The Berry phase: A topological test for the spectrum structure of frustrated quantum spin systems

The nature of the low energy spectrum of frustrated quantum spin systems is investigated by means of a topological test introduced by Hatsugai [Y. Hatsugai, J. Phys. Soc. Jpn. 73 (2004) 2604; Y. Hatsugai, J. Phys. Soc. Jpn. 74 (2005) 1374; Y. Hatsugai, J. Phys. Soc. Jpn. 75 (2006) 123601] which enables to infer the possible existence or absence of a gap between the ground state and excited states of these systems. The test relies on the determination of an order parameter which is a Berry phase. The structure of the spectra of even and odd-legged systems in 2d and 3d is analysed. Results are confronted with previous work.