How Many Quantum Phase Transitions Exist Inside the Superconducting Dome of the Iron Pnictides?

Recent experiments on two iron-pnictide families suggest the existence of a single quantum phase transition inside the superconducting dome despite the fact that two separate transition lines—magnetic and nematic—cross the superconducting dome at $T_c$. Here we argue that these two observations are actually consistent. We show, using a microscopic model, that each order coexists with superconductivity for a wide range of parameters, and both transition lines continue into the superconducting dome below $T_c$. However, at some $T_{\mathrm{merge}}<T_c$, the two transitions merge and continue down to $T=0$ as a single simultaneous first-order nematic-magnetic transition. We show that superconductivity has a profound effect on the character of this first-order transition, rendering it weakly first order and allowing strong fluctuations to exist near the quantum phase transition.

Figure 1

Schematic phase diagram summarizing our main results. The spin-density wave and nematic transition lines, $T_m$ and $T_n$, separately cross the superconducting transition line, $T_c$, and coexist with superconductivity immediately below $T_c$. As temperature is lowered, one of the transitions becomes first order (dashed line). At a smaller $T=T_{\mathrm{merge}}$ the two transition lines merge onto a single simultaneous weakly first-order transition, which persists down to $T=0$ and gives rise to a single QPT inside the dome. The back bending of the lines below $T_c$ may or may not take place (see Refs. 4, 14, 15). Inset: schematic Fermi surface.

Figure 2

(a) The regions of coexistence between mean-field SDW and superconductivity (light-blue region) and nematicity and superconductivity (red region) in the (${\delta }_0$, ${\delta }_2$) parameter space, in $d=2$. In the red region both SDW and nematic order coexist with SC order. (b) Color plot of $\alpha =u_m/g_m$ inside the SC state for different ${\delta }_0/\Delta$ and ${\delta }_2/\Delta$ ($\Delta$ is the SC gap).

Figure 3

The effective action $S_{\mathrm{eff}}[\phi ]$ at $T=0$ and $d_{\mathrm{eff}}=3$, as a function of the nematic order parameter $\phi$ for various ${\overline{a}}_m$, which measure the distance to the pure SDW $T=0$ transition. From top to bottom, ${\overline{a}}_m(4{\pi }^2/9u_m^2)=0.0345$, 0.0320, 0.0310, 0.03034, and 0.0295. The dashed line $\phi ={\phi }_0$ separates the region where nematic order does not induce magnetic order ($|\phi |<{\phi }_0$) from the regions where magnetic order is simultaneously induced ($|\phi |>{\phi }_0$). We set $u_m/g_m=5$.