Diverging nematic susceptibility, physical meaning of $T^{*}$ scale, and pseudogap in the spin fermion model for the pnictides

Using Monte Carlo simulations with a tunable uniaxial strain, the nematic susceptibility of the spin fermion model for the pnictides is calculated. The results are in excellent agreement with the experiments by Chu et al. [Science 337, 710 (2012)]. Via a Ginzburg-Landau analysis, our study suggests a nematicity in the spin fermion model primarily originating in magnetism, but with the lattice/orbital also playing a key role by boosting up critical temperatures and separating the structural $T_S$ and Néel $T_N$ transitions. At $T>T_S$, Curie-Weiss behavior is observed with a characteristic temperature $T^{*}$ being the $T_N$ of the purely electronic system. In this temperature regime, short-range magnetic order with wave vectors $(\pi ,0)\text{−}(0,\pi )$ induce local nematic fluctuations and a density-of-states pseudogap, compatible with several experiments. The present analysis relies on the study of a particular model for the iron superconductors; thus further studies are needed to conclusively establish the driver of nematicity in real materials.

Figure 1

Monte Carlo spin-nematic order parameter, at $\tilde{g}=0.16$ and $\tilde{\lambda }=0.12$. (a) $\Psi$ vs $T$ and $\epsilon$, measured at a fixed lattice distortion $\epsilon$ for each temperature (restricted MC). Shown are the $T^{*}$ (see text) and $T_S$ $\left(\sim T_N\right)$ temperatures. Data are for an $8\times 8$ cluster with TCA+TBC (PBC $8\times 8$ clusters with ED give similar results). Red points are the equilibrium values using unrestricted MC with ED and PBC $8\times 8$ clusters. (b) $\Psi$ vs $\epsilon$ at fixed temperatures, illustrating their nearly linear relation in unrestricted MC (red), and also the linear slopes of the restricted MC curves (green/blue) close to $T_S$. Results are obtained with ED/PBC $8\times 8$ clusters. Note that the number of green/blue points vastly outnumbers the number of red points, highlighting how much more demanding this effort has been than in Ref. [14].

Figure 2

Nematic susceptibility ${\chi }_s$ of the spin fermion model vs temperature $T$ (circles, triangles, and squares) obtained from Fig. 1, at the realistic couplings $\tilde{g}=0.16$ and $\tilde{\lambda }=0.12$ $(\alpha =\tilde{g}/a_0)$. Two MC techniques were employed: “PBC $L=8$” is the standard MC method with ED in the fermions at every step, using $8\times 8$ clusters with PBC. “TCA $L=8$” and “TCA $L=16$” correspond to the TCA+TBC method on $L\times L$ clusters. Size effects are small. Also shown are two GL fits: the light blue (thick) line displays the Curie-Weiss equation ${\chi }_s\approx \frac{\tilde{g}}{a_0(T-T^{*})}$, indicating a divergence at a lower temperature $T^{*}$, characteristic of the electronic sector alone. At $T\le T_S$, the lattice follows the electronic sector. The black (thin) line is Eq. (A40) with the $3T_S{\Psi }^2$ correction (see text) [29].

Figure 3

(a) The spin and (b) orbital nematic susceptibility obtained from Monte Carlo simulations for $\tilde{\alpha }=0.0011,\tilde{g}=0.16$, and $\tilde{\lambda }=0.12$. The continuous curves in light blue are the fittings obtained from the numerically guided Ginzburg-Landau approach.

Figure 4

(a) The MC structural transition temperature $T_S$ vs the orbital-lattice coupling $\tilde{\lambda }$, at fixed $\tilde{g}=0.16$. The continuous line is the fit in Eq. (5), from the GL equations. (b) Spin structure factor $S\left(\mathbf{k}\right)$ vs temperature $T$ for the two magnetic wave vectors of relevance. Results were obtained via MC simulations on PBC $8\times 8$ clusters. $T_{\mathrm{PG}}$ is the pseudogap temperature [Fig. 4]. (c) Density of states $N\left(\omega \right)$ (symmetrized) from unrestricted MC simulations on $8\times 8$ clusters $(\tilde{g}=0.16$; $\tilde{\lambda }=0.12)$, at various temperatures. Results at $T_S=158$ K are in red. $T_{\mathrm{PG}}\sim 174$ K (blue) is the crossover temperature where the pseudogap opens at the Fermi level (i.e., at $\omega -\mu =0.0)$ upon cooling.

Figure 5

Spin-nematic susceptibility ${\chi }_s$ vs temperature $T$ (red circles) obtained from Fig. 6 (at $\tilde{g}=0.16$ and $\tilde{\lambda }=0.84)$. The standard MC technique on an $8\times 8$ cluster with PBC was employed (involving ED of the fermions at every MC step). Also shown are two GL fits, as also employed in Fig. 2. The blue (thick) line indicates a divergence at a temperature $T^{*}$ (lower than $T_S)$ characteristic of the electronic sector alone. In the range $T\le T_S$, the lattice follows the electronic behavior. The black (thin) line and black tilted square points are a fit including the $3T_S{\Psi }^2$ correction (see text in the previous section of this appendix). The fitting parameters are $T^{*}=105$ K and $T_S=304$ K. The actual Néel temperature for $\tilde{g}=0.16$ and $\tilde{\lambda }=0.84$ is not shown.

Figure 6

Spin-nematic order parameter from the MC simulations, at $\tilde{g}=0.16$ and $\tilde{\lambda }=0.84$. (a) $\Psi$ vs $T$ and $\epsilon$, measured at a fixed lattice distortion $\epsilon$ for each temperature (restricted MC). Shown are the $T^{*}$ temperature (see text) and $T_S$. Results shown are for an $8\times 8$ cluster with TCA+TBC, but PBC $8\times 8$ clusters with ED give similar results. Red points are the equilibrium values using unrestricted MC with ED and PBC $8\times 8$ clusters. (b) MC results illustrating the relation between $\Psi$ and $\epsilon$ in unrestricted MC (red) and the restricted MC curves (green/blue), parametric with temperature. Results are obtained with ED/PBC $8\times 8$ clusters. Note that $\Psi$ vs $\epsilon$ (red squares) is no longer linear which is expected because Eq. (A18) is valid only for $\tilde{\lambda }=0$ (and approximately valid for small $\tilde{\lambda })$.