# Subspace expansion to get excited states

Hi,

I recently implemented single-site DMRG with subspace expansion (https://arxiv.org/abs/1501.05504) using ITensor. It works great for ground state. However, I am not sure about the proper expansion step for the excited states. Right now, I am using the same expansion procedure as the ground-state search, and it is working good provided I employ the bare single-site DMRG at the end. It would be great if somebody has some idea about the correct expansion term in the excited state search.

With regards,
Titas

Hi Titas,

Can you extend the subspace expansion trick from that paper to excited states by replacing the use of H by H - |psi0><psi0|, where |psi0> is the ground state you previously found?

It looks like the expansion term they use just depends on the current guess for the MPS you are optimizing and the MPO you are finding the ground state of, so you should be able to extend the formulas in the paper for the case when the MPO is a sum of MPOs, i.e. the Hamiltonian plus the projector onto the ground state.

Cheers,
Matt

commented by (290 points)
Hi Matt,

Initially I thought the same. But the expansion term depends on the MPO bond dimension, and for the projectors it would be huge even if I treat them in a sparse way, as similar as the ITensor DMRG code. I may miss something.

Regards,
Titas
commented by (290 points)
Hi Matt,

For Heisenberg chain, if I work in bond dimension 100, then the expansion term due to $H$ is a tensor of (2, 100, 5x100) dimension, which is not much. But for each projector, the same would be a tensor of (2, 100, 100x100), which is too large, and will grow ballistically with bond dimension. I just tried with this, but ram consumption is too high and the code is slow. Is there any other workaround to do it that you know? I didn't find any 'sparse' way do it, that does not involve any tensor product of bond indices involved in the projector.

With regards,
Titas
commented ago by (4.8k points)
Hi Titas,

Sorry for the late response. I have to admit I am not very familiar with subspace expansion techniques.

It does seem like you should be able to take advantage of the sparsity of the tensor network structure to deal efficiently with the projector (since the expansion term looks a lot like terms that show up in normal DMRG calculations). Could you maybe show a minimal working example of what you have tried so we can see where it goes wrong?

Another suggestion is that you could use 2-site DMRG in order to grow the bond dimension, and then when you are at the desired bond dimension you can use 1-site DMRG.

Beyond that you may have to delve into the literature to see if anyone has come up with subspace expansion techniques more suitable for excited states.

Cheers,
Matt