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arXiv:2603.15768

Latent symmetry in a minimal non-Hermitian trimer

Paulo A. Brandão

quant-phSubmitted Mar 16, 2026• Announced Mar 18, 2026• 0 citations
v1 · Mar 16, 2026v2 · May 22, 2026
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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.

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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 PT\mathcal{PT}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 PT\mathcal{PT}PT symmetry, and exceptional-point physics naturally coexist.
Exceptional points (EPs), indicative of parity-time (PT) symmetry breaking, play a central role in non-Hermitian physics, yet most studies begin from deliberately engineered effective Hamiltonians whose parameters are tuned to exhibit exceptional behavior. In realistic open quantum systems, however, dynamics are governed by Lindblad superoperators whose spectral structure is high-dimensional, symmetry-constrained, and not obviously reducible to minimal non-Hermitian models. A general framework for discovering exceptional dynamics directly from microscopic dissipative models has been lacking. Here we introduce a symmetry-resolved approach for identifying and characterizing exceptional points directly from the full Liouvillian generator. Correlated dissipation induces graph symmetries that decompose Liouville space into low-dimensional invariant sectors, within which minimal non-Hermitian blocks govern the onset of EPs and PT-breaking behavior. We further introduce a numerical diagnostic - the exceptional-point strength E\mathcal{E}E - based on eigenvector conditioning, which quantifies proximity to defective dynamics without requiring analytic reduction. Applied to tight-binding models with correlated dephasing and relaxation, the method reproduces analytically predicted exceptional seams and reveals universal scaling of E\mathcal{E}E near EP manifolds. More broadly, the framework enables systematic discovery of hidden exceptional structure in complex or high-dimensional open systems and is naturally compatible with matrix-free and tensor-network implementations for scalable many-body applications.
Combining non-Hermiticity and interactions yields unique effects in open quantum many-body systems. Here, we develop the generalized Schrieffer-Wolff transformation, discuss its symmetry properties, and derive the effective Hamiltonian suitable for various quasidegenerate non-Hermitian systems. We apply our results to an exemplary PT\mathcal{P}\mathcal{T}PT-symmetric circuit QED composed of two non-Hermitian qubits embedded in a lossless resonator. We consider a resonant quantum circuit as $|{\ensuremath{\omega}}_{r}\ensuremath{-}\mathrm{\ensuremath{\Omega}}|\ensuremath{\ll}{\ensuremath{\omega}}_{r}$, where $\mathrm{\ensuremath{\Omega}}$ and ${\ensuremath{\omega}}_{r}$ are qubits and resonator frequencies, respectively, providing well-defined groups of quasidegenerate resonant states. For such a system, using direct numerical diagonalization we obtain the dependence of the low-lying eigenspectrum on the interaction strength between a single qubit and the resonator ggg and the gain (loss) parameter $\ensuremath{\gamma}$, and compare that with the eigenvalues obtained analytically using the effective Hamiltonian of resonant states. We identify PT\mathcal{P}\mathcal{T}PT-symmetry broken and unbroken phases, trace the formation of exceptional points of the second and the third order, and provide a complete phase diagram $g\ensuremath{-}\ensuremath{\gamma}$ of low-lying resonant states. We relate the formation of exceptional points to the additional P\mathcal{P}P-pseudo-Hermitian symmetry of the system and show that non-Hermiticity mixes the ``dark'' and the ``bright'' states, which has a direct experimental consequence.
Lines of exceptional points are robust in the three-dimensional non-Hermitian parameter space without requiring any symmetry. However, when more elaborate exceptional structures are considered, the role of symmetry becomes critical. One such case is the exceptional chain (EC), which is formed by the intersection or osculation of multiple exceptional lines (ELs). In this Letter, we investigate a non-Hermitian classical mechanical system and reveal that a symmetry intrinsic to second-order dynamical equations, in combination with the source-free principle of ELs, guarantees the emergence of ECs. This symmetry can be understood as a non-Hermitian generalized latent symmetry, which is absent in prevailing formalisms rooted in first-order Schrödinger-like equations and has largely been overlooked so far. We experimentally confirm and characterize the ECs using an active mechanical oscillator system. Moreover, by measuring eigenvalue braiding around the ELs meeting at a chain point, we demonstrate the source-free principle of directed ELs that underlies the mechanism for EC formation. Our Letter not only enriches the diversity of non-Hermitian exceptional point configurations, but also highlights the new potential for non-Hermitian physics in second-order dynamical systems.
PT\mathcal{PT}PT and anti-PT\mathcal{PT}PT phase transitions in a trimerized Su--Schrieffer--Heeger chain with nonreciprocal Rashba spin-orbit coupling
We theoretically investigate a one-dimensional trimerized Su--Schrieffer--Heeger chain with three sublattices per unit cell subjected to a nonreciprocal Rashba spin-orbit coupling. Invoking a spin-flip symmetry, the non-Hermitian Hamiltonian decomposes into two independent spin sectors, enabling a spin-resolved analysis of non-Hermitian skin effects and system symmetries. We identify a rich phase diagram consisting of four bulk phases and two edge-state phases. The bulk phases include fully PT\mathcal{PT}PT-unbroken (real-spectrum) and fully anti-PT\mathcal{PT}PT-unbroken (imaginary-spectrum) regimes, as well as two mixed phases where one band remains on the real or imaginary axis while the other two form complex-conjugate pairs. The two edge-state phases correspond to topological edge modes with either PT\mathcal{PT}PT-unbroken (real) or anti-PT\mathcal{PT}PT-unbroken (imaginary) energies. Using non-Bloch band theory and Cardano's method, we derive closed-form expressions for phase boundaries and establish the bulk-edge correspondence for each spin sector. Calculations of Berry phase and directional inverse participation ratios confirm our analytical predictions. Our results provide a minimal platform for realizing spin-resolved non-Hermitian topology and edge-selective symmetry preservation, in which the bulk and edge states belong to distinct symmetry classes.
Non-Hermitian topological phases are typically engineered through gain and loss, nonreciprocity, or interaction with an environment. Here we show that they can instead emerge purely by projecting a fully Hermitian, topologically trivial parent lattice onto an embedded subsystem. The mechanism is general: when a zero mode of the eliminated degrees of freedom couples to the retained subsystem, the embedding self-energy develops a pole, the zero-frequency description becomes singular, and topology is carried by the finite-frequency projected Green's function. We realize the mechanism exactly in a trivial nearest-neighbor square lattice with an embedded one-dimensional zig-zag brane. In the periodic transverse geometry, the parity of the eliminated complement selects the outcome: even sectors reduce to a regular Schur complement and yield conventional SSH-type descendants, whereas odd sectors host a sublattice-imbalance zero mode and follow the resonant route. There, the complex bands braid through isolated finite-frequency exceptional points (EPs), while a parity symmetry inherited from the embedding, together with TRS†\mathrm{TRS}^{\dagger}TRS†, induces conjugated pseudo-Hermiticity and quantizes the complex Berry phase. The stable bulk invariant of the nondegenerate phases is this quantized complex Berry phase; adjacent sectors are separated by parity-paired exceptional points whose half-integer vorticities encode the local exchange of complex-energy strands.The absence of the non-Hermitian skin effect ensures that the invariant is defined directly on the ordinary Brillouin zone. A topolectrical implementation of the projected response predicts momentum-resolved transmission minima at the exceptional-point transition frequencies together with a characteristic low-frequency resonant admittance, providing an experimentally testable signature of the mechanism.
We study when block-coupled regular graphs can realize prescribed complex quantum-like (QL) bit states as exact synchronized eigenstates. Two regular subgraphs GAG_AGA​ and GBG_BGB​ supply normalized all-ones eigenvectors VAV_AVA​ and VBV_BVB​, and algebraically regular bipartite couplings reduce the full graph-supported operator exactly to a 2×22\times 22×2 effective block on S=span⁡{∣0⟩,∣1⟩}\mathcal S=\operatorname{span} \{ \lvert 0\rangle, \lvert 1\rangle \}S=span{∣0⟩,∣1⟩}. Within this reduction we prove that two natural symmetric complexifications are not universal for state synthesis under a real-spectrum requirement: complex symmetric coupling with real diagonal regularities forces the target computational basis amplitude ratio r=ω2/ω1r=ω_2/ω_1r=ω2​/ω1​, for ∣ψ⟩=ω1∣0⟩+ω2∣1⟩\lvert ψ\rangle = ω_1\lvert 0\rangle + ω_2\lvert 1\rangle∣ψ⟩=ω1​∣0⟩+ω2​∣1⟩, to satisfy r2∈Rr^2\in\Rr2∈R, while real symmetric coupling with complex diagonal regularities forces r+1/r∈Rr+1/r\in\Rr+1/r∈R. Replacing complex symmetry by Hermitian coupling removes this phase obstruction. For any nonbasis target state, any prescribed real eigenvalue, and any prescribed nonzero signed spectral gap, a Hermitian weighted coupling realizes the target exactly. Additionally, an independently tuned directed-coupling model gives a second universality mechanism. We then pass from continuous effective parameters to finite weighted graphs with entries in {0,±1,±i}\{0, \pm1, \pm i\}{0,±1,±i} (the fourth roots of unity and zero), characterize the balanced discrete coupling lattice by perfect matchings, and show that exact discrete Hermitian realizations are dense in the synchronized pure-state space. These results give a state-synthesis universality taxonomy for complex QL-bits and identify Hermitian conjugate pairing as the robust structural mechanism that supports arbitrary complex amplitudes with real two-level spectra.