Continuous matrix product states for non-relativistic quantum fields: a lattice algorithm for inhomogeneous systems

TL;DR

This paper introduces a lattice algorithm combining continuous matrix product states with standard MPS optimization to approximate ground states of non-relativistic quantum fields, effective for both homogeneous and inhomogeneous systems.

Contribution

It develops a novel method to directly approximate continuum quantum field ground states using a sequence of discretized Hamiltonians and lattice MPS techniques, bridging continuum and lattice approaches.

Findings

Successfully applied to inhomogeneous bosonic systems in periodic potentials.

Achieved accurate ground state approximations through iterative lattice refinement.

Demonstrated the method's effectiveness for non-uniform quantum fields.

Abstract

By combining the continuous matrix product state (cMPS) representation for quantum fields in the continuum with standard optimization techniques for matrix product states (MPS) on the lattice, we obtain an approximation $|\Psi\rangle$, directly in the continuum, of the ground state of non-relativistic quantum field theories. This construction works both for translation invariant systems and in the more challenging context of inhomogeneous systems, as we demonstrate for an interacting bosonic field in a periodic potential. Given the continuum Hamiltonian $H$, we consider a sequence of discretized Hamiltonians $\{H(\epsilon_{\alpha})\}_{\alpha=1,2,\cdots,p}$ on increasingly finer lattices with lattice spacing $\epsilon_1 > \epsilon_2 > \cdots > \epsilon_p$. We first use energy minimization to optimize an MPS approximation $|\Psi(\epsilon_1)\rangle$ for the ground state of $H(\epsilon_1)$. Given the MPS $|\Psi(\epsilon_{\alpha})\rangle$ optimized for the ground state of $H(\epsilon_{\alpha})$, we use it to initialize the energy minimization for Hamiltonian $H(\epsilon_{\alpha+1})$, resulting in the optimized MPS $|\Psi(\epsilon_{\alpha+1})\rangle$. By iteration we produce an optimized MPS $|\Psi(\epsilon_{p})\rangle$ for the ground state of $H(\epsilon_p)$, from which we finally extract the cMPS approximation $|\Psi\rangle$ for the ground state of $H$. Two key ingredients of our proposal are: (i) a procedure to discretize $H$ into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\rangle$ and one boson $|1\rangle$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.