Charge transport, not interaction range, bounds the Lieb-Schultz-Mattis twist cost

Abstract

Geometric interaction range can overestimate the energy cost of a Lieb-Schultz-Mattis twist. We prove that, under the stated assumptions, every sufficiently large admitted periodic ring with L>3R has an explicit O(1/L) upper bound on its full spectral gap. The bound is governed by a decomposition-independent seminorm measuring the charge transport performed by an interaction, rather than by its geometric span alone. We derive finite-angle and wrapped-polarization formulations, evaluate the transport quantity exactly for the XY chain, and obtain direction-resolved bounds for periodic cells, cylinders, ladders, and long-range dipolar tails. The results provide a unified quantitative framework for charge transport in long-range Lieb-Schultz-Mattis constructions.