Exactly Solvable Counting Statistics in Open Weakly Coupled Interacting Spin Systems

We study the full counting statistics for interacting quantum many-body spin systems weakly coupled to the environment. In the leading order in the system-bath coupling, we derive exact spin current statistics for a large class of parity symmetric spin-$1/2$ systems driven by a pair of Markovian baths with local coupling operators. Interestingly, in this class of systems the leading-order current statistics are universal and do not depend on details of the Hamiltonian. Furthermore, in the specific case of a symmetrically boundary driven anisotropic Heisenberg ($XXZ$) spin-$1/2$ chain, we explicitly derive the third-order nonlinear corrections to the current statistics.

Figure 1

The spin transport model. Exemplar parity-symmetric system is coupled locally to two spin baths (represented by boxes), at spin sites 1 and $n$. The baths act on the system via jump operators (3) with the corresponding rates ($a$, $b$, $c$, $d$). The arrows beneath the jump rates indicate the direction in which the spin current is being driven.

Figure 2

The first four current cumulants obtained numerically for the $XXZ$ spin chain and staggered field $XXZ$ spin model, $H=H_{XXZ}+\sum _{j=1}^{n}h(\pm 1{)}^j{\sigma }_j^z$, with field strength $h$, for $n=4$, $\mathrm{\Delta }=0.5$, $\mu =0.5$, $ϵ=0.1$. Dot-dashed (dashed) lines indicate analytical results up to second (fourth) order in $ϵ$.