Using the sum of exponentials to fit long-range interactions

[LieBUPT]

Several years ago, the idea of fitting long-range interactions with sums of exponentials was discussed on the ITensor forumhttps://itensor.org/support/270/constructing-long-range-hamiltonians-with-mpos. The reason I am opening this issue is that I would like to learn more about the technical details of how this approach is implemented in ITensorMPS. My question may be a bit roundabout to explain, but I’ll try to make it as clear as possible below.

Example

Let me use an example to illustrate my question. If I want to construct the MPO of a simple Hamiltonian $H=\sum_{i<j} (1/|j-i|^\alpha) \sigma_i^z \sigma_j^z$, I can approximate it by the sum of several Hamiltonians

$$H = \sum_{m=1}^M x_m H_m = \sum_{m=1}^M x_m [ \sum_{i<j} \lambda_k^{|i-j|} \sigma_{i}^{z} \sigma_{j}^{z} ]$$，

where the coefficients ${x_m}$ can be determined by an optimization procedure. The question is: how should one construct the Hamiltonian with exponentially decaying interactions using ITensor?

Direct coding approach

One straightforward way is to explicitly write code such as:

H_m = OpSum()

for i in 1:L
    for j in 1:i-1  
        xi = (i - 1) % L
        xj = (j - 1) % L
        Vij = lambda^abs(xi-xj)
        H_m += Vij, "Sz", i, "Sz", j
    end
end

H_MPO_m = MPO(Float64, H_m, sites)

Then, one could combine all ${H_{\text{MPO},m}}$ with the coefficients ${x_m}$ to obtain the MPO for $H$.

My question

However, according to Pirvu, Murg, Cirac, and Verstraete, New J. Phys. 12, 025012 (2010), the MPO representation of each $H_m$ has a compact analytical form with small bond dimension.

• Does the MPO constructor in ITensor automatically produce such a compact form, or will it generate a more redundant representation?
• Is there a specialized function in ITensor (similar to fit_with_sum_of_exp in TeNPy) that can directly construct MPOs with exponentially decaying interactions?
• If not, does one need to manually construct the MPO by explicitly writing out the $B$ matrices (as shown in the figure I attached below)?

[emstoudenmire]

Thanks for your interest, but since this seems to be purely a question or a discussion, could you please post it on our ITensor Discussion Forum? That would be a great place for such discussions.

As a short answer here, the approach used by our OpSum to MPO conversion system in ITensorMPS incorporates the idea of fitting long-range interactions by a sum of exponentials by viewing that as a special case of a more sophisticated algorithm that does a sequence of SVD's across each bond of the network. It can achieve the same bond dimensions or better as the fitting with exponentials approach.