My correlation functions from MPS.correlation_function and MPSEnvironment.expectation value disagree

How do I use this algorithm? What does that parameter do?
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xaver
Posts: 10
Joined: 23 Apr 2021, 07:51

My correlation functions from MPS.correlation_function and MPSEnvironment.expectation value disagree

Post by xaver »

Hello,

I have issues, calculation correlation functions of 2-site operators in different ways. Before posting a lot of specific code, let me pls. ask in a more general context:
  • Let \(|\psi\rangle\) be an MPS of finite even length L
  • Let \(A_i B_{i+1}\) denote a nearest neighbor 2-site operator, i.e. some npc.Array shape=(2, 2, 2, 2) labels=['p0', 'p0*', 'p1', 'p1*'], which is composed from two 1-site operators \(\), and \(\) where - to simplify matter - both, \(A\) and \(B\) do also have string labels \(Astr\) and \(Bstr\) (Eg. one could just have Astr=Bstr='Sz' on some spin model).
  • To not have any questions about canonical forms later, let the 2-site op be unitary. (I don't know if this is relevant, but anyway)
  • Eval \( C_a = \langle\psi|A_{L/2} B_{L/2+1} A_i B_{i+1}|\psi\rangle \) i.e. Ca = psi.correlation_function([Astr,Bstr],[Astr,Bstr], [L/2])
  • Now, set \(|\phi\rangle = A_{L/2} B_{L/2+1} |\psi\rangle\), i.e. phi = copy.deepcopy(psi); phi.apply_local_op(L/2,A B)
  • Define an MPSEnvironment like bk = MPSEnvironment(phi,psi) and eval the expectation value \(C_b = \langle\phi|A_i B_{i+1}|\psi\rangle\) with Cb = bk.expectation_value(AB)
My naive expectation was that \(C_a = C_b\), which however it is not for me. By that I mean they just disagree completely.
So what is it that I do not understand?
And, how to properly eval correlation functions using MPSEnvironment?


PS.: If I play the same game with only a single site operator, then \(C_a = C_b\) as I expected.
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Johannes
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Location: TU Munich

Re: My correlation functions from MPS.correlation_function and MPSEnvironment.expectation value disagree

Post by Johannes »

The issue is confusion about the (first two) arguments of correlation_function.

Giving a list of operators as arguments does not mean that you evaluate correlations between multi-site terms, but that you evaluate different correlation functions on even vs odd sites:
psi.correlation_function(['A','B'], ['C', 'D']) evaluates
\[\begin{cases} \langle A_i C_j\rangle & \text{for even $i$, even $j$} \\ \langle B_i C_j\rangle & \text{for odd $i$, even $j$} \\ \langle A_i D_j\rangle & \text{for even $i$, odd $j$} \\ \langle B_i D_j\rangle & \text{for odd $i$, odd $j$} \\ \end{cases}\]

If you want to evaluate \( \langle A_i B_{i+1} C_j D_{j+1} \rangle \), you need to use the term_correlation_function_right or term_correlation_function_left, respectively, which does exactly that!
xaver
Posts: 10
Joined: 23 Apr 2021, 07:51

Re: My correlation functions from MPS.correlation_function and MPSEnvironment.expectation value disagree

Post by xaver »

Aah! Now I do get \(C_a = C_b\) for my system. Great. Thx.

(I was trying to interpret the list of strings from the 2nd figure in tenpy.networks.mps.MPS.correlation_function following the description of correlation_function, which kind of suggested to someone=me, lost in the very helpful docs, that the elements of an optional list of opstr might be consecutively attached to the physical legs ... :oops: )
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