Figure 1

Phase diagram of the Yang-Lee quantum critical system. The quantum critical point (QCP) is located at $\varphi =\pi /4$ and ${\beta }^{-1}=0$. In the $\mathcal{PT}$-unbroken phase (i.e., $\left|\varphi \right|<\pi /4$), the $\mathrm{\Delta }\varphi$ dependencies of physical quantities obey the conventional scaling laws for the Yang-Lee edge singularity [6] (see Appendix pp1) in two successive limits, i.e., ${\beta }^{-1}\to 0$ followed by $\varphi \to \pi /4-0$. In the $\mathcal{PT}$-broken phase (i.e., $\left|\varphi \right|>\pi /4$), physical quantities diverge periodically and the corresponding limits cannot be defined by two sequential limits of ${\beta }^{-1}\to 0$ followed by $\varphi \to \pi /4+0$. Unconventional scaling laws about the dependence on the temperature ${\beta }^{-1}$ are obtained if the limit ${\beta }^{-1}\to 0$ is taken after $\varphi \to \pi /4$. In the $\mathcal{PT}$-unbroken phase, a crossover between the two limiting behaviors occurs near the dashed curve given by ${\beta }^{-1}=(2/\pi )R\sqrt{cos2\varphi }$.