## Implementing non on-site symmetries

Discussing the best way to implement feature X
Umberto Borla
Posts: 18
Joined: 23 Jul 2018, 09:23
Location: Technical University Munich

### Implementing non on-site symmetries

Hello,

I am planning some large scale iDMRG simulations of a 2D spin model on the cylinder. The model has a global U(1) symmetry which could be exploited, corresponding to on-site particle number conservation in an analogous fermionic formulation of the model. In the spin language, however, the number operator maps into a star operator (up to constants):

$$N=\sum_i n_i=\sum_{i}\frac{1-\sigma^x_{i,i+\hat{x}}\sigma^x_{i,i-\hat{x}}\sigma^x_{i,i+\hat{y}}\sigma^x_{i,i-\hat{y}}}{2}$$

and therefore it is not on-site. I was wandering if implementing such multi-site conserved charges is something that can be done, and how much effort it would require given the current status of TeNPy. Of course I would be happy to contribute to the task, but at the moment I do not have enough knowledge of how np_conserved works to even tell if this is feasible, so I would like to hear the opinion of someone more expert then me on the matter.

Thanks!
Johannes
Posts: 357
Joined: 21 Jul 2018, 12:52
Location: TU Munich

### Re: Implementing non on-site symmetries

Ugh, that's a really tough one!

First of all: Is it really necessary to do the mapping from fermions to bosons, or couldn't you maybe directly implement the fermionic model instead?
(I'd guess not, but just to make sure...)

My first reaction definitely is "That's impossible".
TeNPy's np_conserved really needs the structure of the conserved quantity to be something like sum_i diag_op(i), because this directly ensures that the Schmidt states have definite charge values, and that the "charge rule", which determines the non-zero entries of the tensor,
takes the form given in eq. 47 of the TeNPyNotes.

Thinking a little bit, I though that you could maybe generalize this charge rule. Doing this will imply to reimplement at least tensordot, svd, and so on, which are base on it.
However, I think that still can't work, because if you think about it, a schmidt state left to a given bond won't have a definite charge value.
So really you need to do global basis change to exploit that symmetry.

In any case, I think that exploiting this charge would require a more or less complete reimplementation of np_conserved, in particular tensordot, svd etc. Even if it might be possible, the faster computing time probably doesn't pay off the precious human time you would need to invest to make this work. Sorry
Umberto Borla
Posts: 18
Joined: 23 Jul 2018, 09:23
Location: Technical University Munich

### Re: Implementing non on-site symmetries

Don't worry, from what I could understand I was expecting it to be a really challenging task.

Just to provide some context, the original model is a gauge theory with fermionic matter. The mapping eliminates all gauge redundancy, so that the mapped model has a smaller number of d.o.f. and so it is in principle easier to study, although since in the original model one can conserve N, there is indeed a trade off. Besides, in the original model the Gauss law that selects the physical states has to be implemented energetically in the Hamiltonian, while it is automatically satisfied in the spin model.

Thanks anyway!
Johannes