Quenching across a quantum critical point: dependence of scaling laws on spatial periodicity

We study the quenching dynamics of a quantum many-body system in one dimension described by a Hamiltonian having spatial periodicity. Specifically, we consider a spin-1/2 $XX$ chain subject to a periodically varying magnetic field in the $\hat z$ direction or, equivalently, a tight-binding model of spinless fermions having a periodic local chemical potential. If the strength of the magnetic field (or chemical potential) is varied slowly in time at a rate $1/\tau$ so as to take the system across a quantum critical point, we find that the density of excitations thereby produced scales as a power of $1/\tau$.Remarkably, the power depends on the spatial periodicity of the field and deviates from the $1/\sqrt{\tau}$ scaling that is ubiquitous to a range of systems. This behavior is analyzed by mapping the slow quenching problem to a collection of fermionic two-level systems, labeled by the lattice momentum $k$, for which the effective Hamiltonians vary as a power of the time close to the quantum critical point. For a magnetic field described by multiple periodicities, the power depends on the smallest period for very large values of $\tau$. Finally, we find that if there are interactions between the fermions, the power varies continuously with the interaction strength.