Enhancement of electronic spin susceptibility in Pauli-limited unconventional superconductors

We calculate the wave-vector-dependent electronic spin susceptibility ${\chi }_{\alpha \beta }(\mathbf{q},{\mathbf{H}}_0)$ of a $d$-wave superconductor in uniform magnetic field ${\mathbf{H}}_0$ with Pauli pair-breaking. We find that the transverse component of the susceptibility tensor can be greater than its normal state value; the longitudinal component also slightly increases but in a very limited range of $q$'s. We identify several wave vectors $\{{\mathbf{q}}_{\perp },{\mathbf{q}}_{\parallel }\}$ that correspond to the maxima of either ${\chi }_{\perp }$ or ${\chi }_{\parallel }$. We compare our results with available data on the high-field phase in heavy-fermion CeCoIn${}_5$.

Figure 1

Phase diagram of a Pauli-limited superconductor, with the $Q$ phase [9] sketched. We consider a circular Fermi surface, and $d$-wave order parameter ${\Delta }_{\mathbf{k}}={\Delta }_0(T,H)sin2{\varphi }_{\mathbf{k}}$. The magnetic field has a large uniform component ${\mathbf{H}}_0$ and a spatially varying perturbation $\delta {\mathbf{H}}_{\mathbf{q}}$ with wave vector $\mathbf{q}$.

Figure 2

(a), (b) Magnetic field produces pockets of low-energy spin-down excitations near the nodes of the $d$-wave order parameter. The largest enhancement of ${\chi }_{\perp }$ susceptibility occurs when a magnetic ordering vector connects ends of quasiparticle pockets with opposite signs of ${\Delta }_{\mathbf{k}}$ (a), or ends with the same sign ${\Delta }_{\mathbf{k}}$ for ${\chi }_{\parallel }$ (b). (c) The magnitude of the ordering vectors as a function of the field at zero temperature, from (a) and (b). (d) Effects of temperature, from $T=0$ to $T=0.3T_c$ indicated by arrows, on $q_{\parallel ,\perp 1}$ vectors, and angle ${\varphi }_{\parallel 1}$ between ${\mathbf{q}}_{\parallel 1}$ and the $x$ axis (inset).

Figure 3

The $T=0$ normalized susceptibility in the superconducting and normal states (flat line at small $q$) as a function of $q$. We set ${\mu }_{\mathrm{B}}H=0.5{\Delta }_0$, close to the Pauli-limiting field ${\mu }_{\mathrm{B}}{H}_{\mathrm{P}}/{\Delta }_0=0.56$, and ${\Delta }_0/{\epsilon }_f=0.005$ (red) or ${\Delta }_0/{\epsilon }_f=0.1$ (black). Transverse susceptibility shows enhancement over the normal state ${\chi }^N\left(q\right)$ with two peaks at $q_{\perp 1,3}$ for nodal-$\mathbf{q}$ direction (solid), and one peak for $\mathbf{q}\left|\right|{\mathbf{q}}_{\perp 2}$ (dashed), in accordance with Figs. 2 and 2. The lower pane shows zoomed-in $\delta {\chi }_{\perp }\left(q\right)={\chi }_{\perp }^{SC}\left(q\right)-{\chi }_{\perp }^N\left(q\right)$ for ${\Delta }_0/{\epsilon }_f=0.005$. The maximal enhancement $\delta {\chi }_{\perp }\left(q\right)$ occurs at wave vectors ${\mathbf{q}}_{\perp 1,3}$ and is of the order $\delta {\chi }_{\perp }/{\chi }_0\sim {\Delta }_0/{\epsilon }_f$.

Figure 4

Contour lines of maximal enhancement of transverse susceptibility ${\chi }_{\perp }$ in the $T$-$H$ phase diagram for ${\Delta }_0=0.005{\epsilon }_f$. Different contours correspond to relative enhancements $\delta \chi \left(q\right)/{\chi }_0$, given in percents. The three panels correspond to $\mathbf{q}$ vectors in Fig. 2. The dotted line is the first-order Pauli-limiting phase transition.