Universal routes to spontaneous $\mathcal{P}\mathcal{T}$-symmetry breaking in non-Hermitian quantum systems

$\mathcal{PT}$-symmetric systems can have a real spectrum even when their Hamiltonian is non-Hermitian, but develop a complex spectrum when the degree of non-Hermiticity increases. Here we utilize random-matrix theory to show that this spontaneous $\mathcal{PT}$-symmetry breaking can occur via two distinct mechanisms, whose predominance is associated to different universality classes. Present optical experiments fall into the orthogonal class, where symmetry-induced level crossings render the characteristic absorption rate independent of the coupling strength between the symmetry-related parts of the system.

Figure 1

(a) Sketch of a non-Hermitian $\mathcal{PT}$-symmetric system, where a region with absorption rate $\mu$ (and mean level spacing $\Delta$, left) is coupled symmetrically via a tunnel barrier (supporting $N$ channels with transmission probability $T$) to an amplifying region with a matching amplification rate (right). Below this is the scattering description of the system. (b) Two routes to spontaneous $\mathcal{PT}$-symmetry breaking, depending on whether the Hermitian limit $\mu =0$ is $\mathcal{T}$ symmetric (orthogonal class displaying level crossings, left) or not (unitary class displaying avoided crossings, right). Shown are real eigenvalues of a random Hamiltonian $\mathcal{H}$ [Eq. (4)] as function of $T$ for fixed $\mu =0$ (left of dashed line), and then as a function of $\mu$ for fixed $T=1$ (right of dashed line). Complex-valued levels (formed by level coalescence at $\mu >0$) are not shown. Here ${\mu }_0=\sqrt{N}\Delta /2\pi$, and we set $N=10$.

Figure 2

Gradient plots of the ensemble-averaged fraction $f(\mu ,T)$ of complex energy levels (among all levels within a range of energies over which the mean spacing $\Delta$ can be assumed constant), for the orthogonal class (left) and the unitary class (right). In (a) and (b), $\mu$ is scaled to ${\mu }_0$. The bottom panels show same data with $\mu$ scaled to ${\mu }_0^{\prime }={\mu }_0/\sqrt{1+1/\mathit{NT}}$ (c) and ${\mu }_T^{\prime }=\sqrt{T}{\mu }_0$ (d). Numerical results with $N=50$.