# Time evolution of a discrete 1-d lattice of spin-(1/2) particles under a given Hamiltonian, or special cases thereof

I am trying to get some feel for the dynamics induced on a discrete 1-d lattice of spin-(1/2) quantum particles by the following Hamiltonian

$\hat{H} = \sum_{i, j} r_{i j} \left[ \mathrm{sin}(\alpha_{i j}) \left(\hat{S}^{x}_{i} \hat{S}^{x}_{j} - \hat{S}^{y}_{i} \hat{S}^{y}_{j} \right) + \mathrm{cos}(\alpha_{i j}) \left(\hat{S}^{x}_{i} \hat{S}^{y}_{j} + \hat{S}^{y}_{i} \hat{S}^{x}_{j} \right) \right]$

where $i$ and $j$ are particle labels, $r_{i, j}$ quantifies the overall interaction strength between particles $i$ and $j$, $\alpha_{i j}$ quantifies the relative strength of the two terms for these particles, and $\hat{S}^{\{x, y\}}_i$ are the $x$ and $y$ spin-(1/2) angular momentum operators on particle $i$.

Can anyone point me to some literature on this family of Hamiltonians or, if anyone recognizes this as a general case of a more well-characterized family, some literature on that special case? Thank you in advance!

• Do you know anything about $r_{ij}$? (If not, why is it a 1D lattice?) – Norbert Schuch Feb 26 '15 at 18:58
• Actually, that's sort of what I am asking. What constraints can I put on the $r_{ij}, \alpha_{ij}$ (e.g. translational invariance) to arrive at a well-characterized model? Sorry for making that unclear. – Adrian Feb 26 '15 at 20:57
• This makes this a rather broad question. If you only have nearest neighbor interactions, this is can be transformed to the XX model (or directly solved by mapping it to free fermions). If the $r_{ij}$ only act between even and odd sites, you can transform this to a model of hopping particles (but this can be hard to solve). --- In any case: Why do you call this a 1-d lattice? The way you write it there is no 1-d structure. – Norbert Schuch Feb 26 '15 at 21:35