Gaplessness from disorder and quantum geometry in gapped superconductors

TL;DR

The paper shows that quantum geometry, encoded in the Fubini-Study metric of Bloch states, can extend the localization length of disorder-induced Andreev bound states at π-junctions in a nominally gapped superconductor. Using a 2D lattice model with tunable geometry, self-consistent mean-field theory, and a disorder-structured pairing interaction, the authors reveal a crossover to a Bogoliubov Fermi surface of subgap states as ζ increases, imparting gapless, dirty-nodal-like behavior while preserving superfluid order. They quantify this via the inverse participation ratio, DoS, ρ_s(T), and momentum-resolved spectral functions, showing that quantum geometry enhances inter-junction coupling and subgap transport. The results connect to moiré graphene experiments and suggest experimental probes (optical, thermal) to distinguish localized subgap states from true nodal excitations, with potential delocalization transitions in certain symmetry classes.

Abstract

It is well known that disorder can induce low-energy Andreev bound states in a sign-changing, but fully gapped, superconductor at $π-$junctions. Generically, these excitations are localized. Starting from a superconductor with a sign-changing and nodeless order parameter in the clean limit, here we demonstrate a mechanism for increasing the localization length associated with the low-energy Andreev bound states at a fixed disorder strength. We find that the Fubini-Study metric associated with the electronic Bloch wavefunctions controls the localization length and the hybridization between bound states localized at distinct $π-$junctions. We present results for the inverse participation ratio, superfluid stiffness, site-resolved and disorder-averaged spectral functions as a function of increasing Fubini-Study metric, which indicate an increased tendency towards delocalization. The low-energy properties resemble those of a dirty nodal superconductor with gapless Bogoliubov excitations. We place these results in the context of recent experiments in moire graphene superconductors.

Figures (11)

• Figure 1: (a) Schematic illustration of disorder-induced islands (green and gray) with sign-changing gaps; a $\pi-$junction hosts localized Andreev bound states (ABS). Quantum metric controls the minimal spread of the Bloch wavefunctions across many lattice sites (white dots). (b) ABS localization length $\xi$ at a single $\pi-$junction normalized by $\xi_0\sim v^\star_{\rm F}/\Delta$ as a function of $\zeta$; see Eq. \ref{['eq:H_chiral_band']}. The green dots are obtained from a self-consistent mean-field calculation for the model defined in Eqs. \ref{['model']}, \ref{['eq:H_int']} on an $18\times18$ lattice, with a domain wall in $V(\mathbf{r})=V_{0}+V_{d}\Theta(x)$ ($V_{0} = 0.5t$, $V_{d} = 1.6t$); gray curve is based on Eq. \ref{['eq:H_BdG_1D']}. Inset: DoS in the uniform system, showing extended $s$-wave SC gaps ($\Delta_1\pm\Delta_0$). (c) IPR for the correlated-disordered model ($V_{1}=0.5t,\:V_{2}=2.1t$) averaged over a fixed energy interval $E/L=0.012t$ for a system of size $L\times L$ for different values of $\zeta$, and fixed $\ell=0.1$. (d) Snapshots of the wavefunctions (spatial probability distributions) at small and large $\zeta$. The other parameters, fixed for all simulations are $t' = 0.18t$, and $U = 0.5t$.
• Figure 2: Superfluid stiffness $\rho_s$ as a function of temperature $T$, for three different value of $\zeta$, calculated self-consistently on an $18\times18$ lattice. For all values of $\zeta$, we find a low-temperature power-law behavior of the stiffness indicating gapless superconductivity. The dashed lines are power-law fits, with the exponents varying between $3.7$ at $\zeta = 0.1$, to $2$ at $\zeta = 1.2$, at low temperatures. Inset: DoS for the three $\zeta$ values, showing the low-energy gapless excitations. Other parameters are $t' = 0.18t, U=0.5t, V_{1}=0.5t, V_{2}=2.1t$, and disorder correlation length $\ell=0.1$.
• Figure 3: Bogoliubov spectral function ${\cal A}(\mathbf{k},\omega)$ evaluated at $\zeta=1.2$ on a 22$~\times~$22 lattice (in arbitrary units), for $\zeta=0.1$ and $\zeta=1.2$. (a), (c) Line cuts of ${\cal A}(\mathbf{k},\omega)$ along high-symmetry lines in the Brillouin zone. Below the clean pairing gap [dashed lines; see the inset of Fig. \ref{['fig:general_picture']}], the localized gapless excitations with enhanced localization length nearly mimic an impurity-band like excitation spectrum. (b), (d) Momentum-resolved spectral function at zero energy ${\cal A}(\mathbf{k},\omega=0)$, which suggests that the disorder-induced subgap excitations form a "Fermi surface". Other parameters are $t' = 0.18t, U=0.5t, V_{1}=0.5t, V_{2}=2.1t$, and $\ell=0.1$.
• Figure 4: Example of the $s$-wave pairing fields (onsite, $x$ nearest neighbor, $y$ nearest neighbor) solved self-consistently on a $22 \times 22$ lattice. The self-consistent solutions show that $|\Delta_{x,y}|\gg\Delta_{0}$, i.e., the extended components dominate, and that $\Delta_{x,y}$ change sign throughout the system. Results are shown for three different values of the disorder correlation length $\ell$, demonstrating the emergence of larger and larger puddles as $\ell$ grows. The red borders in $\Delta_{x,y}$ mark the boundaries between regions of different sign --- the $\pi-$junctions. Microscopic parameters used in the simulations are $t' = 0.18t, U=0.5t, V_{1}=0.5t, V_{2}=2.1t$, and fixed $\zeta=0.1$.
• Figure 5: Dependence of the inverse participation ratio scaling with system size on the number of disorder realizations. All other parameters are the same as in Fig. \ref{['fig:general_picture']} of the main text.