# A question about time evolution in tutorial/finiteT/metts.cc

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Hello,

In ITensor3 tutorial/finiteT/metts.cc, what puzzled me is "psi.ref(1) /= norm(psi(1));". Does this relate to the probability {<cps|exp(-0.5betaH)|cps>}^{-0.5}(New J. Phys. 12(5), 055026(2010))?　If this is true, how to get the result and why not the overlap of two MPS? Does this apply to different models?

println("Doing time evolution");
for(int tt = 1; tt <= nt; ++tt)
{
psi = applyMPO(expH,psi,args);
psi.ref(1) /= norm(psi(1));
}


Thanks
Zong-Sheng Zhou

## 1 Answer

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answered by (70.1k points)

Hi Zong-Sheng,
Thanks for the question. Yes, that line of code is just normalizing the MPS to have norm equal to 1. The reason it only accesses the first MPS tensor (.ref(1)) is that the applyMPO function guarantees that the returned MPS will be in right-canonical gauge, so that the norm of the whole MPS equals the norm of the first MPS tensor.

The expression of the paper is the norm that the (unnormalized) METTS would have if one did the imaginary time evolution all in one big step. However, here the expH MPO is just for a step of tau, and so the norm afterward will be different (the same expression but with tau instead of beta). Dividing it off at every step or dividing it off at the very end will give the same final normalized state.

Hope that helps,
Miles

commented by (270 points)
Hi Miles.

Thanks for your answer. That is helpful and I think I got it now.

Best regards!
Zong-Sheng Zhou