  In recent years, the infinite time-evolution block decimation (iTEBD) method
has been demonstrated to be one of the most efficient and powerful numerical
schemes for time-evolution in one-dimensional quantum many-body systems.
However, a major shortcoming of the method, along with other state-of-the-art
algorithms for many-body dynamics, has been their restriction to one spatial
dimension. We present an algorithm based on a \textit{hybrid} extension of
iTEBD where finite blocks of a chain are first locally time-evolved before an
iTEBD-like method combines these processes globally. This in turn permits
simulating the dynamics of many-body systems in the thermodynamic limit in
$d\geq1$ dimensions including in the presence of long-range interactions. Our
work paves the way for simulating the dynamics of many-body phenomena that
occur exclusively in higher dimensions, and whose numerical treatments have
hitherto been limited to exact diagonalization of small systems, which
fundamentally limits a proper investigation of dynamical criticality. We expect
the algorithm presented here to be of significant importance to validating and
guiding investigations in state-of-the-art ion-trap and ultracold-atom
experiments.
