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PowerLaw

Introduction

A type of long-range interaction that is commonly studied is where each site in the system is coupled to every other site in the system with a strength proportional to some (negative) power of the distance between them. For example, the (infinite) Haldane--Shastry model [1,2] has the inverse-square interaction {$$H = \sum_{i<j} \frac{1}{(j-i)^2} \mathbf{S}_i \cdot \mathbf{S}_j.$$}

However, power-law interactions cannot be exactly represented as an MPO with a finite bond dimension. One approach could be to write the full MPO for some large cutoff in distance, and them use an MPO compression algorithm to reduce the MPO bond dimension to a computationally feasible size [3]. Another approach is to recognise that exponentially decaying interactions can be represented by an MPO with bond dimension 3, and a sum of {$N$} exponentials has bond dimension {$N+2$}. We can then approximate the power law by a weighted sum of {$N$} exponential functions [4] {$$\frac{1}{n^2} \approx \sum_{i=1}^N \alpha_i \exp (\beta_i n).$$} We can then optimise this fit by minimising the error below some maximum length {$L$}. (In fact, the MPO compression approach will end up producing a sum of exponential functions as well, although the exact expressions chosen by each method may be different.)

This optimisation can be done using the following Python script:

This should produce something like

which can then be used as an operator expression for mp-idmrg-s3e, for example:

The exact energy density of the ground state is {$-\pi^2/24 = 0.4112335167120566\ldots$}, which we can check with

References