Quadratic dispersion relations in gapless frustration-free systems

TL;DR

This paper proves that in gapless frustration-free systems with local qubits on cubic lattices, low energy excitations are generally quadratic or softer, establishing a fundamental limitation on their dispersion relations.

Contribution

It provides a general proof that gapless frustration-free Hamiltonians with local qubits cannot have linearly dispersive excitations, extending previous case studies.

Findings

Low energy excitations are often quadratic or softer in such systems.

Linear dispersion is impossible in these frustration-free models.

Examples show some states are not low energy excitations.

Abstract

Recent case-by-case studies revealed that the dispersion of low energy excitations in gapless frustration-free Hamiltonians is often quadratic or softer. In this work, we argue that this is actually a general property of such systems. By combining a previous study by Bravyi and Gosset and the min-max principle, we prove this hypothesis for models with local Hilbert spaces of dimension two that contains only nearest-neighbor interactions on cubic lattice. This may be understood as a no-go theorem realizing gapless phases with linearly dispersive excitations in frustration-free Hamiltonians. We also provide examples of frustration-free Hamiltonians in which the plane-wave state of a single spin flip does not constitute low energy excitations.