SpectralGapProperty

The spectral gap plays a crucial role in determining the behavior of quantum mechanical systems. In particular, gapped Hamiltonians have a well-defined lowest energy state (ground state) and a finite energy difference between this ground state and the first excited state, which gives rise to important physical phenomena such as the energy gap in solid-state physics.

Moreover, the exponential decay of correlations in the ground state of gapped Hamiltonians is a manifestation of the fact that physical observables become effectively local in such systems, which greatly simplifies their study and allows for powerful mathematical tools to be applied, such as the theory of matrix product states and tensor networks. In contrast, gapless systems typically exhibit power-law decay of correlations and are much more difficult to analyze.