Conventional quantum phase transition driven by a complex parameter in a non-Hermitian $\mathcal{PT}-\mathrm{symmetric}$ Ising model

A conventional quantum phase transition (QPT) can be accessed by varying a real parameter at absolute zero temperature. Motivated by the discovery of the pseudo-Hermiticity of non-Hermitian systems, we explore the QPT in the non-Hermitian $\mathcal{PT}-\mathrm{symmetric}$ Ising model, which is driven by a staggered complex transverse field. Exact solution shows that the Laplacian of the ground-state energy density, with respect to real and imaginary components of the transverse field, diverges on the boundary in the complex plane. The phase diagram indicates that the imaginary transverse field has the effect of shrinking the paramagnetic phase. In addition, we also investigate the connection between the geometric phase and the QPT.

Figure 1

Phase diagram for the ground state of the Ising ring in a a staggered complex transverse field. The heavy lines represent the boundary which separates three quantum phases. Phases I and III are paramagnet, while II is ferromagnet.

Figure 2

The Laplacian of the ground-state energy density ${\varepsilon }_g$ as a function of the field for the case $N=300$. The peaks mark the clear regions of criticality.

Figure 3

(a) The curvature density $\mathcal{C}$ as a function of the field for the Hamiltonians with the parameters of (a) Eq. (52) and $N=300$. (b) The inversion of (a). The dips and peaks indicate the quasicritical lines. (c) Contour map of $\mathcal{C}$. The red dashed lines indicate the phase diagram in Fig. 1.

Figure 4

The curvature density $\mathcal{C}$ as a function of the field for the Hamiltonians with the parameters of Eq. (56) and $N=300$. The peaks only indicate the quasicritical lines of the circle in Fig. 1, but not the straight lines on the $\xi$ axis.