We prove an entanglement area law for a class of 1D quantum systems involving
    infinite-dimensional local Hilbert spaces. This class of quantum systems
    include bosonic models such as the Hubbard-Holstein model, and both U(1) and
    SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new
    results concerning the robustness of the ground state and spectral gap to the
    truncation of Hilbert space, applied within the approximate ground state
    projector (AGSP) framework from previous work. In establishing this area law,
    we develop a system-size independent bound on the expectation value of local
    observables for Hamiltonians without translation symmetry, which may be of
    separate interest. Our result provides theoretical justification for using
    tensor network methods to study the ground state properties of quantum systems
    with infinite local degrees of freedom.

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