Zeeman Effect in Superconducting Two-Leg Ladders: Irrational Magnetization Plateaus and Exceeding the Pauli Limit

The effect of a parallel magnetic field on superconducting two-leg ladders is investigated numerically. The magnetization curve displays an irrational plateau at a magnetization equal to the hole density. Remarkably, its stability is fundamentally connected to the existence of a well-known magnetic resonant mode. Once the zero-field spin gap is suppressed by the field, pairs acquire a finite momentum characteristic of a Fulde-Ferrell-Larkin-Ovchinnikov phase. In addition, $S^z=0$ triplet superconducting correlations coexist with singlet ones above the irrational plateau. This provides a simple mechanism in which the Pauli limit is exceeded as suggested by recent experiments.

Figure 1

Magnetization curves at fixed hole density $\delta =0.0625$ for different system sizes. Three plateaus are visible for $m=0$, $\delta$, and $1-\delta$. $H_c$ is the superconducting critical field (see text) and results in a discontinuity of the slope in the magnetization curve. The undoped case is shown for comparison. Inset: finite-size scaling of the two critical fields corresponding to the boundaries of the $m=\delta$ plateau for various densities $\delta =0.0625$ (○), 0.125(□), 0.1875(⋄), 0.25(▵).

Figure 2

Plateau phase diagram in the ($\mathbf{H}$, $\delta$) plane. The expected Pauli limit $H_p$ (see text) is shown in dashed lines. The superconducting critical field $H_c$ is also shown (the error bars indicate the uncertainty in determining the transition).

Figure 3

Evolution of local densities for increasing magnetization $m$ in a system with 6 holes and $L=48$. The spin density is normalized by $2\delta /m$ so that spin and hole densities always have the same mean value.

Figure 4

Pairing energy ${\Delta }_p$ (see definition in text) as a function of $\mathbf{H}$ for $\delta =0.125$. The dashed line indicates the transition to the FFLO state ($q=0\to q=\pi m$).

Figure 5

(a) Superconducting correlations in real space showing $q=\pi m$ oscillations for rung-rung singlet and leg-leg triplet ($S^z=0$) correlations while rung-rung triplet ($S^z=+1$) correlations decay faster (continuous lines are fits). (b) Same in log-log scale showing the algebraic decaying with exponent $\alpha$ (straight lines are envelopes of fits).

Figure 6

Inverse of the superconducting correlations exponent $\alpha$ as a function of $\mathbf{H}$ for two densities. Singlet and $S^z=0$ triplet correlations coexist above the $m=\delta$ plateau. Error bars are estimates of uncertainties from fitting.