Continuous matrix product states for nonrelativistic quantum fields: A lattice algorithm for inhomogeneous systems

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 $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$, directly in the continuum, of the ground state of nonrelativistic 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 ${{H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})}}_{\ensuremath{\alpha}=1,2,\ensuremath{\cdots},p}$ on increasingly finer lattices with lattice spacing ${\ensuremath{\epsilon}}_{1}>{\ensuremath{\epsilon}}_{2}>\ensuremath{\cdots}>{\ensuremath{\epsilon}}_{p}$. We first use energy minimization to optimize an MPS approximation $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{1})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{1})$. Given the MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})\ensuremath{\rangle}$ optimized for the ground state of $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}})$, we use it to initialize the energy minimization for Hamiltonian $H({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})$, resulting in the optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{\ensuremath{\alpha}+1})\ensuremath{\rangle}$. By iteration we produce an optimized MPS $|\mathrm{\ensuremath{\Psi}}({\ensuremath{\epsilon}}_{p})\ensuremath{\rangle}$ for the ground state of $H({\ensuremath{\epsilon}}_{p})$, from which we finally extract the cMPS approximation $|\mathrm{\ensuremath{\Psi}}\ensuremath{\rangle}$ for the ground state of $H$. Two key ingredients of our proposal are as follows: (i) a procedure to discretize $H$ into a lattice model where each site contains a two-dimensional vector space (spanned by vacuum $|0\ensuremath{\rangle}$ and one boson $|1\ensuremath{\rangle}$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.

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