The Jordan-Wigner transformation is a powerful approach to studying one-dimensional spin models. The following dictionary between spin operators and creation/annihilation operators for fermions allows us to recast spin models in terms of fermions:

$$S\_{j}^{+} = c\_{j}^{\dagger}\exp\left[i\pi\sum\_{k<j}n\_{k}\right],\hspace{12pt} S\_{j}^{z} = n\_{j}-\frac{1}{2}$$

For example, a Heisenberg spin-1/2 chain in a magnetic field becomes

$$H = J\sum\_{j}\vec{S}\_{j}\cdot\vec{S}\_{j+1} - h\sum\_{j}S\_{j}^{z} \longrightarrow \sum\_{j}\left[\frac{J}{2}\left(c\_{j}^{\dagger}c\_{j+1}+c\_{j+1}^{\dagger}c\_{j}\right) + J\left(n\_{j}-\frac{1}{2}\right)\left(n\_{j+1}-\frac{1}{2}\right) -hn\_{j}\right].$$

The original spin model has a $U(1)$-symmetry associated with rotations about the magnetic field axis; in the fermion model, this is translated into conservation of total fermion number. There are several consequences of this for, e.g., quantum critical points induced by an applied magnetic field. Moreover, this is why the fermion theory is local in spite of the apparent non-locality in the Jordan-Wigner dictionary.

My question is, what can we say about such a model in which the $U(1)$ symmetry is broken from the beginning?

There are many choices which can accomplish this, but one simple choice is to include a Heisenberg anisotropy. This destroys our freedom to choose $h\propto\hat{z}$, so the spin model will look something like

$$H = J\sum\_{j}\left(\vec{S}\_{j}\cdot\vec{S}\_{j+1} + \lambda S\_{j}^{y}S\_{j+1}^{y}\right) - \sum\_{j}\vec{h}\cdot\vec{S}\_{j}$$

The $U(1)$ symmetry is certainly broken now; moreover, if $h$ has components in the x or y directions, the corresponding fermion model appears to have a nonlocal structure, since a single operator $S^{\pm}$ gets turned into a long-range string of fermion operators. Are there worked out examinations of such models in the literature? All examples of the Jordan-Wigner mapping I have seen feature number conservation and spatial locality, but it seems to me violations of these constraints are very common.