Quantum Geometric Helical Superconductivity

Abstract

Several physical phenomena in superconductors, such as helical superconductivity and the diode effect, rely on breaking time-reversal symmetry. This symmetry-breaking is usually accounted for via the Lifshitz invariant, a contribution to the free energy which is linear in the phase gradient of the order parameter. In dispersive single-band superconductors with conventional pairing, the Lifshitz invariant can be computed from the asymmetries of the spectrum near the Fermi surface. We show that in multi-band superconductors, the quantum geometry also contributes to the Lifshitz invariant, and this is the dominant contribution when the low-energy bands are flat. We also analogously demonstrate quantum-geometry-driven commensurate-incommensurate transitions in charge and pair density waves.

Figures (4)

• Figure 1: [Left] The phase gradient $q$ of the superconducting order parameter $\Delta$ is odd under time-reversal symmetry $\mathcal{T}$. Accordingly, for time-reversal symmetric systems (blue), the free energy $F$ is an even function of $q$, and thus is quadratic in phase gradients; this quadratic coefficient is the superfluid stiffness. However, if time-reversal symmetry is broken (red), then the free energy can have a linear coefficient called the Lifshitz invariant, which quantifies [top-right] helical superconductivity and [bottom-right] the superconducting diode effect.
• Figure 2: To characterize superconductivity in terms of abelian quantum geometry, the bandstructure needs to decompose into low-energy bands (red) near the Fermi level $\mu$ (blue), and high-energy bands which are far enough from it to be integrated out. The approximations are illustrated in green: in the simplest case (left), the low-energy bands are degenerate with each other and isolated from other bands by a large gap. These requirements can be relaxed somewhat (right); the methods presented here are a good approximation provided both the high-energy bands and any energy-splittings between the low-energy bands are located far from the Fermi energy compared to $U$ and $k_BT$.
• Figure 3: Instead of evaluating the quantum geometry in the usual $(k_x,k_y)$-space, we evaluate it in $(k_x,k_y,\alpha)$-space, where $\alpha$ is a perturbative $\mathcal{T}$-breaking parameter. The gray plane at $\alpha=0$ indicates the $\mathcal{T}$-symmetric case, the red plane indicates at finite $\mathcal{T}$-breaking $\alpha=\bar{\alpha}$, and the blue plane is the time-reverse of the red plane (by assumption on the definition of $\alpha$). For $\mathcal{T}$-symmetric SCs, $P_k$ and $\mathcal{T} P_{-k'}\mathcal{T}^{-1}$ differ only in $k$-space (by momentum vector $q$ from the gradient expansion), but without this symmetry, they also differ via $\alpha\rightarrow -\alpha$. Including this parameter in the quantum metric (and the quantum geometric coefficients of all higher-order corrections in $q$) then accounts for this difference.
• Figure 4: (a) 1d model with a bipartite flat band (one unit cell highlighted). Gray dots are A sites, red dots are B and C sites. Time-reversal symmetry is broken by the magnetic flux pattern in blue. (b) Sample spectra with no flux; the gap closes at $t_++t_-=t_0$. (c) Helical wavevector of the flat band, normalized by the flux-induced hopping phase $\theta$, as a function of hopping amplitudes $t_{\pm}$.