Anomalous Superfluid Density in Pair-Density-Wave Superconductors

TL;DR

This work investigates the stability of two-dimensional unidirectional PDW superconductors by computing the superfluid density $n_s(T)$ within a lattice model with finite-momentum pairing. The authors show a large intrinsic instability region where the longitudinal stiffness $n_s^{xx}$ becomes negative, and identify a stable PDW regime with pronounced in-plane anisotropy and two experimental fingerprints: a severely suppressed $n_s^{xx}$ and an unconventional $T$-dependence of $n_s$ due to a Van Hove singularity in the gapless Bogoliubov band. The analysis decomposes $n_s$ into a fermionic part and a negative collective (Higgs) contribution, with destructive interference from finite pairing momentum suppressing transport along the PDW direction and boosting transverse stiffness. At low $T$, $n_s^{yy}$ increases as $+T^2$ whereas $n_s^{xx}$ decreases as $-T^2$, a result tied to the curvature of the current-weighted DOS near a PDW-induced VHS; these features provide clear experimental signatures. Collectively, the findings highlight the fragility of finite-temperature PDW superconductivity due to the combined effects of interference and collective modes, offering concrete guidance for identifying and stabilizing PDW states in real materials.

Abstract

Pair-density-wave (PDW) states are a long-sought-after phase of quantum materials, with the potential to unravel the mysteries of high-$T_c$ cuprates and other strongly correlated superconductors. Yet, surprisingly, a key signature of stable superconductivity, namely the positivity of the superfluid density, $n_s(T)$, has not yet been demonstrated. Here, we address this central issue by calculating $n_s(T)$ for a generic model two-dimensional PDW superconductor. We uncover a surprisingly large region of intrinsic instability, associated with negative $n_s(T)$, revealing that a significant portion of the parameter space thought to be physical cannot support a pure PDW order. In the remaining stable regime, we predict two striking and observable fingerprints: a small longitudinal superfluid response and an unusual temperature dependence for $n_s(T)$. These generally model-independent, as well as experimentally relevant findings suggest that the fragility of the superfluid density poses a significant problem for the formation of stable, finite temperature PDW superconductivity.

Figures (5)

• Figure 1: Zero-temperature phase diagram as a function of interaction strength ($V_1$) and chemical potential ($\mu$). The PDW state is stable only in a limited region (blue), separated from the uniform BCS phase by a first-order transition (solid line). A large portion of the phase diagram is an unstable regime (gray) characterized by a negative longitudinal superfluid density ($n_s^{xx} < 0$). There is a boundary (brown dashed) seperating stable and unstable PDW and determined by zero superfluid density. This finding guides the experimental search for stable PDW phases by identifying the strong-pairing, relatively low-density quadrant as the most promising regime.
• Figure 2: Anisotropic superfluid density in the stable PDW phase for parameters $(-V_1, -\mu) = (5.9t, 0.4t)$. (a) The longitudinal component, $n_s^{xx}$, is dramatically suppressed near zero, in stark contrast to the robust transverse component, $n_s^{yy}$. The superfluid density component $n_s^{xx}$ vanishes at $T_0 \approx 0.15t$, which sets an upper bound for the critical transition temperature $T_c$. Therefore $T_c$ is much smaller than the order parameter at zero temperature. (b) This suppression is a two-fold mechanism: first, the fermionic contribution $n_0^{xx}$ is itself suppressed to a small positive value by destructive interference from the large pairing momentum $\mathbf{Q}$. Second, this remaining modest contribution is near-perfectly cancelled by the negative term from the collective Higgs mode, $n_H^{xx}$. This dramatic and fragile anisotropy is a key experimental fingerprint of the PDW state.
• Figure 3: A second key experimental fingerprint of the PDW state: its anomalous temperature dependence, shown here for parameters $(-V_1, -\mu) = (5.8t, 0.5t)$. (a) At low temperatures, the superfluid density exhibits a highly unconventional $T^2$ behavior, increasing along the transverse direction ($n_s^{yy}$) while decreasing along the longitudinal one ($n_s^{xx}$). (b) This striking sign-change is a direct probe of the unique, gapless electronic structure. The opposite signs of the $T^2$ correction are dictated by the opposite curvatures of the longitudinal ($D_x$) and transverse ($D_y$) current-weighted density of states (DOS) at the Fermi level ($E=0$), a feature which arises from a nearby van Hove singularity. For comparison: the homogeneous $s$-wave superfluid density exhibits an exponentially weak $T$-dependence, while the homogeneous $d$-wave shows a linear-in-$T$ decrease at low $T$; both are isotropic with $n_{xx}=n_{yy}$.
• Figure 4: The order parameter is plotted versus the interaction strength $V_1$ at a fixed chemical potential $\mu = -0.5t$. The critical interaction separating the $d$-wave superconducting and Pair-Density Wave (PDW) phases occurs at approximately $V_{1c} = -5.95t$.
• Figure 5: The dependence of the energy of superconducting states on the pairing momentum, $Q_x$, is calculated for a chemical potential of $\mu=-0.5$ and an interaction potential of $V=-6.0$. The results indicate the existence of two local minima, with the global, most stable minimum found at a non-zero value of $Q_x$.