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The second law of thermodynamics dictates that heat simultaneously flows fromthe hot to cold bath on average. To go beyond this picture, a range of works inthe past decade show that, other than the average dynamical heat fluxdetermined by instantaneous thermal bias, a non-trivial flux contribution ofintrinsic geometric origin is generally present in temporally driven systems.This additional heat flux provides a free lunch for the pumped heat and couldeven drive heat against the bias. We review here the emergence and developmentof this so called ``geometric heat pump'', originating from the topologicalgeometric phase effect, and cover various quantum and classical transportsystems with different internal dynamics. The generalization from the adiabaticto the non-adiabatic regime and the application of control theory are alsodiscussed. Then, we briefly discuss the symmetry restriction on the heat pumpeffect, such as duality, supersymmetry and time-reversal symmetry. Finally, weexamine open problems concerning the geometric heat pump process and elucidatetheir prospective significance in devising thermal machines with highperformance.

Geometric Heat Pump: Controlling Thermal Transport with Time-dependent Modulations