Induced spin-triplet pairing in the coexistence state of antiferromagnetism and singlet superconductivity: Collective modes and microscopic properties

The close interplay between superconductivity and antiferromagnetism in several quantum materials can lead to the appearance of an unusual thermodynamic state in which both orders coexist microscopically, despite their competing nature. A hallmark of this coexistence state is the emergence of a spin-triplet superconducting gap component, called a $\pi$ triplet, which is spatially modulated by the antiferromagnetic wave vector, reminiscent of a pair density wave. In this paper, we investigate the impact of these $\pi$-triplet degrees of freedom on the phase diagram of a system with competing antiferromagnetic and superconducting orders. Although we focus on a microscopic two-band model that has been widely employed in studies of iron pnictides, most of our results follow from a Ginzburg-Landau analysis, and as such should be applicable to other systems of interest, such as cuprates and heavy fermion materials. The Ginzburg-Landau functional reveals not only that the $\pi$-triplet gap amplitude couples trilinearly with the singlet gap amplitude and the staggered magnetization magnitude but also that the $\pi$-triplet $d$-vector couples linearly with the magnetization direction. While in the mean-field level this coupling forces the $d$-vector to align parallel or antiparallel to the magnetization, in the fluctuation regime it promotes two additional collective modes—a Goldstone mode related to the precession of the $d$-vector around the magnetization and a massive mode, related to the relative angle between the two vectors, which is nearly degenerate with a Leggett-like mode associated with the phase difference between the singlet and triplet gaps. We also investigate the impact of magnetic fluctuations on the superconducting-antiferromagnetic phase diagram, showing that due to their coupling with the $\pi$-triplet order parameter the coexistence region is enhanced. This effect stems from the fact that the $\pi$-triplet degrees of freedom promote an effective attraction between the antiferromagnetic and singlet superconducting degrees of freedom, highlighting the complex interplay between these two orders, which goes beyond mere competition for the same electronic states.

Figure 1

Schematics of the staggered magnetization $\mathit{M}$ and the triplet unit vector $\widehat{\mathit{d}}$. We choose a spherical coordinate system with the $\rho$ vector along $\mathit{M}$, i.e., $\mathit{M}=M{\widehat{e}}_{\rho }$, and define the angle between the staggered magnetization and the $d$-vector as ${\alpha }_{\mathrm{m}d}$. The free-energy density does not depend on the angle $\beta$; it only depends on $\widehat{\mathit{M}}·\widehat{\mathit{d}}=cos{\alpha }_{\mathrm{m}d}$. Therefore, $f$ is invariant with respect to rotations of the $d$-vector around the staggered magnetization vector $\mathit{M}$.

Figure 2

Phase diagram in the (a) ($T,{\delta }_0$) and (b) ($T,{\delta }_2$) planes. The green (orange) curve is the singlet SC (AFM) critical temperature. $M_0$ and ${\mathrm{\Delta }}_{\mathrm{s}0}$ are both nonzero in the coexistence region located between the green and the orange curves. Therefore, in this region, $|{\mathrm{\Delta }}_{\mathrm{t}0}|\propto M_0\left|{\mathrm{\Delta }}_{\mathrm{s}0}\right|$ is also nonzero. The black dots denote the tetracritical points. Their coordinates are (a) $({\delta }_0^{*},T^{*})=(1.669,1)T_{\mathrm{c},0}$ and (b) $({\delta }_2^{*},T^{*})=(3.105,1)T_{\mathrm{c},0}$.

Figure 3

The behaviors of the singlet SC, triplet SC, and AFM order parameters (${\mathrm{\Delta }}_{\mathrm{s}0}$, ${\mathrm{\Delta }}_{\mathrm{t}0}$, and $M_0$, respectively) as a function of temperature $T$ [(a) fixed ${\delta }_0$ ] and ${\delta }_0$ [(b) fixed temperature] in the phase diagram of Fig. 2. The condensation energies of the pure SC phase $f_s$, of the pure magnetic phase $f_m$, and of the AFM-SC coexistence phase $f_{\mathrm{coex}}$ are shown in the insets. In order to show all quantities in the same plots, we multiplied ${\mathrm{\Delta }}_{\mathrm{t}0}$ by multiplicative factors, as indicated in the figure.

Figure 4

(a) Fluctuation-corrected phase diagram in the ($T,{\delta }_0$) plane. The black, light gray, dark gray, and white regions are the pure AFM, pure SC, coexistence AFM-SC, and normal phases, respectively. The green and orange curves represent the phase diagram without fluctuations [same lines as in Fig. 2]. Clearly, magnetic fluctuations shrink the AFM region. The “new” multicritical point, represented by the red triangle, is still a tetracritical point. (b) The AFM order parameter for the same parameters of panel (a) and ${\delta }_0=1.20T_{\mathrm{c},0}$ with (dashed line) and without (solid line) the inclusion of magnetic fluctuations. The solid circles denote the positions of the SC critical temperatures. (c) The effect of the triplet SC order parameter on the boundaries of the coexistence region: the greater the value of $a_{\mathrm{t}}$, the smaller ${\mathrm{\Delta }}_{\mathrm{t}}$ becomes. Solid curves are for $a_{\mathrm{t}}\to \infty$ (${\mathrm{\Delta }}_{\mathrm{t}}\equiv 0$), dash-dotted curves are for $a_{\mathrm{t}}=0.2{\mathrm{meV}}^{-1}$ [as in panel (a)], and dashed curves are for $a_{\mathrm{t}}=0.02{\mathrm{meV}}^{-1}$.