Gap generation and flat band catalysis in dice model with local interaction

The gap generation in the dice model with local four-fermion interaction is studied. Due to presence of two valleys degenerate electron states, there are main types gaps. intra- and intervalley describes hole pairing same different valleys, respectively. We found that while intravalley takes place only supercritical regime, generated for an arbitrary small coupling. physical reason absence critical coupling catalysis by flat band spectrum model. completely quenched kinetic energy when integrated over momentum equation leads extremely large proportional area Brillouin zone.