  We investigate the dynamics of two Jordan Wigner solvable models, namely, the
one dimensional chain of hard-core bosons (HCB) and the one-dimensional
transverse field Ising model under coin-toss like aperiodically driven
staggered on-site potential and the transverse field, respectively. It is
demonstrated that both the models heat up to the infinite temperature ensemble
for a minimal aperiodicity in driving. Consequently, in the case of the HCB
chain, we show that the initial current generated by the application of a twist
vanishes in the asymptotic limit for any driving frequency. For the transverse
Ising chain, we establish that the system not only reaches the diagonal
ensemble but the entanglement also attains the thermal value in the asymptotic
limit following initial ballistic growth. All these findings, contrasted with
that of the perfectly periodic situation, are analytically established in the
asymptotic limit within an exact disorder matrix formalism developed using the
uncorrelated binary nature of the coin-toss aperiodicity.
