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

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 $|\Psi ⟩$, 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 $|\Psi ({\epsilon }_1)⟩$ for the ground state of $H({\epsilon }_1)$. Given the MPS $|\Psi ({\epsilon }_{\alpha })⟩$ 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 $|\Psi ({\epsilon }_{\alpha +1})⟩$. By iteration we produce an optimized MPS $|\Psi ({\epsilon }_p)⟩$ for the ground state of $H({\epsilon }_p)$, from which we finally extract the cMPS approximation $|\Psi ⟩$ 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 $|0⟩$ and one boson $|1⟩$ states), and (ii) a procedure to map MPS representations from a coarser lattice to a finer lattice.

11+8 pages, 10 figures

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• Density-Matrix Renormalization Group for Continuous Quantum Systems
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• Entanglement renormalization for gauge invariant quantum fields
• Numerical continuum tensor networks in two dimensions
• Continuous matrix-product states in inhomogeneous systems with long-range interactions
• Entanglement Renormalization of the class of Continuous Matrix Product States
• Generalised ansatz for continuous Matrix Product States
• Holographic Tensor Networks as Tessellations of Geometry