Variational Optimization of Continuous Matrix Product States

Just as matrix product states represent ground states of one-dimensional quantum spin systems faithfully, continuous matrix product states (cMPS) provide faithful representations of the vacuum of interacting field theories in one spatial dimension. Unlike the quantum spin case, however, for which the density matrix renormalization group and related matrix product state algorithms provide robust algorithms for optimizing the variational states, the optimization of cMPS for systems with inhomogeneous external potentials has been problematic. We resolve this problem by constructing a piecewise linear parameterization of the underlying matrix-valued functions, which enables the calculation of the exact reduced density matrices everywhere in the system by high-order Taylor expansions. This turns the variational cMPS problem into a variational algorithm from which both the energy and its backwards derivative can be calculated exactly and at a cost that scales as the cube of the bond dimension. We illustrate this by finding ground states of interacting bosons in external potentials and by calculating boundary or Casimir energy corrections of continuous many-body systems with open boundary conditions.

Automated summary

Matrix product states and continuous matrix product states are faithful representations of ground states of one-dimensional quantum spin systems and interacting field theories in one spatial dimension, respectively. By constructing a piecewise linear parameterization and using high-order Taylor expansions, we are able to optimize the continuous matrix product states efficiently for systems with inhomogeneous external potentials. This method allows for the calculation of exact reduced density matrices and exact computation of energy and its backwards derivative.