Enhancing the Hyperpolarizability of Crystals with Quantum Geometry

We demonstrate that higher-order electric susceptibilities in crystals can be enhanced and understood through nontrivial topological invariants and quantum geometry, using one-dimensional <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mi>π</a:mi></a:mrow></a:math>-conjugated chains as representative model systems. First, we show that the crystalline-symmetry-protected topology of these chains imposes a lower bound on their quantum metric and hyperpolarizabilities. Second, we employ numerical simulations to reveal the tunability of nonlinear, quantum geometry-driven optical responses in various one-dimensional crystals in which band topology can be externally controlled. Third, we develop a semiclassical picture to deliver an intuitive understanding of these effects. Our findings offer a firm interpretation of otherwise elusive experimental observations of colossal hyperpolarizabilities and establish guidelines for designing topological materials of any dimensionality with enhanced nonlinear optical properties.