Global phase diagram for magnetism and lattice distortion of iron-pnictide materials

We study the global phase diagram of magnetic orders and lattice structure in the Fe-pnictide materials at zero temperature within one unified theory tuned by both electron doping and pressure. On the low doping and high-pressure side of the phase diagram, there is one single transition, which is described by a $z=2$ mean-field theory with very weak run-away flows; on the high doping and low-pressure side, the transition is expected to split to two transitions, with one O(3) spin-density wave transition followed by a $z=3$ quantum Ising transition at larger doping. The fluctuation of the strain field of the lattice will not affect the spin-density wave transition but will likely drive the Ising nematic order transition a mean-field transition through a linear coupling, as observed experimentally in ${\text{BaFe}}_{2-x}{\text{Co}}_x{\text{As}}_2$.

Figure 1

(Color online) The schematic phase diagram of Ginzburg-Landau mean-field theory in Eq. (1) plotted against $r=r_{\sigma }+r_{\varphi }$ and $\Delta r=r_{\varphi }-r_{\sigma }$. $r$ is linear with temperature $T$, while $\Delta r$ is tuned by anisotropy ratio $J_z/J_{in}$. When $\Delta r$ is small, the interaction between ${\vec{\varphi }}_1$ and ${\vec{\varphi }}_2$ induces a strong first-order transition, which corresponds to the undoped 122 materials with more isotropic electron kinetics; when $\Delta r$ is large, the transition is split into two transitions, with an Ising transition followed by an SDW transition at lower temperature, and this is the case in the 1111 materials with quasi-two-dimensional dispersions. The multicritical point $\Delta r_c$ is determined by $\tilde{u}$.

Figure 2

(Color online) (a) The schematic two-dimensional Fermi pockets of 1111 materials, the concentric circles are two very close hole pockets, the ovals are electron pockets. (b) and (d) The relative position of hole and electron pockets after translating by $\left(\pi ,0\right)$ and $\left(0,\pi \right)$ in the momentum space, in the low doping, critical doping, and high doping regimes.

Figure 3

(Color online) The plot of $u$ in Eq. (8) against anisotropic dispersion coefficient $\gamma$ between the isotropic limit $\gamma =0$ to anisotropic dispersion with $\gamma =1.95$.

Figure 4

(Color online) The solution of the RG equation (8). All three quartic perturbations decrease first then increase and finally become nonperturbative. The run-away flow is weaker than marginally relevant perturbations.

Figure 5

(Color online) The global phase diagram of quasi-two-dimensional materials, with applications for 1111 materials. The finite temperature transition is always split to an Ising nematic transition and a SDW transition. The zero-temperature transitions depend on the doping and pressure. In the high doping and low-pressure side, the transition is split to two, as observed in experiments; in the low doping and high-pressure side, there is one single transition very close to the mean-field solution. A multicritical point, where the three transition lines merge is identified, which is expected to be a strongly coupled fixed point.

Figure 6

Numerical results of $r_c$ of ${\vec{\varphi }}_a$ due to coupling to electrons. $x$ axis is the electron doping. The peak of this curve corresponds to the critical doping $x_c=7.6\%$, where electron and hole pockets touch each other after translating the hole pockets by the SDW wave vector. The two pockets intersect (separate) if doping is smaller (larger) than this critical doping. If $x>x_c$, the transition is split into two transitions by quantum fluctuations; if $x<x_c$, the transition is a $z=2$, $d=2$ transition with a very weak run-away flow in $2d$.

Figure 7

(Color online) The plot of the hole and electron pockets after translating the hole pockets by the SDW wave vector, at the critical doping $x=7.6\%$. The green circle is electron pocket located around $\left(0,\pi \right)$ and the blue one is electron pocket located around $\left(\pi ,0\right)$.