Two-dimensional transverse-field $XY$ model with the in-plane anisotropy and Dzyaloshinskii-Moriya interaction: Anisotropy-driven transition

The two-dimensional (2D) quantum spin-$S=1\mathrm{/}2$ $XY$ model with the transverse-field $H$, in-plane-anisotropy $\gamma$, and Dzyaloshinskii-Moriya (DM) $D$ interactions was investigated by means of the exact diagonalization method, which enables us to treat the $D$-mediated complex-valued Hermitian matrix elements. According to the preceding real-space renormalization group analysis at $H=0$, the $\gamma$-driven phase transition occurs generically for $D\mathrm{\ne }0$ in contrast to the 1D $XY$ model where both $\gamma$- and $D$-induced phases are realized for $\gamma >D$ and $\gamma <D$, respectively. In this paper, we evaluated the $\beta$ function $\beta (\gamma )$, namely, the differential of $\gamma$ with respect to the concerned energy scale, and from its behavior in proximity to $\gamma =0$, we observed an evidence of the $\gamma$-driven phase transition; additionally, $\gamma$’s scaling dimension is estimated from $\beta (\gamma )$'s slope. It was also determined how the value of the DM interaction influences the order-disorder phase boundary $H_c(\gamma )$ around the multi-critical point, $\gamma \to 0$.