Long-range interactions: What is the limit?

Hi everyone,

I am trying to solve for the dynamics using TDVP in the Peierls-Kane-Mele model

H = \sum_{i\sigma}\varepsilon_{i\sigma}^{\phantom{\dagger}} c_{i\sigma}^\dagger c_{i\sigma}^{\phantom{\dagger}} -\sum_{i\sigma}c_{i\sigma}^\dagger c_{i+1\sigma}^{\phantom{\dagger}}(t_0+t_1(a_i^\dagger+a_i^{\phantom{\dagger}})) + \lambda\sum_{i\sigma\sigma'}c_{i\sigma}^\dagger(\vec{v}\cdot\vec{\sigma})_{\sigma\sigma'}c_{i+2\sigma'}^{\phantom{\dagger}},

where c,c^\dagger are electronic and a,a^\dagger are bosonic operators, as usual, and \vec{v} is a pseudo magnetic field coupling to the electrons’ spin-dependent next-nearest neighbor hopping.

If t_1=0 and \lambda\neq 0 (electron-sites only), then the TDVP result agrees perfectly with ED.
If t_1\neq 0 and \lambda=0 (MixedSiteSet), results agree with ED as well.
But when t_1\neq 0 and \lambda \neq 0, the results deviate a lot from ED. Changing the sweep parameters (maxdim, cutoff, niter), or the parameters in the global subspace expansion (which I now tried doing at each timestep instead of only for the first couple ones), or the parameters in the TDVP itself did not result in any improvement of the results. In fact, if I would have no ED results to rely on, I would’ve concluded that the TDVP simulations were converged.

Is this behavior to be expected?

In the first two cases, the longest coupling is between next-to-nearest core tensors (say for a MPS M_1M_2M_3, the \lambda term would couple M_1 with M_3 only. In the first case, M_i would represent electronic DOF, while in the second case M_1,M_3 are electronic and M_2 a bosonic DOF).
In the case where both terms are treated simultaneously, the longest coupling is between a core tensor and the tensor at the fourth place next to it (for a MPS M_1M_2M_3M_4M_5, \lambda\neq 0 and t_1\neq 0 the longest coupling would be between M_1 with M_5, assuming electrons and phonons are aranged in alternating order).

Any suggestions on how to solve this issue would be very appreciated. In Simulation of Hamiltonian evolution of long-range interaction using MPS, @miles mentioned extra techniques that need to be done in combination with TDVP. Would these be helpful for the Hamiltonian I am looking at? If so, I would really appreciate some more information on that.


Sorry to hear you’re experiencing this issue! I don’t have an immediate answer or guess as to what is going on, other than TDVP can sometimes be tricky without using various convergence helping methods. The global subspace expansion is the one I’m most familiar with.

Would you be willing to share both your TDVP and ED codes (could be by email) so we can try some things out ourselves? I’d like to make sure our TDVP code is state of the art. I can’t promise we will be very quick in testing this but I’d like to work on it, thanks.

Thanks for your reply! After performing some more tests, I realized that there was a bug in a part of the Hamiltonian, hence the issue is not related to TDVP. After fixing this bug, the result agrees well with ED.

That’s great to hear and glad you found the bug.