Nonunitary Criticality & Non-Hermitian Transitions

• Nonunitary criticality is a universality class defined by non-Hermitian dynamics that leads to negative central charges and complex scaling dimensions.
• Models employ PT-symmetric gain/loss potentials, non-Hermitian impurities, and postselected dynamics to trigger exceptional points and transitions between topological phases.
• Experimental platforms in photonics, atomic systems, and superconductivity validate these theories, offering insights into quantum circuitry and many-body localization.

Nonunitary criticality defines a broad universality class of critical phenomena characterized by underlying non-Hermitian dynamics and the violation of quantum mechanical unitarity. In contrast to conventional (unitary) quantum critical points, which are governed by Hermitian Hamiltonians and conformal field theories (CFTs) with positive central charge, nonunitary critical points arise in open quantum systems, systems with parity-time ($\mathcal{PT}$) symmetry, or upon analytic continuation to complex couplings, yielding negative or complex scaling dimensions, negative central charges, and exceptional-point singularities. These nonunitary criticalities are realized in a diverse range of settings, including free-fermion chains with local non-Hermitian impurities, non-Hermitian Su-Schrieffer-Heeger (SSH) models, dissipative quantum circuits, random tensor networks, quasi-periodic non-Hermitian lattices, and paradigmatic Yang-Lee edge singularities.

1. Emergence and Models of Nonunitary Criticality

Nonunitary criticality emerges when the constraints of Hermiticity and unitarity are relaxed, either by explicit introduction of non-Hermitian terms in the Hamiltonian, by $\mathcal{PT}$-symmetric gain/loss potentials, or through postselected (non-trace-preserving) quantum measurement processes. A typical minimal setting is a (1+1)-dimensional free-fermion chain perturbed by a local non-Hermitian, $\mathcal{PT}$-symmetric defect, $H = H_0 + H_{\rm imp}$, with $H_0$ the critical (e.g., SSH point or half-filled tight-binding) chain and $H_{\rm imp}$ a local term such as $i\gamma c^\dagger_0 c_0$ or its sublattice-extended variant (Li et al., 18 Feb 2025). The presence of such an impurity can induce an exceptional point (EP) at a critical parameter (e.g., impurity strength $\lambda = 1$), at which the non-Hermitian eigenmode permeates the entire system.

In non-Hermitian SSH or generalized fermion chains, staggered imaginary potentials ($i\gamma$) or complex-valued hopping break Hermiticity but preserve overall $\mathcal{PT}$ symmetry (Zhou et al., 2024, Chou et al., 11 Sep 2025). These models exhibit transitions between distinct symmetry-protected topological phases and non-Hermitian critical regimes, with the latter characterized by negative central charge and topological degeneracy tied to non-unitarity.

Open quantum systems and postselected dynamics, as realized in $\mathcal{PT}$0-symmetric experiments or measurement-induced quantum circuits, realize nonunitary CFTs by quantum–classical correspondences (e.g., mapping 1D Ising with imaginary field to non-Hermitian single-qubit Hamiltonians (Gao et al., 2023, Matsumoto et al., 2020)) and generate nonunitary critical exponents and scaling laws.

2. Signatures and Diagnostics: Central Charge, Entanglement, and Spectrum

Nonunitary critical points are rigorously characterized by negative central charge $\mathcal{PT}$1, anomalous entanglement scaling, and finite-size energy spectra exhibiting non-standard scaling behavior. In the free-fermion chain with a non-Hermitian impurity, fitting the von Neumann or Rényi entanglement entropy $\mathcal{PT}$2 for a block of length $\mathcal{PT}$3 in a ring of size $\mathcal{PT}$4 to CFT predictions,

$\mathcal{PT}$5

gives an effective central charge $\mathcal{PT}$6 at the EP regimen under periodic boundary conditions (PBC) (Li et al., 18 Feb 2025). This negative central charge is robust under PBC but is destroyed by open or twisted boundary conditions, which revert the entanglement and spectral properties to unitary scaling with $\mathcal{PT}$7. The many-body ground state energy likewise reveals $\mathcal{PT}$8 when fitting to the standard finite-size formula,

$\mathcal{PT}$9

again with $\mathcal{PT}$0 in the nonunitary critical phase (Li et al., 18 Feb 2025, Zhou et al., 2024). The excited state splittings align with the conformal dimensions of the corresponding nonunitary CFT.

For non-Hermitian SSH models with $\mathcal{PT}$1 symmetry under open boundary conditions (OBC), the finite-size ground state energy flow reveals a universal transition from $\mathcal{PT}$2 (Dirac CFT) to $\mathcal{PT}$3 (nonunitary CFT), with explicit scaling functions $\mathcal{PT}$4 encoding the crossover as a function of tuning parameters (Zhou et al., 2024).

3. Exceptional Points, Boundary Conditions, and Topological Enrichment

Exceptional points serve as the loci at which nonunitary criticality is realized. At these points, eigenvalues and eigenvectors coalesce, and the many-body ground state includes the non-Hermitian EP mode. The critical wavefunction and its correlations are permeated by this defect, leading to nonunitary scaling in both spectrum and entanglement.

Boundary conditions play a nontrivial, categorically non-Hermitian role: periodic, open, or twisted boundary conditions belong to different "universality classes." Under PBC, the global EP supports $\mathcal{PT}$5 entanglement scaling and anomalous ground state energy, while OBC typically suppress the EP. In special cases, boundary parameter tuning can recover a nonunitary mode even in OBC, but the extracted central charge may differ between entanglement and spectrum (Li et al., 18 Feb 2025). Twists or infinitesimal variation in the PBC phase immediately quenches the nonunitary EP (Li et al., 18 Feb 2025).

$\mathcal{PT}$6-symmetry enriches nonunitary critical points with explicit topological structure. In non-Hermitian SSH-type models, topologically robust edge modes exist at EPs, protected by a generalized mass inversion of the continuum Dirac Hamiltonian, not present in Hermitian cases. The topological degeneracy at criticality is encoded in the imaginary part of the entanglement entropy scaling, $\mathcal{PT}$7, with $\mathcal{PT}$8 and $\mathcal{PT}$9 the bulk winding number (Chou et al., 11 Sep 2025).

4. Universality Classes and Critical Exponents

The nonunitary critical regime arising from non-Hermitian defects or extended imaginary fields is governed by nonunitary CFTs, notably the minimal Virasoro models $H = H_0 + H_{\rm imp}$0: for the Yang-Lee edge singularity, $H = H_0 + H_{\rm imp}$1 (ordinary), $H = H_0 + H_{\rm imp}$2 (tricritical), and $H = H_0 + H_{\rm imp}$3 for symplectic-fermion or $H = H_0 + H_{\rm imp}$4 ghost CFT (Lencsés et al., 2022, Zhou et al., 2024, Chou et al., 11 Sep 2025). Effective central charges are computed as $H = H_0 + H_{\rm imp}$5, with $H = H_0 + H_{\rm imp}$6 negative in nonunitary theories, and the nonunitary $H = H_0 + H_{\rm imp}$7-theorem establishes monotonicity of $H = H_0 + H_{\rm imp}$8 under $H = H_0 + H_{\rm imp}$9-symmetric RG flows (Lencsés et al., 2022).

Scaling exponents at nonunitary critical points are anomalous and distinct from unitary universality classes. Representative exponents include (Li et al., 18 Feb 2025, Gao et al., 2023, Li et al., 2022):

• Correlation length: $H_0$0 with typical $H_0$1 (free-fermion) or $H_0$2 (Yang-Lee-type),
• Magnetization: $H_0$3, with $H_0$4 (Yang-Lee edge),
• Susceptibility: $H_0$5, with $H_0$6,
• Entanglement entropy: negative logarithmic scaling coefficient,
• Dynamical exponent: $H_0$7,
• Anomalous dimension: $H_0$8 (for nonunitary BCS Yang-Lee universality class).

Finite-size scaling and dynamical scaling further characterize these transitions, and explicit finite-size scaling functions encode the flow from unitary to nonunitary criticality with universal singularities, such as the rise–dip–rise pattern at topological EPs (Zhou et al., 2024).

5. Quasiperiodic and Random Nonunitary Critical Phases

Nonunitary criticality is not confined to homogeneous or impurity-driven models. In exactly solvable non-Hermitian quasicrystals (e.g., modulated Hatano-Nelson chains), "quasiperiodic skin criticality" emerges: all eigenstates share exactly the same multifractal spatial structure, characterized by a global fractal dimension $H_0$9 that is strictly independent of energy and set only by the global phase of the modulation (Chen et al., 30 Jan 2026). The nonunitary gauge transformation maps the system to a uniform chain; the skin effect and multifractality represent a unique universality class distinct from Hermitian criticality or standard non-Hermitian skin effects.

In random nonunitary quantum circuits and Gaussian tensor networks, there is a full correspondence between random nonunitary evolution in $H_{\rm imp}$0 dimensions and unitary physics in $H_{\rm imp}$1 dimensions in the Altland-Zirnbauer symmetry classes. Critical entanglement properties are inherited from these effective higher-dimensional metallic phases, with boundary logarithmic scaling of entropy and power-law decay of correlation functions (Jian et al., 2020).

6. Experimental Realizations and Physical Relevance

Nonunitary criticality, particularly the Yang-Lee edge singularity and its generalizations, has been experimentally realized in a range of photonic and atomic platforms. All-optical simulations using heralded single photons directly probe the partition function zeros, critical exponents, and dynamical scaling of the non-Hermitian Ising model (Gao et al., 2023). Rydberg atomic arrays with laser-induced loss allow for matrix-product-state simulations and experimental proposals that realize kinked dynamical magnetization responses and recover nonunitary CFT exponents (Shen et al., 2023). Open-system dynamics, including measurement-induced nonunitarity and postselected quantum circuits, naturally realize the Yang-Lee universality class, and the critical behavior is experimentally accessible via observable response functions (Matsumoto et al., 2020).

In nonunitary superconductivity, as in UTe$H_{\rm imp}$2, the nonunitary triplet pairing and its critical tuning by the easy-axis magnetization manifest in experimentally measurable thermodynamic anomalies, including nonstandard upper critical field curvature and point nodes in the gap structure (Kittaka et al., 2020).

7. Open Problems and Theoretical Directions

Current research on nonunitary criticality highlights several open directions:

• Renormalization group irreversibility and the structure of nonunitary $H_{\rm imp}$3-functions (boundary entropy), especially given the sign flip and negative central charge (Li et al., 18 Feb 2025, Lencsés et al., 2022).
• The precise classification and field-theoretical description of nonunitary defect lines and their endpoint fixed points.
• The universality of scaling functions and singularities (e.g., rise–dip–rise) at non-Hermitian topological transitions, and their relation to edge-state quantization (Zhou et al., 2024, Chou et al., 11 Sep 2025).
• Extensions to interacting models, higher-dimensional nonunitary critical phases, many-body localization in nonunitary regimes, and the interplay with disorder.
• Generalization and stability of these nonunitary universality classes beyond free or Gaussian theories, including in multicritical and multiclass settings (Lencsés et al., 2022).

Nonunitary criticality thus forms a landscape of universality classes, diagnostic tools, and physical mechanisms distinct from unitary paradigms, with foundational implications for both condensed matter and quantum statistical mechanics.