A unifying framework for sum rules and bounds on optical, thermoelectric and thermal transport from quantum geometry

Abstract: We present a geometric formulation of optical, thermoelectric, and thermal linear response in clean, zero temperature band insulators based on a single object: a generalized time-dependent quantum geometric tensor (g-tQGT) built from correlations of projected particle and heat polarization operators. Within this framework, the AC transport tensors admit compact expressions that make their geometric content explicit. The response splits into a Berry curvature contribution that remains finite in the DC limit and a frequency correction governed by the quantum metric, implying geometry driven effects even in topologically trivial insulators. At equal times, the g-tQGT recovers the usual integrated QGT and yields energy-weighted thermal analogs whose antisymmetric parts are fixed by orbital and heat magnetization. Importantly, in the thermal channel, a thermal quantum geometric tensor is obtained. Casting the theory in a Hilbert-Schmidt inner product form yields a bound on the trace of the thermal QGT, an uncertainty relation on the projected polarization operators and a purely geometric upper bound on the finite-time accumulated response. The latter is used in the optical channel to derive a geometric upper bound on the electric current. Finally, time derivatives of the g-tQGT are used to generate a hierarchy of generalized thermoelectric and thermal sum rules, and bounds on these sum rules are obtained. These bounds are used to find inequalities between different physical objects such as the optical mass, susceptibility functions and magnetizations.