Gapped Hamiltonians: Eigenvalues Move Freely Within Energy Gaps

Because gapped Hamiltonians have energy gaps, their eigenvalues can move freely as long as they don't cross energy levels. This is due to the existence of a forbidden band in the energy spectrum, where no energy levels are present. The eigenvalues can vary freely above or below the energy gap without crossing it.

This ability to move freely reflects the protective nature of the energy gap. The presence of an energy gap implies that the energy levels of the system are completely isolated and do not mix or cross within a certain energy range. This protective property allows the energy levels within the gap to evolve freely under appropriate conditions, without being affected by external perturbations.

In topological phases, the protective nature of the energy gap and the structure of energy levels are crucial. Because the energy levels within the gap cannot transform or mix with each other, they can exhibit unique topological properties, such as topological boundary states or the quantum Hall effect. These topological properties are significant for various applications, including topological quantum computing and quantum memory.

Therefore, the presence of an energy gap allows the eigenvalues of gapped Hamiltonians to move freely without crossing the forbidden band. This movement is protected and stable, making the energy gap a crucial concept in studying and utilizing topological phases.