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Continuous matrix product states for non-relativistic quantum fields: a lattice algorithm for inhomogeneous systems
Martin Ganahl, Guifre Vidal
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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 |Ψ⟩, 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(ϵ_α)}_(α=1,2,⋯,p) on increasingly finer lattices with lattice spacing ϵ₁ > ϵ₂ > ⋯> ϵₚ. We first use energy minimization to optimize an MPS approximation |Ψ(ϵ₁)⟩ for the ground state of H(ϵ₁). Given the MPS |Ψ(ϵ_α)⟩ optimized for the ground state of H(ϵ_α), we use it to initialize the energy minimization for Hamiltonian H(ϵ_(α+1)), resulting in the optimized MPS |Ψ(ϵ_(α+1))⟩. By iteration we produce an optimized MPS |Ψ(ϵₚ)⟩ for the ground state of H(ϵₚ), from which we finally extract the cMPS approximation |Ψ⟩ 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.
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