Quantum geometry and impurity sensitivity of superconductors without time-reversal symmetry: application to rhombohedral graphene and altermagnets

TL;DR

This work shows that quantum geometry of Bloch states can render nonmagnetic impurities pair-breaking in superconductors whose normal-state Hamiltonians break time-reversal symmetry. By deriving a general Tc-suppression framework involving disorder matrix elements and geometry-weighted commutators, the authors connect gauge fixing to the maximal localization of the Cooper-pair wavefunction and reveal regimes where disorder can even enhance Tc via kinetic effects. Across concrete models of rhombohedral graphene, twisted MoTe$_2$, and altermagnets, they demonstrate non-monotonic Tc versus disorder and the potential stabilization of topological (triplet) pairing due to quantum geometry. The results offer a versatile diagnostic: controlled disorder can help identify pairing symmetry and reveal finite-momentum pairing tendencies in TRS-broken superconductors, with implications for emerging materials in this class.

Abstract

Analyzing the consequences of the quantum geometry induced by the momentum dependence of Bloch states has emerged as a very rich and active field in condensed matter physics. For instance, for the superfluid stiffness or the pairing mechanism, these geometric aspects can play an important role. We here demonstrate that quantum geometry can also be essential for the disorder sensitivity of a superconductor, in particular when time-reversal symmetry is broken in the normal-state Bloch Hamiltonian. We derive a general expression for the behavior of the critical temperature $T_c$ involving weighted (anti-)commutators of the superconducting order parameter and impurity matrix elements, which includes both wave-function effects and kinetic pair breaking due to broken time-reversal symmetry in the dispersion. We analyze how the former effects lead to "quantum geometric pair breaking", where any superconductor becomes susceptible to microscopically non-magnetic impurities, and formally relate it to the maximum possible localization of two-particle Wannier states. Further, in the presence of kinetic pair breaking, impurities can also enhance pairing, leading to an overall more complex, non-monotonic behavior of $T_c$ with impurity concentration. We also analyze the fate of finite-momentum pairing. Our results are directly relevant to rhombohedral graphene, twisted MoTe$_2$, and superconducting altermagnets.

Figures (7)

• Figure 1: Diagrammatics. (a) shows an exact diagrammatic representation of the disordered particle-particle bubble, expressed in terms of the full Green's function (double line) and the renormalized vertex (colored triangle). In the limit $k_{\mathrm{}{F}} l \gg 1$, the dressed Green's function and the vertex reduce to the noncrossing diagrams shown in (b) and (c), respectively.
• Figure 2: Gauge choice, disorder sensitivity, and localization of Cooper-pair wave function. Here, (a-d), (e-f), and (g-h) refer to $n=4$ (R$4$G) in Eq. (\ref{['MatrixElements']}), $n=3$ (R$3$G), and altermagnetism, respectively. The dependence of (a) $m_{\mathrm{opt}}$ defining the optimal gauge and superconducting Chern numbers, and of the corresponding $\zeta_{\mathrm{opt}}$ (b) on $\eta^2$. (c) shows the normalized measure of the localization of the Cooper pair wavefunction $C_{w}(\mathbf{x}=0)$, and (d) the spatial profile of this function for $\eta^2=1/4$ for the indicated $m$. As opposed to $n=4$ in (a-d), we find for $n=3$ (e) that $\zeta_{\text{opt}}$ differs in the shaded region between gauge (green) and $\Delta_{\mathbf{k}}$ (red) optimization, where the most stable superconductor is a triplet state ($\Delta_{\mathbf{k}} = -\Delta_{-\mathbf{k}}$). (f) Shows $\zeta_{\mathrm{opt}}$ for the spinless case in R$3$G, where again optimal gauge choice and superconductor coincide, revealing an emergent Anderson theorem for $\eta^2=1/2$. (g) shows the optimal $\zeta$ for a checkerboard lattice model of altermagnetism as a function of spin-orbit coupling $\alpha$ and altermagnetic strength $\Phi$. To illustrate the symmetry of the pairing state, which is the same for all $\alpha,\Phi \neq 0$, we show the angular dependence of the singlet component $\Delta_s$ of the optimal pairing state for $\alpha=\Phi=0.5$ in (h); for clarity, the imaginary part is multiplied by 15. There is also a non-unitary, in-plane triplet component (not shown).
• Figure 3: Kinetic effects and their interplay with quantum geometry. (a) Dependence of the critical temperature on the inverse lifetime for different values of the trigonal warping amplitude in the simplified RnG dispersion, which the kinetic pair-breaking $a_{\mathbf{k}} = (\xi_{\mathbf{k}} - \xi_{-\mathbf{k}})/2$. (b) Illustrates the interplay of kinetic and geometric effects on the behavior of the critical temperature in the full continuum R4G model for different disorder impurities $w_A^2 + w_B^2 = 1$.
• Figure 4: Finite momentum pairing. (a) Position of the maximum of the particle-particle bubble $\chi^{C}(\mathbf{q})$ as a function of the disorder strength for different values of the cutoff frequency $\omega_D$. For $\omega_D = 4 \text{ meV}$, the 2D map of $\chi^{C}(\mathbf{q})$ is shown in the clean limit (b) and disordered limit (c) when the finite-momentum pairing is fully suppressed.
• Figure 5: Blocks entering the disordered particle-particle bubble.