Dzyloshinskii-Moriya interactions in valence bond systems II*

We investigate the effect of Dzyaloshinskii-Moriya interactions on the low temperature magnetic susceptibility for a system whose low energy physics is dominated by short-range valence bonds (singlets). Our general perturbative approach is applied to specific models expected to be in this class, including the Shastry-Sutherland model of the spin-dimer compound SrCu$_2 $(BO$_3$)$_2$ and the antiferromagnetic Heisenberg model of the recently discovered $S=1/2$ kagome compound ZnCu$_3$(OH)$_6 $Cl$_2$. The central result is that a short-ranged valence bond phase, when perturbed with Dzyaloshinskii-Moriya interactions, will remain time-reversal symmetric in the absence of a magnetic field but the susceptibility will be nonzero in the zero temperature limit. Applied to ZnCu$_3$(OH)$_6$Cl$_2$, this model provides an avenue for reconciling experimental results, such as the lack of magnetic order and lack of any sign of a spin gap, with known theoretical facts about the kagome Heisenberg antiferromagnet.