{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/continuous-matrix-product-states-for-non","title":"Continuous matrix product states for non-relativistic quantum fields: a lattice algorithm for inhomogeneous systems","arxiv_id":"1801.02219","date":"2018-01-07","proceeding":null,"authors":["Martin Ganahl","Guifre Vidal"],"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.","url_abs":"http://arxiv.org/abs/1801.02219v1","url_pdf":"http://arxiv.org/pdf/1801.02219v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"continuous-matrix-product-states-for-non","repo_url":"https://github.com/mganahl/PyTeN","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}