Tensor Exponential

Question

How do you manage exponential tenor?

Of course, one could do it, but what is the best way?

Consider this example:

T_{ijmn}

I want to combine indices i by m and j by n, then put it into a matrix form and take exponential,

Matrix M=T_{im;Jn} ; M=exp(M);

then put M back into original T

M_{p;q}-------------->T1

It should have the same indices as original T.

T1------------------------->T_{ijmn}

Is it any function to put Itensor into a matrix and vice versa?

The same is the case for eigval decomposition (for general matrices) and....

Answer 1

Hi, so we have the expHermitian function:
http://itensor.org/docs.cgi?page=classes/decomp

For non-Hermitian matrices we don't have a function set up to automatically exponentiate tensors for you.

About putting an ITensor into the form of a matrix, we have combiners, which are special tensors (still of type ITensor or IQTensor, so actually a special type of tensor storage) that combine multiple indices into a single index:
http://itensor.org/docs.cgi?page=classes/combiner

If those don't cover your use case, please tell me more specifically about your case and I can see if ITensor has the tools you will need.

Miles

Comments

Hi,


I want to make transfer matrix of an iPEPS tensor and obtain its eigenvalues. It's not necessary Hermitian.

I have another question:
How could I calculate A+A^{T}?
A_{ij}+A_{ji}?
Do I have to use delta function?

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Hi, so you're in luck on this one: when adding two ITensors (or IQTensors) the library automatically permutes the indices for you. So if you have two tensors A and B, as long as they have the same indices (in whatever order) you can just do:
auto C = A + B;

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You are right. I understand what you mean. But, I want to make C hermitian, not simply adding them.

What about transfer matrix and obtaining its eigenvalues? what is your advise?

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Ok yes, then to make C Hermitian you have to think a bit differently about ITensors vs matrices. With ITensor we don't have the notion of a transpose. Instead we have primed indices. So a tensor that has indices like T_{i i'} is like a square matrix. Then to "transpose" it you do auto TT = swapPrime(T,0,1); which swaps the prime and unprimed indices.

So if you did: auto TS = 0.5*(T + swapPrime(T,0,1)); you will get a symmetrized version of T.

To get a Hermitian version, do: auto TH = 0.5*(T+dag(swapPrime(T,0,1)); where the "dag" also takes the conjugate of the indices and the tensor elements.

About the transfer matrix, I studied it once by writing an Arnoldi routine for iteratively getting the leading eigenvalues of a non-Hermitian matrix. You could use the ITensor davidson routine as an example of how to write a sparse iterative code using ITensor. I should have my old Arnoldi code lying around also. Maybe I should update it for ITensor v2 but it's not something I could do very soon.

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Excellent! Thanks.