In an integrable quantum system (say XXZ model), where there is an extensive number of conserved charges, does the set of local conserved charges obtained from expanding the log of the transfer matrix form a complete commuting set? In other words, does specifying all conserved charges obtained from the transfer matrix specify a unique state of the system?
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2 Answers
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"Remarks on the notion of quantum integrability" (JS Caux and J Mossel, Journal of Statistical Mechanics: Theory and Experiment 2011, no. 02, 2011), Sec. 5 mentions with regard to the charges obtained from the expansion of the log of the transfer matrix that it is "reasonable to assume that the set of charges so obtained is maximal, though it remains a conjecture", so this seems to be unproven in general (at least at the time of the publication of that paper).
I could not find stronger statement for the special case of the XXZ model in the literature, but would be very interested in this in case someone has a better answer!
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$\begingroup$ For a representation-theoretic perspective see my answer to a closely related question: physics.stackexchange.com/a/806771 $\endgroup$ Commented Apr 28 at 20:05
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I think the keywords you looking for are the generalised Gibbs ensemble (GGE) and the set of conserved charges known as quasilocal charges, which are complete in the sense from the last sentence of the OP. See also the last paragraph of my answer at a related question, and especially the two references therein to the work of Ilievski et al for the Heisenberg XXZ chain.
