Latent symmetry in a minimal non-Hermitian trimer

Abstract

We study a minimal non-Hermitian trimer with latent symmetry formed by a cospectral pair of sites embedded in a three-site network with nonreciprocal couplings. We show that the model admits an exact decomposition into dark and bright sectors: the dark mode is spectrally isolated and retains a complex eigenvalue, while the bright sector reduces to an effective non-Hermitian dimer. For a suitable choice of parameters, this reduced subsystem becomes $\mathcal{PT}$-symmetric and exhibits partial spectral reality, with two real eigenvalues coexisting with the complex dark eigenvalue. At the critical point, the bright sector hosts an embedded second-order exceptional point, which renders the full trimer defective and gives rise to the characteristic Jordan-block dynamics. These results establish the non-Hermitian trimer as a minimal analytically solvable setting in which latent symmetry, sector-resolved $\mathcal{PT}$ symmetry, and exceptional-point physics naturally coexist.

Figures (3)

• Figure 1: Minimal non-Hermitian trimer formed by three coupled sites, $\ket{1}$, $\ket{2}$, and $\ket{3}$, with complex onsite energies $\Omega_j = \omega_j+i\gamma_j$. The pair ($\ket{1}$, $\ket{2}$) becomes cospectral when $\Omega_1=\Omega_2$ and $g_{13}g_{31}=g_{23}g_{32}$, giving rise to a latent symmetry, while site $|3\rangle$ plays the role of a singlet site.
• Figure 2: Imaginary parts of the eigenvalues $\lambda_0$, $\lambda_+$ and $\lambda_-$ as a function of $\gamma$. The exceptional points are located at $\gamma_c = \pm 1$ for the set of parameters $\omega = 0$, $\mu = 1$ and $\kappa = 1/\sqrt{2}$.
• Figure 3: Dynamics in the bright sector ($\ket{\psi(0)} = \ket{B})$ below the $\mathcal{PT}$ phase transition point. Local occupations $P_j = |\braket{j | \psi(t)}|^2$ for (a) $\chi = 0$ and (b) $\chi = 1/5$. Other parameters are the same as in Fig. \ref{['fig2']} and $\gamma = 1/2$.