  We develop a time-dependent real-space renormalization-group approach which
can be applied to Hamiltonians with time-dependent random terms. To illustrate
the renormalization-group analysis, we focus on the quantum Ising Hamiltonian
with random site- and time-dependent (adiabatic) transverse-field and
nearest-neighbour exchange couplings. We demonstrate how the method works in
detail, by calculating the off-critical flows and recovering the ground state
properties of the Hamiltonian such as magnetization and correlation functions.
The adiabatic time allows us to traverse the parameter space, remaining near-to
the ground state which is broadened if the rate of change of the Hamiltonian is
finite. The quantum critical point, or points, depend on time through the
time-dependence of the parameters of the Hamiltonian. We, furthermore, make
connections with Kibble-Zurek dynamics and provide a scaling argument for the
density of defects as we adiabatically pass through the critical point of the
system.
