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 $J_2approx 0.5$ (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 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.