## Cell choice and long range correlations in a 2D iDMRG calculation

How do I use this algorithm? What does that parameter do?
yanghao
Posts: 2
Joined: 12 Jul 2022, 08:29

### Cell choice and long range correlations in a 2D iDMRG calculation

Hi,

I am new to the field. I am wondering how we choose the "Lx" parameter in a 2D iDMRG calculation. It seems like in the literature there are not much discussion on it. For example in https://journals.aps.org/prb/pdf/10.110 ... .94.121111 Ly is reported but Lx is not specified. Is this because we always assume translational invariance along the x-axis so Lx=1? If this is the case, how do we compute the NN correlations beyond the x=0 cell such as those shown in Fig. 1 in the above work or even long-range correlations in Tenpy? Also, is it possible to compute the entanglement spectrum along a cut at some finite x value with Lx=1 only?

Many thanks!
Yang-hao
Johannes
Posts: 337
Joined: 21 Jul 2018, 12:52
Location: TU Munich

### Re: Cell choice and long range correlations in a 2D iDMRG calculation

If you expect translation invariance along the cylinder direction (and you Hamiltonian itself is also translation invariant in x-direction), Lx=1 is certainly a natural choice. Using Lx=1 does not hinder the calculation of correlation lengths - in general, if you have an infinite MPS, it just implies that you have a unit cell that repeats, but you can insert operators to be measured anywhere. The trick is to insert as many unit-cells as needed, and use the properties of the canonical form to collapse the (infinite) contractions on the left/right.
As an example, consider a Lx=1, Ly=3 square lattice, where you would have a three-site MPS unit cell with (right-canonical) tensor B0, B1, B2.
Say you want to measure $$\left\langle X_{x=5,y=1} Y_{x=7,y=0} \right\rangle =_{TI} \left\langle X_{x,y=1} Y_{x+2,y=0} \right\rangle$$, then you would get a tensor network diagram looking like this:

Code: Select all

x,y=  -1,0 -1,1 -1,2  0,0  0,1  0,2  1,0  1,1  1,2  2,0  2,1  2,2  3,0  3,1  3,2

... ---B0---B1---B2---B0---B1---B2---B0---B1---B2---B0---B1---B2---B0---B1---B2---  ...
|    |    |    |    |    |    |    |    |    |    |    |    |    |    |
|    |    |    |    X    |    |    |    |    Y    |    |    |    |    |
|    |    |    |    |    |    |    |    |    |    |    |    |    |    |
... ---B0*--B1*--B2*--B0*--B1*--B2*--B0*--B1*--B2*--B0*--B1*--B2*--B0*--B1*--B2*--  ...

--S--B1---B2---B0---B1---B2---B0---
(can. form)          |    |    |    |    |    |    |   |
=              |    X    |    |    |    |    Y   |
|    |    |    |    |    |    |   |
--S--B1*--B2*--B0*--B1*--B2*--B0*--


Here, the second line used the canonical form properties to collabse contractions of B's from the right into identities, and the left to the squared singular values.