The interplay between band topology and nonlinear optical responses presents a promising platform for discovery, particularly for second-harmonic generation (SHG), which requires broken inversion symmetry. Here, we theoretically investigate the topological constraints on SHG in “essentially noncentric insulators” in which the topologically nontrivial phase of matter exists only in crystal classes without a center of inversion. Using the Floquet formalism combined with the Keldysh Green's function method, we derive explicit analytic expressions for the interband contributions to the SHG response driven by linearly polarized light in a 2D two-band model. Our analysis identifies that the transverse SHG response is intrinsically characterized by the shift vector. Based on the theory of “essentially noncentric insulators”, we show that the shift obstruction relation imposes a topological constraint, namely, that in a nontrivial phase the shift vector cannot be continuously tuned to zero across the entire Brillouin zone. As a result, the $T1a$ component of the transverse SHG response, which depends on this shift vector, is topologically constrained rather than quantized and cannot be globally eliminated. We further obtain the topological phase diagram for a representative 2D two-band model and compute its SHG response using both our formalism and Sipe's method. Our numerical results are in good agreement with Sipe's method and reveal a positive correlation trend between the SHG response and the average intercellular shift. These findings suggest that the identified topological invariants may serve as a design principle for engineering enhanced nonlinear optical responses.

Close

PrevNext

View figure in article

• Nonlinear optics
• Floquet systems
• Nonequilibrium Green's function
• Tight-binding model