Gapped Hamiltonians: Eigenvalue Movement and Zero Energy Crossing

When a system's Hamiltonian possesses an energy gap, it implies the existence of a forbidden band in the energy spectrum where no energy levels reside. In such scenarios, the Hamiltonian's eigenvalues can freely change above or below this gap without crossing the point of zero energy.

The presence of an energy gap signifies that the system's energy levels within the forbidden band are separated, preventing transitions or mixing. As long as these levels remain above or below the gap, they can move and vary freely without being influenced by external conditions or changes within the system.

However, crossing the zero energy point by energy levels alters the system's energy structure, potentially leading to significant changes in its properties. Therefore, in Hamiltonians with energy gaps, energy levels can only move freely within the gap to preserve system stability and protect its characteristics.

This capability of energy levels to move freely is a crucial feature of gapped Hamiltonians, playing a significant role in explaining and studying topological phases, topological insulators, and other topological states. The freedom of movement allows energy levels within the gap to exhibit unique topological properties, leading to applications in various fields.