Time Evolution and Deterministic Optimisation of Correlator Product States

We study a restricted class of correlator product states (CPS) for a spin-half chain in which each spin is contained in just two overlapping plaquettes. This class is also a restriction upon matrix product states (MPS) with local dimension $2^n$ ($n$ being the size of the overlapping regions of plaquettes) equal to the bond dimension. We investigate the trade-off between gains in efficiency due to this restriction against losses in fidelity. The time-dependent variational principle formulated for these states is numerically very stable. Moreover, it shows significant gains in efficiency compared to the naively related matrix product states - the evolution or optimisation scales as $2^{3n}$ for the correlator product states versus $2^{4n}$ for the unrestricted matrix product state. However, much of this advantage is offset by a significant reduction in fidelity. Correlator product states break the local Hilbert space symmetry by the explicit selection of a local basis. We investigate this dependence in detail and formulate the broad principles under which correlator product states may be a useful tool. In particular, we find that scaling with overlap/bond order may be more stable with correlator product states allowing a more efficient extraction of critical exponents - we present an example in which the use of correlator product states is several orders of magnitude quicker than matrix product states.