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In one dimension (1D), a general decaying long-range interaction can be fitto a sum of exponential interactions $e^{-\lambda r_{ij}}$ with varyingexponents $\lambda$, each of which can be represented by a simple matrixproduct operator (MPO) with bond dimension $D=3$. Using this technique,efficient and accurate simulations of 1D quantum systems with long-rangeinteractions can be performed using matrix product states (MPS). However, theextension of this construction to higher dimensions is not obvious. We reporthow to generalize the exponential basis to 2D and 3D by defining the basisfunctions as the Green's functions of the discretized Helmholtz equation fordifferent Helmholtz parameters $\lambda$, a construction which is valid forlattices of any spatial dimension. Compact tensor network representations canthen be found for the discretized Green's functions, by expressing them ascorrelation functions of auxiliary fermionic fields with nearest neighborinteractions via Grassmann Gaussian integration. Interestingly, this analyticconstruction in 3D yields a $D=4$ tensor network representation of correlationfunctions which (asymptotically) decay as the inverse distance ($r^{-1}_{ij}$),thus generating the (screened) Coulomb potential on a cubic lattice. Thesetechniques will be useful in tensor network simulations of realistic materials.

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