Hidden exceptional point, localization-delocalization phase transition in Hermitian bosonic Kitaev model

TL;DR

The paper identifies hidden exceptional points (EPs) within a Hermitian bosonic Kitaev model and shows they drive a localization‑delocalization–like transition in a mapped single‑particle basis. By projecting the Hamiltonian onto a $BCS$‑like pairing basis, the authors reveal a non‑Hermitian core matrix $h_k$ hosting an EP at $μ = |Δ_k|$, which governs nontrivial dynamical behavior despite overall Hermiticity. They further map to an invariant subspace to obtain a single-particle chain $H_{eq}^k$ and quantify localization via the mean inverse participation ratio (MIPR), establishing a phase boundary between localized and extended states. Finally, they propose an experimentally accessible detection in a Dicke model through quench dynamics and monitoring the average photon number $N_P(t)$, with a dynamical signature $D_μ$ peaking at the EP.

Abstract

Exceptional points (EPs), a unique feature of non-Hermitian systems, represent degeneracies in non-Hermitian operators that likely do not occur in Hermitian systems. Nevertheless, unlike its fermionic counterpart, a Hermitian bosonic Kitaev model supports a non-Hermitian core matrix, involving a quantum phase transition (QPT) when an exceptional point appears. In this study, we examine QPTs by mapping the Hamiltonian onto a set of equivalent single-particle systems using a Bardeen-Cooper-Schrieffer (BCS)-like pairing basis. We demonstrate the connection between the hidden EP and the localization-delocalization transition in the equivalent systems. The result is applicable to a Dicke model, which allows the experimental detection of the transition based on the measurement of the average number of photons for the quench dynamics stating from the empty state. Numerical simulations of the time evolution reveal a clear transition point at the EP.

Figures (5)

• Figure 1: Plots of two eigenstates $\left\vert \psi \right\rangle$ of the eqivalent Hamiltonian $H_{\mathrm{eq}}^{k}$, given by Eq. ( \ref{['H_k_eq']}), with two typical system parameters (a) $\Delta _{k}=1$ and $\mu =2$, and (b, c) $\Delta _{k}=2$ and $\mu =1$, respectively. Here, the red or black bars represent the real or imaginary parts of the amplitudes, respectively, and all are analytically determined. (a) The profile of the vacuum state ($\left\vert \psi \right\rangle =\left\vert \mathrm{Vac}\right\rangle$) is obtained from Eq. (\ref{['Vac']}), which is also the ground state with energy $E_{\mathrm{Vac}}^{k}=-0.5359$. It decays exponentially and then a localized state. (b, c) is the profile of the eigenstate ($\left\vert \psi \right\rangle =\left\vert \varphi _{k}\right\rangle$) with energy $E_{k}=0.0$, which is obtained from Eq. (\ref{['E=0 state']}). (b) is the plot on the same scale as (a) for comparison, while (b) is the same plot but on longer scale. We can see form (c) that it becomes to a constant for large $j$, indicating a delocalized state. These results imply that the EP at $\Delta _{k}=\mu$ is a transition point from localization to delocalization in the Fock space.
• Figure 2: Plot of MIPR, which represents the arithmetic mean of the IPRs for a set of eigenstates of the Hamiltonian $H_{\mathrm{eq}}^{k}$, given by Eq. (\ref{['H_k_eq']}). The system parameters are $\Delta =1$ and $N$ indicated in the panel. The selected set of states consists of the lowest $400$ eigenstates. It is evident that there is a transition point at $\mu =1$ for every given $N$. Furthemore, the value of $\mathrm{MIPR}$ becomes stable as $N$ increases. For $\mu <1$, the asymptotic value of $\mathrm{MIPR}$ approaches zero as the system size becomes sufficiently large, whereas it remains finite for $\mu >1$.
• Figure 3: Plots of the average photon number $N_{\text{P}}\left( t\right)$, given by Eq. (\ref{['Np(t)']}), for the quench process in the Dicke model with the Hamiltonian $H_{\text{D}}$, as defined by the Eq. (\ref{['H_D']}). The numerical simulation is performed by tracking the time evolution of the initial state $\left\vert \Psi \left( 0\right) \right\rangle$, which is detailed in the main text preceding Eq. (\ref{['Np(t)']}). The quanity $N_{\text{P}}\left( t\right)$ for a given $\mu$ is obtained by exact diagonalization of the finite-dimensional matrix representation of $H_{\text{D}}$, under the truncation of the photon number. The system parameters are an atom number $N_{\text{atom}}=64$ and $\Delta =1$. (a) A 3D plot of $N_{\text{P}}$ as a function of time $t$ and $\mu$. (b, c) The profiles of $N_{\text{P}}\left( t\right)$ for two typical values of $\mu$.
• Figure 4: Plots of two eigenstates of the effective model Eq. (\ref{['Hab']}), specifically the vacuum state, as given by Eq. (\ref{['Dicke_vac']}), with the system parameters $\Delta=1$ and $\mu=3$ for figures (a) and (b). In these plots, the red or black bars represent the real or imaginary parts of the amplitudes, respectively, and all are analytically determined.
• Figure 5: Plots of $D_{\mu }$, the derivative of the average photon number over a period of time $T$ with respect to $\mu$ ($\Delta =1$), as defined by Eq. (\ref{['Dm']}). The numerical results are obtained through exact diagonalizations of the matrix representations for both the Dicke Hamitonian Eq. (\ref{['H_D']}) and the effective Hamiltonian Eq. (\ref{['Hab']}), under the truncation of the photon number. The time internval $T$ and the number of atoms $N_{\text{atom}}$ are indicated in the panel. The results show that each $D_{\mu }$ has a minimum near the EP at $\mu =2$. The valleys become deeper as $T$ or $N_{\text{atom}}$ increases, implying that $D_{\mu }$ is divergent at the EP in the case of infinite $T$ and $N_{\text{atom}}$.