Efficient matrix factorisation of the modular path integral for extended systems

The modular path integral (MPI) formulation offers a numerically exact, versatile, and highly efficient approach to the quantum dynamics of extended systems characterised by a single-file arrangement of units with short-range interactions, such as hydrocarbons, molecular aggregates with exciton couplings, or spin chains. Rather than propagating the many-particle wavefunction or density matrix in time in the usual stepwise fashion, the MPI scheme proceeds by sequentially integrating over each unit after linking its quantum paths to those of its neighbour and leads to linear scaling with system length. This paper shows that the linking matrix can be further decomposed into a diagonal matrix and a product of low-dimensional matrices, which can be applied sequentially. This factorisation changes the cost scaling from L ² to (Formula presented.), where L is the number of quantum paths, leading to dramatic savings that are analogous to those attained by the fast Fourier transform algorithm and allowing application of the MPI procedure to systems with larger units and to longer propagation times. Applications to spin chains and to electron-vibration dynamics in a large molecular aggregate illustrate the efficiency of the algorithm.