Exactness of the band-projected moment generating function representation

Prove that for any projector P_B onto a subset of Bloch bands and the position operator r, the band-projected exponential M_B(lambda) = P_B exp(lambda · r) P_B equals exp(lambda_i nabla_i + sum_{n=2}^∞ (1/n!) lambda_{i1...in} T^{(n)}_{i1...in}) for all lambda, where nabla_i = ∂_{k_i,B} − A_i is the covariant derivative within the projected subspace (with ∂_{k_i,B} the k-derivative operator restricted to the projected bands and A_i the Berry-connection operator in that subspace), and T^{(n)}_{i1...in} are the fully symmetric rank-n tensors obtained from the recursive geometric operators Q^{(n)} defined by Q^{(n)}_{i1...in} = (−i)^n P_B [r_{i1}, P_B][r_{i2}, [ ... [r_{in}, P_B] ... ]] P_B. In particular, establish that this identity exactly generates all band-projected position moments via M_B(lambda).

Background

The work develops an operator-based, gauge-invariant framework for electron dynamics in crystals under inhomogeneous electromagnetic fields using band-projected operators. Central to the approach is expressing products of projected position operators in terms of covariant derivatives and geometric tensors (quantum metric, Berry curvature, and higher-order geometric objects).

These structures are summarized by a compact moment generating function identity for the band-projected exponential of the position operator, which would generate all projected moments and encode the full hierarchy of geometric tensors in the exponent. While lower-order structures are derived explicitly, the authors do not provide a complete proof of the closed-form identity and explicitly conjecture its exactness. Establishing this identity would provide a unified and exact operator formula for all band-projected position moments, enabling systematic higher-order expansions in field inhomogeneity.