Definition of conserved quantities in integrable system

Question

This question is about the definition of conserved quantities integrable systems.

Using Algebraic Bethe ansatz,a family of commuting operators $F(\lambda)$ can be contructed by taking a partial trace (over the auxiliary space) of the monodromy matrix $T(\lambda)$

$$F(\lambda) = tr(T(\lambda)),$$

$$[F(\lambda),F(\mu)] = 0,$$

as stated in Faddeev's paper.

Furthermore, they should be "functionally" independent; this concept is clear in classical integrable systems but I can't find a good definition for the quantum case.

In principle, one can construct an extensive number of conserved quantities for a finite dimensional Hilbert space by considering projectors onto each eigenspace $P_{n} = |n\rangle \langle n|$

$$[P_n,H]=0,$$

this argument is independent of $H$.Therefore, having an extensive number of conserved quantities should not be a good definition for integrability.

These conserved quantities should probably be local (i.e. sum of few-body terms), but it is not clear if the trace of the monodromy matrix always gives such local operators.

Answer 1

As you point out, there's an issue with the notion of quantum integrability defined, in analogy with the classical case, through the existence of sufficiently many conserved charges. For more on this, see Caux and Mossel, "Remarks on the notion of quantum integrability" https://arxiv.org/abs/1012.3587

So instead one can ask for a underlying algebraic notion, called 'Yang--Baxter integrability', that is well defined and implies the existence of many conserved charges. For more about this, see also https://physics.stackexchange.com/a/780318/.

The commuting charges constructed in this way (as logarithmic derivatives of the transfer matrix) are less and less local, see How local are the conserved charges in a quantum integrable model?. However, one can construct so-called quasi-local charges as the (first) logarithmic derivative of transfer matrices with higher spin in the auxiliary space (rather than higher derivatives of the 'fundamental' transfer matrix with spin-1/2 auxiliary space). See Ilievski, Medenjak and Prosen, "Quasilocal conserved operators in isotropic Heisenberg spin 1/2 chain" (https://arxiv.org/abs/1506.05049) and Ilievski et al, "Complete Generalized Gibbs Ensemble in an interacting Theory" (https://arxiv.org/abs/1507.02993).