Level statistics of a pseudo-Hermitian Dicke model

A non-Hermitian operator that is related to its adjoint through a similarity transformation is defined as a pseudo-Hermitian operator. We study the level statistics of a pseudo-Hermitian Dicke Hamiltonian that undergoes quantum phase transition (QPT). We find that the level-spacing distribution of this Hamiltonian near the integrable limit is close to Poisson distribution, while it is Wigner distribution for the ranges of the parameters for which the Hamiltonian is nonintegrable. We show that the assertion in the context of the standard Dicke model that QPT is a precursor to a change in the level statistics is not valid in general.

Figure 1

(Color online) Phase diagram of $\eta$, which is defined by Eq. (7), in a special case where $\alpha =\gamma$ and $\beta =\delta$. The solid curve corresponds to the critical line $\alpha \beta =1/4$. Solid circles correspond to the level-spacing distributions in Fig. 2.

Figure 2

Level-spacing distributions for the Hamiltonian (8) with $n=1$. (a) Almost Poisson distribution at $\alpha =\gamma =0.1$ and $\beta =\delta =0.2$, (b) intermediate distribution at $\alpha =\gamma =0.5$ and $\beta =\delta =0.3$, (c) Wigner distribution at $\alpha =\gamma =0.5$ and $\beta =\delta =0.7$, and (d) deformed Wigner distribution at $\alpha =\gamma =4$ and $\beta =\delta =4.5$.

Figure 3

(Color online) Phase diagram of $\eta$, which is defined by Eq. (7) for $\gamma =\alpha /n$ and $\delta =\beta /n$; (a) $n=1/2$, (b) $n=2$, (c) $n=1/3$, and (d) $n=3$. The solid curve in each panel is given by $\alpha \beta =n^2/{\left(n+1\right)}^2$. (e) Dependence of $\eta$ plotted against $n$ for $\alpha =1.25$ and $\beta =0.8$. The parameter combination is shown in (a)–(d) by the points $(\times )$.