Dynamics of order parameters near stationary states in superconductors with a charge-density wave

We consider a simple model of a quasi-one-dimensional conductor in which two order parameters (OP) may coexist, i.e., the superconducting OP $\Delta$ and the OP $W$ that characterize the amplitude of a charge-density wave (CDW). In the mean field approximation, we present equations for the matrix Green's functions $G_{ik}$, where the first subscript $i$ relates to the one of the two Fermi sheets and the other, $k$, operates in the Gor'kov-Nambu space. Using the solutions of these equations, we find stationary states for different values of the parameter describing the curvature of the Fermi surface $\mu$, which can be varied, e.g., by doping. It is established, in particular, that in the interval ${\mu }_1<\mu <{\mu }_2$, the self-consistency equations have a solution for coexisting OPs $\Delta$ and $W$. However, this solution corresponds to a saddle point in the energy functional $\Phi (\Delta ,W)$, i.e., it is unstable. Stable states are (1) the W state, i.e., the state with the CDW ($W\ne 0$, $\Delta =0$) at $\mu <{\mu }_2$ and (2) the S state, i.e., the purely superconducting state ($\Delta \ne 0$, $W=0$) at ${\mu }_1<\mu$. These states correspond to minima of $\Phi$. At $\mu <{\mu }_0=({\mu }_1+{\mu }_2)/2$, the state (1) corresponds to a global minimum, and at ${\mu }_0<\mu$, the state (2) has a lower energy, i.e., only the superconducting state survives at large $\mu$. We study the dynamics of the variations $\delta \Delta$ and $\delta W$ from these states in the collisionless limit. It is characterized by two modes of oscillations, the fast and the slow one. The fast mode is analogous to damped oscillations in conventional superconductors. The frequency of slow modes depends on the curvature $\mu$ and is much smaller than $2\Delta /\hslash$ if the coupling constants for superconductivity and CDW are close to each other. The considered model can be applied to high-$T_{\text{c}}$ superconductors where the parts of the Fermi surface near the “hot” spots may be regarded as the considered two Fermi sheets. We also discuss relation of the considered model to the simplest model for Fe-based pnictides.

Figure 1

The considered quasi-one-dimensional model.

Figure 2

The order parameters $\Delta$ (blue solid line) and $W$ (green solid line) on the curvature $\mu$ at $T=0$ as follows from the nontrivial solutions of the self-consistency equations (16) and (17). In (a) the interaction constants are taken to be close, while in (b) ${\lambda }_{\text{CDW}}\gg {\lambda }_{\text{SC}}$. Moreover, the short-dashed lines show the trivial solutions of Eqs. (16) and (17) for the corresponding order parameter. Note that $\mu$, $W$, and $\Delta$ are measured in $W_0$.

Figure 3

The order parameters $\Delta$ (blue solid line) and $W$ (dashed green line) on the curvature $\mu$ at $T=0$ in the case if one of them vanishes. Clearly, as follows from the self-consistency equations (16) and (17), $\Delta$ is independent of $\mu$ in case $W=0$, while, if $\Delta =0$, $W$ resembles the LOFF-type dependence. Note that $\mu$, $W$, and $\Delta$ are measured in $W_0$.

Figure 4

Panels (a)–(c) show the dependence of the order parameters on temperature for different values of $r=exp\left(\gamma \right)$: in (a) $r=1.2$, in (b) $r=exp\left(1\right)$, and in (c) $r=5.0$. The solid lines correspond to superconducting order parameter while the dashed to charge-density wave. The color encoding is as follows. Red and blue lines show the solutions of the self-consistency equations for two values of the curvature $\mu$, the one being close to ${\mu }_1$ and the other to ${\mu }_2$. The solid black line shows the solution for $\Delta \left(T\right)$ in case $W=0$ being independent on $\mu$; the dashed black and green lines show the solution for $W\left(T\right)$ for the two values of the curvature $\mu$, correspondingly to $\mu$ close to ${\mu }_1$ (black) and to $\mu$ close to ${\mu }_2$ (green). Panel (d) shows the dependence of ${\mu }_1$ (solid lines) and ${\mu }_2$ (dashed lines) on temperature $T$. Here, black lines correspond to $r=1.2$, blue lines to $r=exp\left(1\right)$, and red lines to $r=5.0$. Note that the temperature $T$ is normalized to $T_{\text{SC}}$ and $\mu$ is measured in $W_0$. One can see that having fixed $\mu$ in the “coexistence” region one can exit it after a certain value of temperature, as is clear from panel (d). Thus, there are abrupt jumps in the temperature dependencies of the order parameters. The curves for $W$ at $\Delta =0$ in panels (b) and (c) for $\mu$ close to ${\mu }_2$ result from the fact that, in this region, $W$ is a multivalued function of $\mu$; a LOFF-type dependence is found. The values of ${\mu }_1$ and ${\mu }_2$ are, respectively, in (a) $0.385$ and $0.4$; in (b) $0.6$ and $0.74$; in (c) $0.6$ and $0.82$.

Figure 5

Dependence of ${\mu }_1$ and ${\mu }_2$ on $r=exp\left(\gamma \right)=W_0/{\Delta }_0$. The short dashed vertical line marks the value $\gamma =1$ where the function ${\mu }_1\left(r\right)$ has a maximum. The curvature $\mu$ is measured in $W_0$.

Figure 6

Two situations for $\mu <{\mu }_0$ (a) where the minimum of $\Phi$ at $W_0$ is lower than the one at ${\Delta }_0$, and $\mu >{\mu }_0$ (b) where the situation is reverted. In the case (a) the state with $W=W_0\ne 0$ and $\Delta =0$ is favored, while in case (b) one has $\Delta ={\Delta }_0\ne 0$ and $W=0$.

Figure 7

The case $\Delta ={\Delta }_{\text{X}}$ and $W=W_{\text{X}}$. Since the quadratic form corresponding to ${\delta }^2\Phi$ is negative definite at ${\Gamma }_{\text{X}}=({\Delta }_{\text{X}},W_{\text{X}})$, this point is a saddle point.

Figure 8

Qualitative phase diagram for the transition from the CDW into the SC state. The free energy has a minimum at ${\Gamma }_{\text{W}}$ for $\mu <{\mu }_2$ and a saddle point for $\mu >{\mu }_2$, whereas at ${\Gamma }_{\text{S}}$ the minimum exists for $\mu >{\mu }_1$ and a saddle point for $\mu <{\mu }_1$. In the range ${\mu }_1<\mu <{\mu }_2$, the free energy has two minima with one of them being lower up to a value $\mu ={\mu }_0$ and higher after ${\mu }_0$ (cf. Fig 6). Thus, at $\mu ={\mu }_0$, transition takes place from the CDW into the SC state if increasing $\mu$, or vice versa if decreasing. The transition is first order since the point ${\Gamma }_{\text{X}}$ is unstable being a saddle point (cf. Fig. 7).

Figure 9

Near the point ${\Gamma }_{\text{S}}=({\Delta }_0,0)$, the blue line denotes the fast oscillating and damped superconducting order parameter with a high frequency ${\omega }_{\text{high}}=2{\Delta }_0/\hslash$, and the green line denotes the charge-density wave order parameter oscillating at a much lower frequency ${\omega }_{\text{low}}\simeq \sqrt{2({\mu }^2-{\mu }_1^2)}/\hslash$. Near the point ${\Gamma }_{\text{W}}=(0,W_0)$, the behavior is inverted replacing ${\Delta }_0\leftrightarrow W_0$, accordingly adapting the frequencies as ${\omega }_{\text{high}}=2W_0/\hslash$ and ${\omega }_{\text{low}}\simeq \sqrt{2({\mu }_2^2-{\mu }^2)}/\hslash$, respectively.