$s+is$ state with broken time-reversal symmetry in Fe-based superconductors

We analyze the evolution of the superconducting gap structure in strongly hole-doped Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ between $x=1$ and $x\sim 0.4$ (optimal doping). In the latter case, the pairing state is most likely $s\pm$, with different gap signs on hole and electron pockets, but with the same signs of the gap on the two $\Gamma$-centered hole pockets (a $++$ state on hole pockets). In a pure KFe${}_2$As${}_2$ ($x=1$), which has only hole pockets, laser ARPES data suggested another $s\pm$ state, in which the gap changes sign between hole pockets (a $+-$ state). We analyze how a $++$ gap transforms into a $+-$ gap as $x\to 1$. We found that this transformation occurs via an intermediate $s+is$ state in which the gaps on the two hole pockets differ in phase by $\varphi$, which gradually involves from $\varphi =\pi$ (the $+-$ state) to $\varphi =0$ (the $++$ state). This state breaks time-reversal symmetry and has huge potential for applications. We compute the dispersion of collective excitations and show that two different Leggett-type phase modes soften at the two end points of the time-reversal-symmetry-breaking state.

Figure 1

Variation of the relative phase $\varphi$ of the gaps on two hole pockets with $u_{he}$. This phase is zero for $u_{he}>u_{he}^{\max }$, but becomes nonzero at smaller $u_{he}$ and eventually reaches $\varphi =\pm \pi$ at $u_{he}=0$. When $\left|\varphi \right|$ is between 0 and $\pi$, it can be either positive or negative, and the choice breaks $Z_2$ time-reversal symmetry. The width of the TRSB region is controlled by interpocket hole-hole interaction $u_{hh}$ and increases when $u_{hh}$ gets larger.

Figure 2

Qualitative phase diagram for Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ at $x\le 1$. We model the doping dependence by varying the ratio of interpocket electron-hole and hole-hole interactions $u_{he}/u_{hh}$ which roughly scales as $1-x$. The $+-$ state has gaps of opposite signs on the two GCPs and no gap on electron pockets, the $++$ state is an ordinary $s\pm$ state in which the gaps have opposite signs on hole and electron pockets, and between them is the TRSB state. The gap structures are pictorially presented inside each region by vectors placed inside the circles. The magnitudes of the vectors represent $|{\Delta }_i|$ and the angles represent the phases. Cases (a) and (b) are for equal and nonequal intrapocket interactions ($u_{h_1}$ and $u_{h_2}$) for the two hole pockets, respectively. For (a), the TRSB state starts right at $T_c$ and extends into a finite range at $T=0$. For (b), the TRSB region splits off from the $T_c$ line and is only accessible at lower temperatures, while immediately below $T_c$ the $+-$ state gradually evolves into the $++$ state as $u_{he}/u_{hh}$ increases.

Figure 3

The diagrammatic representation of the equations for dispersion of collective modes. The equations for other $\delta {\Delta }_j$ are similar to the one for $\delta {\Delta }_{h_1}$ and are not shown. Wavy lines: interactions $u_{ij}$; chainsaw line: Coulomb interaction $V_q$. The bare vertices are not shown.

Figure 4

Doping evolution of the frequencies of the relevant collective modes at $q=0$. The plasmon mode frequency vanishes at $q=0$ for all $u_{he}$. Mode $M_1$ describes relative phase fluctuation of the two hole gaps. It softens (Refs. 20, 21, and 23) at the transition point between $++$ and TRSB states (at $u_{he}=u_{he}^{\max }$, indicated by the arrow). Mode $M_2$ describes coupled antisymmetric phase fluctuation of the two hole gaps and longitudinal fluctuation of the electron gap. This mode lies below twice the energy of the electron gap and softens at the boundary between TRSB and $+-$ states, at $u_{he}^{\min }=0$. Numerically, the energy of the $M_2$ mode becomes small already for $u_{he}\le u_{he}^{\max }$ because the electron gap rapidly decreases with decreasing $u_{he}$. The circle represents the case discussed in Ref. 23.

Figure 5

The $+-$ to $++$ transition at $T_c$, with increasing $u_{he}$ for the case when the two hole pockets are not equivalent and the interaction $u_{hh}$ has $\cos 4\theta$ angular dependence. The solutions of the linearized gap equations are shown for $u_{he}=0.02$, $u_{he}=0.04$, $u_{he}=0.06$, $u_{he}=0.09$ from left to right and top to bottom, respectively. Other parameters are $\alpha =0.05$, $u_{h_1}=0.2$, $u_{h_2}=0.26$, $u_{hh}=0.2$. Note how one of the hole gap's average value gets smaller as $u_{he}$ increases, goes through zero, and reappears with the opposite sign and with small angular variation at larger $u_{he}$. We expect such behavior immediately below the $T_c$ line in Ba${}_{1-x}$K${}_x$Fe${}_2$As${}_2$ as $x$ decreases from one.

Figure 6

The gap evolution in the TRSB state for angle-dependent interactions and two equivalent GCPs, in a situation when the gap in the $+-$ state has accidental nodes and the gap in the $++$ state is nodeless. The nodes disappear once the system enters the TRSB state, but deep minima (shown by arrows) remain in some range of $\varphi$ (or equivalently $u_{he}$).

Figure 7

Temperatures ($T$) at which the prefactors for the quadratic terms in Ginzburg-Landau expansion for the two critical fields change sign. The parameter $c$ measures the deviation of hole-electron interaction $u_{he}$ from the critical value ($=u_{hh}/\sqrt{2}$). Left panel: two equivalent hole pockets ($y=0$). In this situation, the condensation of one critical field leads to $+-$ order, the condensation of the other leads to $++$ order. The two lines cross at the critical $u_{he}$. In the presence of mode-mode coupling, the emergence of one order tends to prevent the emergence of the other, and the actual temperature, at which the second order emerges, gets smaller (black line). We found (see text) that below the black line the two orders lock into the TRSB state. Right panel: nonequivalent hole pockets ($y=\frac{1}{8}$). The eigenfunctions reduce to pure $+-$ and $++$ only at large $\left|c\right|$, while in the region labeled “mixed” the system gradually transforms form the $+-$ to $++$ order with one of the hole gaps going through zero in-between. The lines at which the prefactors for the quadratic terms vanish now do not cross. Due to mode-mode coupling, the order which appears first now induces second order, i.e., both are present immediately below the actual $T_c$ line, with a relative phase of 0 or $\pi$, i.e., time-reversal symmetry is not broken at $T_c$. The TRSB state still emerges, but at a lower $T$ (below the black line).

Figure 8

Diagrammatic representation of the set of nonlinear equations for the gaps ${\Delta }_{h_1}$ and ${\Delta }_{e_1}$ (viewed as anomalous self-energies). In our case, ${\Delta }_{e_1}={\Delta }_{e_2}\equiv {\Delta }_e$. The equation for ${\Delta }_{h_2}$ is similar and not shown. The double-headed arrows correspond to the anomalous Green's functions. The single and double solid lines and the single and double dotted lines are anomalous Green's functions for fermions near the hole pockets ($h_{1,2}$) and near electron pockets ($e_{1,2}$), respectively.

Figure 9

Diagrammatic representation of the coupled equations for fluctuations of the total density $\delta \rho$ and the SC order parameter $\delta \Delta$ (and $\delta {\Delta }^{*}$) for the one-band case. The solid and dotted wavy lines represent the pairing interaction $u<0$ and unscreened Coulomb interaction $V_q$. The lines with single and double arrows represent the normal (G) and anomalous (F) Green's functions. The coupling is due to GF terms which are nonzero when $\vec{q},\Omega \ne 0$.