Studies of the Loschmidt Echo and Entanglement Spreading in Two Dimensional Anisotropic Spin Systems*

We describe a method for simulating the real time evolution of
extended quantum systems in two dimensions. The method combines the
benefits of integrability and matrix product states in one dimension. In
particular it can be extended to systems whose geometry is that of an
infinitely long cylinder. As a first example application, we present
results for quantum quenches in arrays of coupled quantum Ising chains. In
quenches that cross a phase boundary we find that the return probability
shows non-analyticities in time. We also consider how entanglement
spreads post-quench in the array of chains. We see notable differences in
the spreading on whether the chains are in their ordered or disordered
phase. This difference arises because of the presence of linearly
confined bound states in the ordered phase. As a second example
application, we consider entanglement spreading in anisotropic two
dimensional Heisenberg models.