400-2400$, and $L\times L\times L/20$ with $L=400–1800$. The red curve is the best fit to a quadratic polynomial for the largest system size $2400\times 2400\times 48$ restricted to the temperature region $T\in [0.587,0.61]$.