Breakdown of the adiabatic limit in low-dimensional gapless systems

It is generally believed that a generic system can be reversibly transformed from one state to another by a sufficiently slow change of parameters. Microscopically, this belief is often justified using connections to the quantum adiabatic theorem stating that there are no transitions between different energy levels if the hamiltonian changes slowly in time. Here, we show that in fact the response to such a slow change can be non-trivial in low-dimensional gapless systems. We identify three generic regimes of the response: analytic, non-analytic and non-adiabatic, which are characterized by a different behaviour of the heating induced in the system with the ramp rate. In the last regime, the limits of the ramp rate going to zero and the system size going to infinity do not commute and the adiabatic process does not exist in the thermodynamic limit. We support our results with numerical and analytical calculations.