Interplay between magnetism and superconductivity in the iron pnictides

We consider phase transitions and potential coexistence of spin-density-wave (SDW) magnetic order and extended $s$-wave $\left(s^{+}\right)$ superconducting order within a two-band itinerant model of iron pnictides in which SDW magnetism and $s^{+}$ superconductivity are competing orders. We show that depending on parameters, the transition between these two states is either first order or involves an intermediate phase in which the two orders coexist. We demonstrate that such coexistence is possible when SDW order is incommensurate.

Figure 1

(Color online) (a) Phase diagram for $T_s/T_c=1.5$ as a function of $\delta$ controlled by doping. Thick solid and dashed lines are second-order SDW and SC transitions at $T_s\left(\delta \right)$ and $T_c$, and dotted line—first-order transition between commensurate ${\text{SDW}}_0$ and $s^{+}$ SC. Thin lines—physically inaccessible transitions. The pure magnetic $T_s\left(\delta \right)$ line follows the curve of “paramagnetically limited superconductivity.” The superconducting $T_c$ is independent of $\delta$ in our model. Light lines, denoted ${\delta }_{\Delta }$ and ${\delta }_m$, are instability lines of SC and SDW states. (b) SDW and SC order parameters and (c) free energy for SDW, SC, and unstable mixed phases at $T/T_c=0.1$.

Figure 2

(Color online) (a) Same as in Fig. 1 but for $T_s/T_c=5$. An incommensurate ${\text{SDW}}_q$ order appears below $T_s\left(\delta \right)$ once it becomes smaller than $T_s^{\ast }=0.56T_s>T_c$. Below $T_c$, a new mixed phase appears in which incommensurate ${\text{SDW}}_q$ order coexists with SC. At small $T$, there is no ${\text{SDW}}_q$ state without superconductivity. (b), (c) The transitions into the mixed state are second order from a SC state and first order from a commensurate ${\text{SDW}}_0$ state. The free energy now shows that near $\delta /2\pi T_s=0.2$ a mixed state has lower energy than the two pure states.

Figure 3

(Color online) Same as in the previous two figures, but for intermediate $T_s/T_c=3$. Mixed phase appears only in a tiny region near the point where $T_c$ and $T_s\left(\delta ,q\right)$ cross. At smaller $T$, the system still displays a first-order transition (dotted line) between a commensurate ${\text{SDW}}_0$ state and a SC state.