On the generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

Abstract

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions $e^{-\lambda r_{ij}}$ with varying exponents $\lambda$, each of which can be represented by a simple matrix product operator (MPO) with bond dimension $D=3$. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states (MPS). However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to 2D and 3D by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters $\lambda$, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in 3D yields a $D=4$ tensor network representation of correlation functions which (asymptotically) decay as the inverse distance ($r_{ij}^{-1}$), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.

14 pages, 12 figures

References in corpus (21)

• The density-matrix renormalization group in the age of matrix product states
• A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States
• Matrix Product Density Operators: Simulation of finite-T and dissipative systems
• Criticality, the area law, and the computational power of PEPS
• Ab Initio Quantum Chemistry using the Density Matrix Renormalization Group
• Corner Transfer Matrix Renormalization Group Method
• From density-matrix renormalization group to matrix product states
• Matrix product operator representations
• Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States
• Fermionic Projected Entangled Pair States
• Renormalization of tensor-network states
• Applying matrix product operators to model systems with long-range interactions
• Finite automata for caching in matrix product algorithms
• One-dimensional Continuum Electronic Structure with the Density Matrix Renormalization Group and Its Implications For Density Functional Theory
• Tensor operators: constructions and applications for long-range interaction systems
• Reference electronic structure calculations in one dimension
• Fermionic Implementation of Projected Entangled Pair States Algorithm
• Multigrid Algorithms for Tensor Network States
• Sliced Basis Density Matrix Renormalization Group for Electronic Structure
• Efficient representation of long-range interactions in tensor network algorithms
• Exact bosonic Matrix Product States (and holography)

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• Quantum Simulation of 2D Quantum Chemistry in Optical Lattices
• A simplified and improved approach to tensor network operators in two dimensions
• Projected d-wave superconducting state: a fermionic projected entangled pair state study
• Numerical continuum tensor networks in two dimensions
• Variational Quantum Imaginary Time Evolution for Matrix Product State Ansatz with Tests on Transcorrelated Hamiltonians
• Quantum Phase Transitions in Long-Range Interacting Hyperuniform Spin Chains in a Transverse Field