Superconducting pairing and density-wave instabilities in quasi-one-dimensional conductors

Using a renormalization group approach, we determine the phase diagram of an extended quasi-one-dimensional electron gas model that includes interchain hopping, nesting deviations, and both intrachain and interchain repulsive interactions. $d$-wave superconductivity, which dominates over the spin-density-wave (SDW) phase at large nesting deviations, becomes unstable to the benefit of a triplet $f$-wave phase for a weak repulsive interchain backscattering term $g_1^{\perp }>0$, despite the persistence of dominant SDW correlations in the normal state. Antiferromagnetism becomes unstable against the formation of a charge-density-wave state when $g_1^{\perp }$ exceeds some critical value. While these features persist when both Umklapp processes and interchain forward scattering $\left(g_2^{\perp }\right)$ are taken into account, the effect of $g_2^{\perp }$ alone is found to frustrate nearest-neighbor interchain $d$- and $f$-wave pairing and instead favor next-nearest-neighbor interchain singlet or triplet pairing. We argue that the close proximity of SDW and charge-density-wave phases, singlet $d$-wave, and triplet $f$-wave superconducting phases in the theoretical phase diagram provides a possible explanation for recent puzzling experimental findings in the Bechgaard salts, including the coexistence of SDW and charge-density-wave phases and the possibility of a triplet pairing in the superconducting phase.

Figure 1

RG phase diagram for the quasi-1D electron gas model at incommensurate filling $(g_3=0)$ with intrachain interactions $(g_j^{\perp }=0)$ and nesting deviations, parametrized by $t_{\perp }^{\prime }$. The circles indicate the transition temperature for SDW and the triangles that for SC$d$. All figures in this article are obtained using the bare intrachain interactions ${\tilde{g}}_1=0.32$, ${\tilde{g}}_2=0.64$ and the anisotropy ratio ${\Lambda }_0∕t_{\perp }=15$.

Figure 2

Temperature dependence of the susceptibilities in the normal phase $(g_3=0,g_j^{\perp }=0)$: (a) above the SDW phase $(t_{\perp }^{\prime }=0.04t_{\perp })$. The continuous (dashed) line corresponds to the SDW response at $q_{\perp }=\pi \left(0\right)$; the dashed-dotted line to $\mathrm{SC}d$ correlations. It is interesting to observe that the temperature scale $T\sim t_{\perp }$ below which the $q_{\perp }=0$ curve separates and levels off corresponds to the so-called single particle dimensionality crossover (Ref. 1). Our RG scheme thus captures this effect correctly. (b) Above the SC phase $(t_{\perp }^{\prime }=0.048t_{\perp })$. The continuous line corresponds to the SDW response with transverse modulation $\pi$, the dashed-dotted line to $\mathrm{SC}{d}_{x^2-y^2}$ $\left(\cos k_{\perp }\right)$, the dotted line to $\mathrm{SC}{d}_{xy}$ $\left(r\sin k_{\perp }\right)$, and the dashed line to $\mathrm{SC}g$ $\left(r\sin 2k_{\perp }\right)$ correlations.

Figure 3

$-{\tilde{\Gamma }}_S(-k_{\perp }^{\prime },k_{\perp }^{\prime },k_{\perp },-k_{\perp })$ in the SDW regime $(t_{\perp }^{\prime }=0.04t_{\perp })$ close to the divergence (a), in the $\mathrm{SC}d$ regime $(t_{\perp }^{\prime }=0.048t_{\perp })$ at an intermediate stage of the flow (b) and close to the divergence (c). The vertical scale is arbitrary.

Figure 4

Transition temperatures for different values of ${\tilde{g}}_1^{\perp }$, when ${\tilde{g}}_2^{\perp }=0$ and ${\tilde{g}}_3=0$: (a) ${\tilde{g}}_1^{\perp }=0$ (phase sequence $\mathrm{SDW}\to \mathrm{SC}d$, continuous line), $0.090$ ($\mathrm{SDW}\to \mathrm{SC}d$, dotted line), and $0.120$ ($\mathrm{CDW}\to \mathrm{SC}f$, dashed-dotted line). (b) ${\tilde{g}}_1^{\perp }=0.090$ ($\mathrm{SDW}\to \mathrm{SC}d$, dotted line) and $0.105$ [$\mathrm{SDW}(\to \mathrm{SC}d)\to \mathrm{SC}f$, dashed line]. Note that in the latter case, the $\mathrm{SC}d$ phase is extremely narrow. Circles indicate a SDW phase, squares a CDW phase, triangles $\mathrm{SC}d$ $\left(\cos k_{\perp }\right)$, and crosses $\mathrm{SC}f$ $\left(r\cos k_{\perp }\right)$.

Figure 5

Low temperature phases versus $g_1^{\perp }$, keeping $g_2^{\perp }=0$ and ${\tilde{g}}_3=0$. In the region below the dotted line, spin fluctuations dominate over charge fluctuations in the normal phase.

Figure 6

Temperature dependence of the susceptibilities in the normal phase above the $\mathrm{SC}d$ phase [(a) ${\tilde{g}}_1^{\perp }=0$ and (b) ${\tilde{g}}_1^{\perp }=0.08$] and above the $\mathrm{SC}f$ phase [(c) ${\tilde{g}}_1^{\perp }=0.10$], keeping ${\tilde{g}}_2^{\perp }=0$, ${\tilde{g}}_3=0$, and $t_{\perp }^{\prime }=0.056t_{\perp }$. The continuous line corresponds to SDW, the dotted line to CDW, the dashed line to $\mathrm{SC}d$, and the dashed-dotted line to $\mathrm{SC}f$ correlations.

Figure 7

Low temperature phases for $g_2^{\perp }$ varying, keeping $g_1^{\perp }=0$ and ${\tilde{g}}_3=0$. Circles indicate a SDW phase, triangles $\mathrm{SC}d$, and diamonds $\mathrm{SC}g$ $\left(r\sin 2k_{\perp }\right)$.

Figure 8

Transition temperatures for different values of ${\tilde{g}}_2^{\perp }$, if ${\tilde{g}}_1^{\perp }=0$ and ${\tilde{g}}_3=0$: ${\tilde{g}}_2^{\perp }=0$ ($\mathrm{SDW}\to \mathrm{SC}d$, continuous line), $0.16$ ($\mathrm{SDW}\to \mathrm{SC}d$, dashed line), and $0.20$ ($\mathrm{SDW}\to \mathrm{SC}g$, dotted line).

Figure 9

Low temperature phases for $g_1^{\perp }=g_2^{\perp }$, $g_3=0$. Circles indicate a SDW phase, black squares a CDW phase, triangles $\mathrm{SC}d$ $\left(\cos k_{\perp }\right)$, white diamonds $\mathrm{SC}g$ $\left(r\sin 2k_{\perp }\right)$, and crosses $\mathrm{SC}f$ $\left(r\cos k_{\perp }\right)$. Below the dotted line, spin fluctuations dominate over charge fluctuations in the normal phase.

Figure 10

Transition temperatures when Umklapp processes are taken into account. (a) Without interchain Umklapp scattering: ${\tilde{g}}_3=0$, ${\tilde{g}}_j^{\perp }=0$ ($\mathrm{SDW}\to \mathrm{SC}d$, continuous line), ${\tilde{g}}_3=0.02$, ${\tilde{g}}_j^{\perp }=0$ ($\mathrm{SDW}\to \mathrm{SC}d$, dashed line), ${\tilde{g}}_3=0.02$, ${\tilde{g}}_1^{\perp }={\tilde{g}}_2^{\perp }=0.1$, ${\tilde{g}}_3^{\perp }=0$, ($\mathrm{SDW}\to \mathrm{SC}f$, dotted line). (b) Effect of interchain Umklapp processes: ${\tilde{g}}_3=0.02$, ${\tilde{g}}_1^{\perp }={\tilde{g}}_2^{\perp }=0.1$, ${\tilde{g}}_3^{\perp }=0$ ($\mathrm{SDW}\to \mathrm{SC}f$, dotted line) and ${\tilde{g}}_3^{\perp }=\frac{{\tilde{g}}_1}{{\tilde{g}}_3}{\tilde{g}}_1^{\perp }$ ($\mathrm{SDW}\to \mathrm{SC}d\to \mathrm{SC}f$, continuous line).

Figure 11

Low temperature phases in the presence of Umklapp processes, with $g_3^{\perp }=0$ (a) and $g_3^{\perp }∕g_3=g_1^{\perp }∕g_1$ (b), taking ${\tilde{g}}_2^{\perp }={\tilde{g}}_1^{\perp }$. Circles indicate a site-SDW phase, squares a bond-CDW phase, triangles $\mathrm{SC}d$ $\left(\cos k_{\perp }\right)$ and crosses $\mathrm{SC}f$ $\left(r\cos k_{\perp }\right)$. Below the dotted lines, spin fluctuations dominate over charge fluctuations in the normal phase.

Figure 12

Temperature dependence of the susceptibilities in the normal phase above the $\mathrm{SC}d$ phase [(a) $t_{\perp }^{\prime }=0.152t_{\perp }$, ${\tilde{g}}_1^{\perp }=0.08$] and above the $\mathrm{SC}f$ phase [(b) $t_{\perp }^{\prime }=0.152t_{\perp }$ and (c) $t_{\perp }^{\prime }=0.176t_{\perp }$, both for ${\tilde{g}}_1^{\perp }=0.12$], taking ${\tilde{g}}_3=0.02$ and ${\tilde{g}}_2^{\perp }={\tilde{g}}_1^{\perp }=({\tilde{g}}_1∕{\tilde{g}}_3){\tilde{g}}_3^{\perp }$. The continuous line corresponds to SDW, the dotted line to CDW, the dashed line to $\mathrm{SC}d$ and the dashed-dotted line to $\mathrm{SC}f$ correlations.

Figure 13

Typical pictures of the interactions ${\Gamma }_S(-k_{\perp }^{\prime },k_{\perp }^{\prime },k_{\perp },-k_{\perp })$ (a) and ${\Gamma }_3(-k_{\perp }^{\prime },k_{\perp }^{\prime },k_{\perp },-k_{\perp })$ (b), for initial values ${\tilde{g}}_S={\tilde{g}}_C=0.2$, ${\tilde{g}}_3=0.02$, if the renormalization is restricted to the Peierls channel. The result for ${\Gamma }_C$ is identical to that for ${\Gamma }_S$.