Possible bicollinear nematic state with monoclinic lattice distortions in iron telluride compounds

Iron telluride (FeTe) is known to display bicollinear magnetic order at low temperatures together with a monoclinic lattice distortion. Because the bicollinear order can involve two different wave vectors $(\pi /2,\pi /2)$ and $(\pi /2,-\pi /2)$, symmetry considerations allow for the possible stabilization of a nematic state with short-range bicollinear order coupled to monoclinic lattice distortions at a $T_S$ higher than the temperature $T_N$ where long-range bicollinear order fully develops. As a concrete example, the three-orbital spin-fermion model for iron telluride is studied with an additional coupling ${\tilde{\lambda }}_{12}$ between the monoclinic lattice strain and an orbital-nematic order parameter with $B_{2g}$ symmetry. Monte Carlo simulations show that with increasing ${\tilde{\lambda }}_{12}$ the first-order transition characteristic of FeTe splits and bicollinear nematicity is stabilized in a (narrow) temperature range. In this new regime, the lattice is monoclinically distorted and short-range spin and orbital order breaks rotational invariance. A discussion of possible realizations of this exotic state is provided.

Figure 1

(a) The bicollinear antiferromagnetic spin order with wave vector $(\pi /2,-\pi /2)$. (b) Same as (a) but for the state lattice rotated by 90 degrees with wave vector $(\pi /2,\pi /2)$. (c) Schematic drawing of an iron atom at site $\mathbf{i}$ (filled symbol) and its four Te neighbors (open symbols), projected in the $x-y$ plane in their equilibrium position. The distances ${\delta }_{\mathbf{i},\nu }$ between the irons at site $\mathbf{i}$ and its four neighboring Te atoms are indicated as well. The localized spin ${\mathbf{S}}_{\mathbf{i}}$ is also sketched. (d) Schematic drawing of the Fe lattice equilibrium position in the tetragonal phase (black symbols and lines) and in the monoclinic phase (red symbols and lines). Four Fe atoms are indicated with filled symbols and labeled by their lattice site index.

Figure 2

Diagram of the PTCA setup used to sample the local spin and lattice variables. The lattice is divided into four quadrants and each of four processors generates traveling clusters (indicated with $8\times 8$ squares) and proposes updates for the sites (indicated by small open circles) inside one quadrant.

Figure 3

Diagram of the PTCA setup used to sample the global lattice distortion variables. The lattice is divided into sixteen clusters. Each of the four processors diagonalizes four of the clusters.

Figure 4

Magnetic susceptibility ${\chi }_S$ (squares) and monoclinic lattice susceptibility ${\chi }_{{\delta }_M}$ (circles) evaluated using the PTCA algorithm at ${\tilde{\lambda }}_{12}=1$ employing a $32\times 32$ sites cluster. In this plot, and other plots of susceptibilities shown below, the fluctuations between subsequent temperatures are more indicative of the error bars than the intrinsic errors bars of individual points, which for this reason are not shown.

Figure 5

Magnetic spin structure factor $S(\pi /2,\pi /2)$ (squares) and monoclinic lattice order parameter ${\delta }_M$ (circles) evaluated using the PTCA algorithm for ${\tilde{\lambda }}_{12}=1$ on a $32\times 32$ sites cluster.

Figure 6

(a) Magnetic susceptibility ${\chi }_{S(\pi /2,\pi /2)}$ (red squares) with a maximum at $T_N=165\mathrm{K}$ (dashed line), and the monoclinic lattice susceptibility ${\chi }_{{\delta }_M}$ (blue circles), spin-nematic susceptibility ${\chi }_{\mathrm{\Psi }}$ (orange diamonds), and orbital-nematic susceptibility ${\chi }_{\mathrm{\Phi }}$ (green triangles) all with a maximum at $T_S=193\mathrm{K}$. The susceptibilities were calculated at ${\tilde{\lambda }}_{12}=1$ using $32\times 32$ lattices. (b) Monte Carlo measured order parameters associated to (a). Shown are the magnetic structure factor $S(\pi /2,\pi /2)$ (red squares), monoclinic lattice distortion ${\delta }_M$ (blue circles), spin-nematic order parameter ${\mathrm{\Psi }}_{NNN}$ (orange diamonds), and orbital-nematic order parameter ${\mathrm{\Phi }}_{B_{2g}}$ (green triangles). The transition temperatures were obtained from the susceptibilities in (a) and via numerical derivatives in (b). Both procedures give the same result.

Figure 7

(a) Susceptibilities associated with the magnetic spin structure factor $S(\pi /2,\pi /2)$ (squares) and with the monoclinic lattice distortion (circles) using ${\tilde{\lambda }}_{12}=0.85$ and a $32\times 32$ cluster. Solid lines are guides to the eye. (b) Spin structure factor $S(\pi /2,\pi /2)$ (squares) and monoclinic lattice order parameter ${\delta }_M$ (circles) for the same ${\tilde{\lambda }}_{12}$ and cluster size as in (a).

Figure 8

Phase diagram varying temperature and ${\tilde{\lambda }}_{12}$, for ${\tilde{g}}_{12}=0.24, J_{\mathrm{H}}=0.1$ eV, and $J_{\mathrm{NN}}=J_{\mathrm{NNN}}=0.0$. Note the narrow temperature width of stability of the bicollinear-nematic state, similarly as it occurs for the more standard $(\pi ,0)-(0,\pi )$ nematic state [27]. For values of ${\tilde{\lambda }}_{12}$ smaller than 0.75, our numerical accuracy does not allow us to distinguish between $T_N$ and $T_S$.