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I was studying the algebraic Bethe ansatz for the spin-1/2 XXZ model. In the end one ends up with $2^L$ integrals of motion $Q_k$ that commute with the Hamiltonian, (https://doi.org/10.1103/PhysRevLett.125.090602) and (https://doi.org/10.1088/1751-8121/ac0961).

However the way they are defined is very dense, and I just wanted to know if the integrals of motions are written down explicitly anywhere, at least for a new sites, $n=2,3,4$, etc, in terms of the Pauli/spin matrices. There should be (4, 8, 16) of them, so in principle you should be able to write them down.

I guess I wanted to see if for the few site integrals of motion you can start to see that they are somehow local, and independent from the projectors $\lvert n \rangle \langle n \rvert$.

Qmechanic
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  • Does the question, my answer and the comments at https://physics.stackexchange.com/q/669182/ help? – Jules Lamers Nov 14 '23 at 23:54
  • Hi Jules, thanks for the reply. I already knew that Q_1 is the magnetisation, and Q_2 is the Hamiltonian, but I was hoping to find at least Q_3, the nontrivial conserved quantity. I went through your notes but couldn't find an explicit construction of Q_3. Is it possible to write it down for the XXZ model? – purestate Nov 16 '23 at 05:43
  • The next charge can be e.g. found in Exercise 2.7 of the book C. Gómez, M. Ruiz-Altaba, and G. Sierra, "Quantum Groups in Two-Dimensional Physics" (Cambridge University Press, 1996) – Jules Lamers Nov 17 '23 at 11:01
  • (I'm just talking about the log derivative of the transfer matrix, which gives $L$ charges, if one also includes $S^z$. I have not read the first reference that you cite.) – Jules Lamers Nov 17 '23 at 11:40
  • See also https://physics.stackexchange.com/a/796554/ – Jules Lamers Jan 09 '24 at 01:45
  • Did the above help? – Jules Lamers Mar 22 '24 at 19:00

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