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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.