Role of Quantum Geometry in the Competition between Higgs Mode and Quasiparticles in Third-Harmonic Generation of Superconductors

Abstract

Collective modes in superconductors, such as the Higgs mode, offer deep insights into the nature of condensates. Third-harmonic generation (THG) is a primary tool for probing the Higgs mode, but its signal competes with that of quasiparticle excitations depending on impurity scattering rates. In particular, in the clean regime the standard BCS theory generally predicts the dominance of quasiparticle contributions. Here, we propose and demonstrate that the quantum geometry of electronic bands can be a key mechanism governing this competition. By developing a formalism that explicitly incorporates the quantum metric, and applying it to a tunable model of a dispersive-band superconductor, we show that the quantum metric can dramatically amplify the nonlinear light-Higgs coupling by several orders of magnitude. Our results establish that a large quantum metric can cause the Higgs mode to dominate the THG response, resolving the puzzle of Higgs and quasiparticle competition in the clean regime and identifying band geometry as a crucial ingredient for designing and understanding the nonlinear response of superconductors.

Figures (4)

• Figure 1: Quantum geometric enhancement of the Higgs mode in the THG response. (a) Energy band dispersions of the quadratic band touching model with $t=10$ and $t_b=-5$. (b) Crossover diagram of the THG intensity ratio $|J^{\text{H}}_\mathrm{tot}| / |J^{\text{qp}}_\mathrm{tot}|$ versus the quantum geometry parameter $d_{\mathrm{max}}$ and the gap to band width $\Delta_0/W$, where $W=8t$ is the band width of the upper band. (c) The ratio of the THG peak heights at resonance as a function of $d_{\mathrm{max}}$ for a fixed dispersion, quantifying the crossover from quasiparticle to Higgs dominance. (d) THG spectrum in the Higgs-dominant regime at $d_{\mathrm{max}}=1$. Solid (dashed) lines represent the total (band-only) contributions for the Higgs (red) and quasiparticle (blue) channels. The band-only Higgs contribution is magnified by $\times 500$ for visibility, highlighting that the geometric terms are responsible for the enhancement. Here, the results in (b-d) were calculated with $\mu=-3t$ and in (c,d) we set $\Delta_0/W=0.1$.
• Figure 2: Quantum geometric enhancement of the correlation length for the Higgs mode. (a) The total squared correlation length of the Higgs mode, $\xi_{H,xx}^2$, as a function of the quantum geometry parameter $d_{\mathrm{max}}$ within the quadratic band touching model. (b) The ratio of the geometric to the band contribution to the squared correlation length, $\xi_{\text{geom},xx}^2/\xi_{\text{band},xx}^2$. The parameters $\mu,\Delta_0/W$ are the same as in Fig. \ref{['fig1']}(b,c).
• Figure 3: Real-space lattice models for (a) $d_{\mathrm{max}}=0$ and (b) $d_{\mathrm{max}}=1$.
• Figure 4: Quantum geometric enhancement of the superfluid stiffness. (a) Superfluid weight $D_{xx}$ within the quadratic band touching model. (b) The ratio of the geometric to the band contribution to $D_{xx}$. The parameters $\mu,\Delta_0/W$ are the same as in Fig. \ref{['fig1']}(b,c) of the main text.