Yang-Lee edge singularities at high temperatures

The density of Lee-Yang zeros, in the thermodynamic limit, for classical $n$-vector models and for the quantum Heisenberg model is studied in the asymptotic high-temperature limit. It is shown that the high-temperature series expansions for these models reduce, in this limit, to the corresponding low-density expansions for the monomer-dimer problem with negative dimer activity. If the density of zeros, $g\left(h^{\prime \prime }\right)$, on the imaginary axis of the complex reduced-magnetic-field plane, $h=\frac{H}{k_{B}T}=h^{\prime }+i{h}^{\prime \prime }$, has an algebraic singularity at the edge of the gap in the zero distribution, $g\left(h^{\prime \prime }\right)\sim {\mathrm{}\left[\right|\mathrm{}{h}^{\prime \prime }|-h_0\mathrm{}(T\left)\right]}^{\sigma }$, then $\sigma$ is independent of $n$ in this limit. Analyzing dimer density series on various lattices by means of the ratio test, Dlog Padé, the recursion-relation method, and inhomogeneous differential approximants, we obtain the estimates $\sigma =-0.163\mathrm{}\pm 3$ for $d=2\mathrm{}$ dimensions and $\sigma =0.086\mathrm{}\pm 15$ for $d=3\mathrm{}$.