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Finding new phase is a fundamental task in physics. Landau's theory explainedthe deep connection between symmetry breaking and phase transition commonlyoccurring in magnetic, superconducting and super uid systems. The discovery ofthe quantum Hall effect led to Z topological phases which could be differentfor same symmetry and are characterized by the discrete values of the Berryphases. By studying 1D trimer lattices we report new phases characterized byBerry phases which are piecewise continuous rather than discrete numbers. Thephase transition occurs at the discontinuity point. With time-dependentchanges, trimer lattices also give a 2D phases characterized by very specific2D Berry phases of half period. These Berry phases change smoothly within aphase while change discontinuously at the transition point. We furtherdemonstrate the existence of adiabatic pumping for each phase and gain assistedenhanced pumping. The non-reciprocity of the pumping process makes the system agood optical diode.

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