As an example, suppose the first and last sites of an MPS M are combined into an ITensor as phi = M[1] * M[N]. Then you perform some optimization that changes phi, for instance eigsolve using some other tensor as the "matrix" and phi as the "vector." Now I want to put the updated value of phi back into M[1] and M[N], but this requires me to somehow split the ITensor so that the two resulting ITensor objects have the same index structure as M[1] and M[N], respectively. How do I do this?

In regular DMRG, this is done using the factorize method, but I'm pretty sure this only works when the two things you want to separate have an index in common. For instance, if you do phi = M[1] * M[2], then since M[1] and M[2] share a common index, then the appropriate factorization will "sever" the appropriate bond between the two tensors, thus giving you one tensor with the same index structure as M[1] and one tensor with the same index structure as M[2].

However, when I tried this, it doesn't work. In particular, the code seems to introduce a new index to bind the two tensors together. To show an example using code, I did phi = M[1] * M[2] * M[3] * M[6] * M[7] * M[8] on an 8-site MPS M. Then, after doing some changes to phi using eigsolve, I wanted to separate it into M[1] * M[2] * M[3] and M[6] * M[7] * M[8] (each of which can then be separated using further factorize commands). I tried to do so using the following command:

```
index_list = Index[]
for j=1:3
append!(index_list, inds(M[j]))
end
index_set = IndexSet(index_list)
L,R,spec = factorize(phi, index_set; which_decomp = which_decomp,
tags = tags(linkind(M,3)),
kwargs...)
```

But the result I got was the following:

```
left side of phi:
IndexSet{5} (dim=2|id=16|"S=1/2,Site,n=1") (dim=2|id=978|"S=1/2,Site,n=2")
(dim=2|id=311|"S=1/2,Site,n=3") (dim=1|id=305|"Link,l=3") (dim=8|id=372|"Link,l=3")
right side of phi:
IndexSet{5} (dim=8|id=372|"Link,l=3") (dim=1|id=608|"Link,l=5")
(dim=2|id=514|"S=1/2,Site,n=6") (dim=2|id=558|"S=1/2,Site,n=7")
(dim=2|id=515|"S=1/2,Site,n=8")
```

Notice the new index (dim=8|id=372|"Link,l=3") used to connect the two tensors. I don't want to do this--I just want to separate the two tensors so that they have the same index structure as before. How do I do this? It's possible that factorization isn't the right term to describe this.