Two-particle spectral function for disordered $\mathit{s}$-wave superconductors: Local maps and collective modes

We make testable predictions for the local two-particle spectral function of a disordered $s$-wave superconductor, probed by scanning Josephson spectroscopy (SJS), providing complementary information to scanning tunneling spectroscopy (STS). We show that SJS provides a direct map of the local superconducting order parameter that is found to be anticorrelated with the gap map obtained by STS. Furthermore, this anticorrelation increases with disorder. For the momentum-resolved spectral function, we find the Higgs mode separates from the continuum at arbitrarily small disorder and appears as a nondispersive subgap feature at low momenta, spectrally separated from phase modes for all disorder strengths. The amplitude-phase mixing remains small at low momenta even when disorder is large. Remarkably, even for large disorder and high momenta, the amplitude-phase mixing oscillates rapidly in frequency and hence does not significantly affect the purity of the Higgs and phase-dominated response functions.

Figure 1

Spatial maps for a particular disorder configuration at $V/t=6$ showing strong correlation between (a) the local superconducting order parameter ${\mathrm{\Delta }}_0\left(r\right)$ and (b) the frequency-integrated local two-particle spectral weight $F\left(r\right)$. (c) Spatial map of the corresponding single-particle gap $E\left(r\right)$ obtained from the local one-particle density of states (see Appendix pp3 for details). Notice the strong anticorrelation between (c) and (b). (d) Covariance between $F\left(r\right)$ and $E\left(r\right)$, averaged over disorder realizations, as a function of disorder strength. The anticorrelation increases with disorder. (e) A bar map showing the relative weights of the Higgs (amplitude) and Goldstone (phase) modes in the two-particle spectral weight $F\left(r\right)$ shown in (b). The spectral weight at large disorder is dominated by the phase modes. (f) The integrated amplitude-phase cross correlator $F_{12}\left(r\right)+F_{21}\left(r\right)$ corresponding to the configuration shown in (b). The mixing contribution shows regions with positive and negative values on a scale much smaller than the superconducting coherence length. Here, we present data with $U/t=3$ on a $24\times 24$ lattice, at an average fermion density $\rho =0.875$.

Figure 2

(a)–(d) Density plot of Higgs spectral function $P_{11}(q,\omega )$ with increasing disorder: (a) $V/t=0$ showing no weight at $q=0$; (b) weak disorder $V/t=0.1$ starts showing finite $q=0$ weight of the Higgs mode; (c) $V/t=1$ and (d) $V/t=3$. (e)–(h) Density plot of Goldstone or phase spectral function $P_{22}(q,\omega )$ for (e) $V/t=0$, (f) $V/t=0.1$, (g) $V/t=3$, and (h) $V/t=6$. Note the stability of dispersive modes up to large disorder strength. For $V/t=0.1$, note the absence of phase spectral weight in the region where additional Higgs spectral weight is present. (i) The Higgs threshold ${\omega }_{\mathrm{Higgs}}$, the speed of sound $c_s$, and the two-particle continuum threshold $2E_{\mathrm{gap}}$ (pair-breaking scale) as a function of $V/t$. (j)–(l) The density plot of amplitude-phase cross correlator $P_{12}+P_{21}$ for (j) $V/t=1$, (k) $V/t=3$, and (l) $V/t=6$. The mixing term grows in magnitude but oscillates in sign more rapidly as disorder is increased, leading to cancellations in measurable response functions. Here, we present data with $U/t=3$ on a $24\times 24$ lattice, at an average fermion density $\rho =0.875$, and averaged over 15 disorder realizations.

Figure 3

Energy dependence of the spectral function at low $q$, (a)–(c) $P(q=[0,0],\omega )$, and (d)–(f) $P(q=[\pi /12,0],\omega )$ for increasing disorder $V/t=1,V/t=3$, and $V/t=6$. The decomposition of the contributions from the Higgs mode $P_{11}$, phase mode $P_{22}$, and the mixing $P_{12}+P_{21}$ is also shown. Note the negligible mixing contributions, and the spectral separation of Higgs and phase contributions. The mixing term is weaker than the Higgs weight, and both are much smaller than the weight in the phase mode. Here, we present data with $U/t=3$ on a $24\times 24$ lattice, at an average fermion density $\rho =0.875$.

Figure 4

(a) LDOS as a function of $\omega$ for a particular site. The locations of ${\omega }_p$ and ${\omega }_m$ are indicated in the figure. (b), (c) Covariance between two-particle spectral function $\left(F\right)$ and local single-particle gap $\left(E\right)$ where $F$ is defined as (b) $F\left(r\right)=\frac{1}{0.8\times 2E_{\mathrm{gap}}}{\int }_{0.2\times 2E_{\mathrm{gap}}}^{2E_{\mathrm{gap}}}d\omega P(r,r,\omega )$ and (c) $F\left(r\right)=\frac{1}{0.7\times 2E_{\mathrm{gap}}}{\int }_{0.3\times 2E_{\mathrm{gap}}}^{2E_{\mathrm{gap}}}d\omega P(r,r,\omega )$.