How local are the conserved charges in a quantum integrable model? For the purposes of this question, let us define a quantum integrable model as one solvable by the Bethe Ansatz. That structure endows the model with a set of conserved charges $\{H^{(n)}\}$ whose number is extensive in the system size $N$. My question is about how local are these charges.
Working in the thermodynamic limit $N \to \infty$, let us say that an operator $O$ is local if it is supported on an $\mathcal{O}(1)$ number of sites, and that it is quasilocal if its support decays sufficiently quickly with distance (say faster than $1/r^{d}$ in $d$ dimensions). We might also reasonably apply this terminology to an operator which can be written as a sum of terms with these support properties.
My question was prompted by reading this paper. There the authors study the spin-$\frac{1}{2}$ Heisenberg chain
$$H = \dfrac{J}{4} \sum_{j=1}^{N} \left[\sigma_{j}^{x} \sigma_{j+1}^{x} + \sigma_{j}^{y} \sigma_{j+1}^{y} + \Delta (\sigma_{j}^{z} \sigma_{j+1}^{z} -1) \right],$$
where $J>0$, $\sigma_{j}^{\alpha}$, $\alpha=x,y,z$ are the Pauli matrices, and they consider the regime $\Delta \geq 1$.
At the bottom of p2, the authors identify an infinite set of conserved charges $\{H^{(n)}\}$, which they dub ultra-local. They justify this terminology with the sentence

These charges are ultra-local in the sense that they can be written as $H^{(m)} = \sum_{j=1}^{N} h_{j}^{(m)}$, where the operators $h_{j}^{(m)}$ act nontrivially on a block of at most $m$ sites adjacent to $j$.

Here is where I am confused. If I understand correctly, the index $m$ can be extensive in the system size $N$, so the operators $h_{j}^{(m)}$ could in principle have $\mathcal{O}(N)$ support. If we assume the definition of locality I gave above, which seems fairly reasonable, how then can the authors call these conserved charges ultra-local?
 A: By $H^{(m)}$ the authors denote the usual conserved charge, i.e. the $m$th logarithmic derivative of the transfer matrix at a suitable value of the spectral parameter. For example, for $m=1$ this gives the XXZ spin chain Hamiltonian: a sum of terms $h^{(1)}_j$ that each act at blocks of at most one site adjacent to site $j$, i.e. at nearest neighbours $j,j+1$. This is true for any system size $N$.
If you similarly compute the next ultralocal charge $H^{(2)}$ you will get a sum of terms that act at sites $j,j+1,j+2$. (You might want do this explicitly as an exercise. The result can be found e.g. in my lecture notes on the arXiv.) Again this form is the same for any $N$.
In general, for fixed $m$ the summands act nontrivially at at most $m+1$ adjacent sites, independently of $N$. There's nothing extensive about these summands when you fix $m$.
The only thing that changes with $N$ is how many summands there are (the sum runs up to $N$), and how many such charges you have. Indeed, up to a possible common factor (depending on the normalisation of the R-matrix) the entries of the transfer matrix are polynomials in the spectral parameter of degree at most $N$, so you can in principle compute nontrivial charges for $0\leq m\leq N$. (Here $m=0$ corresponds to the momentum operator, i.e. the log of the translation = cyclic shift operator.)
