Time Evolution and Deterministic Optimisation of Correlator Product States

TL;DR

This paper explores a restricted class of correlator product states for spin chains, demonstrating their efficiency advantages and limitations in fidelity compared to matrix product states, with potential benefits for critical phenomena analysis.

Contribution

It introduces a specific class of correlator product states, analyzes their time evolution and optimization, and compares their efficiency and fidelity to matrix product states.

Findings

Correlator product states offer significant efficiency gains over matrix product states.

Fidelity is reduced when using restricted correlator product states.

Correlator product states can be more effective for extracting critical exponents.

Abstract

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.