Superconducting and density-wave instabilities of low-dimensional conductors with a Zeeman coupling to a magnetic field

In the framework of the weak-coupling renormalization-group technique we examine the possible instabilities of the extended quasi-one-dimensional electron-gas model with both intrachain and interchain electron-electron interactions, including the influence of umklapp scattering and the coupling of spins to a magnetic field. In the limit of purely repulsive intrachain interactions, we confirm the passage from singlet $d$-wave-like superconductivity to an inhomogeneous Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state under magnetic field. The passage is accompanied by an anomalous increase of the upper critical field that scales with the antinesting distance from the quantum critical point joining superconductivity to antiferromagnetism in the phase diagram, as well as the strength of interactions. Adding weak repulsive interchain interactions promotes the passage from singlet to triplet $f$-wave superconductivity, which is expanded under field by the development of a triplet FFLO state with zero angular-momentum projection for the Cooper pairs. The connection between theory and experiment on the anomalous upper critical field in the Bechgaard salts is discussed.

Figure 1

The phase diagram of the quasi-1D electron-gas model with intrachain interactions as a function of magnetic field.

Figure 2

Temperature dependence of the susceptibilities in the normal phase of the superconducting sector of the phase diagram at $t_b^{\prime }/t_b=0.21$. (a): For intrachain couplings only, ${\mathfrak{g}}_i=0$ at zero and finite magnetic fields. (b): For finite interchain couplings ${\mathfrak{g}}_i\ne 0$ at $h/t_b=0.01$.

Figure 3

Low-temperature renormalized effective scattering amplitudes at $t_b^{\prime }/t_b=0.21$ for singlet Cooper pairing $g_{1\perp }(k_b^{\prime },-k_b^{\prime },k_b)+g_{2\perp }(k_b^{\prime },-k_b^{\prime },k_b)$ in the $k_b{k}_b^{\prime }$ plane for the normal phase of (a) SCd $\left(h=0\right)$, (b) umklapp amplitude $g_{3\perp }(k_b^{\prime },-k_b^{\prime },k_b)$ at $h=0$, (c) dFFLO $\left(h/t_b=0.01\right)$.

Figure 4

The variation of the normalized upper critical field $h_{c2}/T_c$ with temperature for (a) various $t_b^{\prime }$ and (b) different amplitudes of umklapp scattering $g_{3\perp }$ at $t_b^{\prime }/t_b=0.22$. The dashed line is mean-field result for the upper critical field in the Pauli limit.

Figure 5

Phase diagram of the extended quasi-electron-gas model at (a) $h=0$ and (b) $h/t_b=0.01$.

Figure 6

Low-temperature effective scattering amplitudes close to $T_c$ in the momentum space at $h/t_b=0.01$, for (a) $g_{2\perp }(k_b^{\prime },-k_b^{\prime },k_b)-g_{1\perp }(k_b^{\prime },-k_b^{\prime },k_b)$ for fFFLO and (b), (c) $g_{\parallel }(k_b^{\prime },-k_b^{\prime },k_b)$ for ${\mathrm{SCf}}_1$.

Figure 7

The evolution of $h_{c2}$ vs temperature for various $t_b^{\prime }$ and as a function of interchain coupling $\mathfrak{g}$.

Figure 8

The phase diagram in the $\mathfrak{g}\text{−}h$ plane for $t_b^{\prime }/t_b=0.2$.