The New Phase due to Symmetry Protected Piecewise Berry Phases; Enhanced Pumping and Non-reciprocity in Trimer Lattices

Finding new phase is a fundamental task in physics. Landau's theory explained the deep connection between symmetry breaking and phase transition commonly occurring in magnetic, superconducting and super uid systems. The discovery of the quantum Hall effect led to Z topological phases which could be different for same symmetry and are characterized by the discrete values of the Berry phases. By studying 1D trimer lattices we report new phases characterized by Berry phases which are piecewise continuous rather than discrete numbers. The phase transition occurs at the discontinuity point. With time-dependent changes, trimer lattices also give a 2D phases characterized by very specific 2D Berry phases of half period. These Berry phases change smoothly within a phase while change discontinuously at the transition point. We further demonstrate the existence of adiabatic pumping for each phase and gain assisted enhanced pumping. The non-reciprocity of the pumping process makes the system a good optical diode.