How one quantum boundary can create nonunitary critical behavior

A boundary that changes the rules

In a familiar quantum system, the mathematical description is Hermitian, a property associated with real energies and the usual probability-conserving rules of quantum mechanics. Non-Hermitian models relax that condition and can display behavior with no direct counterpart in ordinary unitary systems. Much work on nonunitary criticality changes the bulk of a model. Zhou and Ye instead ask whether a boundary alone can do the job.

Their setting is a one-dimensional chain of non-interacting fermions. Hopping between neighboring sites in the bulk is uniform and Hermitian. A separate link joins the chain’s two ends, with hopping strengths in the two directions labelled tR and tL. When those strengths differ, the end-to-end link is non-Hermitian, even though the chain’s bulk remains Hermitian and gapless.

The researchers solve the lattice model analytically. They derive an exact expression for equal-time correlations between sites and examine its long-distance form. They also calculate single-particle reflection and transmission at the boundary. For a many-particle perspective, they use biorthogonal entanglement entropy: in a non-Hermitian system, this quantity is built using both left and right states, rather than the single state used in the standard Hermitian formulation. They study how this entropy changes as the size of a subsystem grows.

Reflection leaves a long-distance trace

The central link is between boundary scattering and correlations. The long-distance, boundary-induced part of the correlation function falls off algebraically, and its strength is set by the single-particle reflection amplitude at the Fermi level—the energy marking the boundary between occupied and unoccupied states in this model. In this way, a scattering property carries information about many-particle behavior far from the boundary.

The boundary parameters divide the model into regimes. When their product is positive, the authors find behavior that can be mapped to a Hermitian boundary with an effective hopping strength. The biorthogonal reflection and transmission coefficients then stay within the familiar bounds for unitary scattering. The boundary is non-Hermitian, but its scattering behavior is unitary-like in this respect.

When the product is negative, that Hermitian mapping is unavailable. The biorthogonal reflection coefficient can exceed one, while the corresponding transmission coefficient becomes negative. These are not ordinary probabilities: they are quantities defined using the model’s left and right states. Their unusual values provide the authors’ proposed scattering signature of the nonunitary regime. At a particular parameter condition, the reflection amplitude diverges, indicating a singular resonance in the model.

The entanglement analysis reveals another contrast. In both regimes, the biorthogonal entanglement entropy retains logarithmic scaling with subsystem size. But the coefficient of that logarithm—an effective central charge—changes from real to complex as the boundary parameters change. In the real regime, the authors relate the result to a unitary defect conformal field theory, a framework for describing critical systems with a local defect. In the complex regime, they report that analytic continuation from the unitary result does not correctly reproduce the real part of the effective central charge. They offer a compact formula that fits their numerical results there, rather than a direct derivation from the correlation matrix’s large-size behavior.

What the result does—and does not—show

The work’s strength is its exact treatment of a simple lattice model: it makes a precise connection between boundary reflection and long-range correlations, then uses entanglement scaling as a second probe. It suggests that nonunitary boundary-critical behavior need not require non-Hermiticity throughout a material; in this model, changing a single end-to-end link is enough.

That is a theoretical result, not an experimental demonstration or evidence that every non-Hermitian boundary will behave this way. The authors identify open questions, including deriving the complex scaling coefficient directly and testing whether the scattering–correlation connection survives with multiple defects or more complicated channels. The model also concerns free fermions, so extending its conclusions to interacting systems is not established here.

This article is based on an arXiv preprint by Zhou and Ye. Preprints are publicly shared research manuscripts and may not have completed peer review. The source presents exact calculations for the stated model, but its broader implications remain proposals for further investigation. Its conclusions are best read as a benchmark and a route for further study, not as a settled general theory of nonunitary criticality.