PT-Symmetry Breaking

• PT-symmetry breaking is defined as the phase transition in non-Hermitian systems where a real eigenvalue spectrum becomes complex due to eigenvalue coalescence at exceptional points.
• The phenomenon is characterized by precise mathematical criteria including PT-commutation and Krein signature analysis, revealing thresholds influenced by degeneracy and system symmetry.
• Applications span quantum mechanics, photonics, and scattering systems, enabling the design of devices with tailored non-Hermitian responses and controlled dynamical behavior.

Parity-Time (PT) symmetry breaking refers to the phenomenon wherein a non-Hermitian, but PT-symmetric operator—typically a Hamiltonian or wave operator—undergoes a phase transition from a regime with an entirely real spectrum to one in which eigenvalues form complex conjugate pairs. This transition is fundamental to non-Hermitian physics and underpins a range of behaviors in quantum mechanics, wave systems, classical statistical mechanics, and open quantum systems. PT-symmetry breaking is distinguished by the interplay between gain and loss channels, non-Hermiticity, symmetry, and spectral topology, and yields both universal and system-dependent features across the physical sciences.

1. Mathematical Framework and General Criteria

A linear operator $H$ is PT-symmetric if it commutes with the combined action of a linear “parity” involution $P$ ($P^2=I$) and the antilinear “time-reversal” involution $T$ ($T^2=I$, reverses $i \mapsto -i$): $[H,PT]=0$. In the context of lattice, wave, and quantum systems, PT symmetry ensures the spectrum of $H$ consists of real eigenvalues and/or complex-conjugate pairs. PT-symmetry breaking refers to the parameter-driven phase transition in which the spectrum ceases to be entirely real at a well-defined threshold (Schomerus, 2010, Zhang et al., 2019, Zhang et al., 2018, Joglekar et al., 2014).

The generic mechanism for PT-symmetry breaking is the coalescence of eigenvalues at an exceptional point (EP). At this critical parameter, two or more real eigenvalues collide in parameter space and split into a complex-conjugate pair as the control parameter is increased further. This transition is closely associated with the onset of dynamical instability in the pseudo-Hermitian (or G-Hamiltonian) sense: PT-symmetric $H$ is always pseudo-Hermitian (Zhang et al., 2019), i.e., there exists a possibly indefinite, Hermitian metric $\eta$ such that $H^\dagger\eta=\eta H$. PT-symmetry breaking is equivalent to the resonance of eigenmodes with opposite Krein signatures—defined as the sign of $\psi^\dagger \eta \psi$—and can only occur when such a resonance is realized at an EP (Zhang et al., 2018).

2. Physical Mechanisms and Classification

PT-symmetry breaking manifests in several universal and model-specific scenarios:

• Spontaneous PT Breaking: For a PT-symmetric $H$, PT symmetry is unbroken when all eigenvectors are also PT-eigenstates and all eigenvalues are real. Beyond a threshold, eigenvalues pairwise coalesce, and the eigenfunctions cease to be PT-invariant, entering the PT-broken phase (Schomerus, 2010, Zhang et al., 2018).
• Exceptional Points (EPs): The critical parameters corresponding to PT-symmetry breaking are mathematically characterized by the vanishing discriminant of the characteristic polynomial (e.g., for $2\times2$ systems, when $(\mathrm{tr} H)^2 - 4\det H = 0$ (Zhang et al., 2019)). The Krein signature framework asserts that PT-breaking occurs only if the colliding eigenmodes have opposite signature; otherwise, higher-order degeneracy or "diagonalizable" EPs may occur without PT breaking (Zhang et al., 2018).
• Systems with Degeneracy and Symmetry: In multi-dimensional or degenerate systems, generic PT-symmetry breaking can be thresholdless: degeneracy together with a generic PT-symmetric perturbation induces immediate complexification of eigenvalues at any nonzero perturbation, unless further symmetry decouples degenerate subspaces (Ge et al., 2014, Fernández et al., 2013).
• Explicit PT Breaking: PT symmetry can be "explicitly" broken by adding non-PT-symmetric (e.g., dissipative) terms, leading to immediate spectral complexification without a finite threshold (Qin et al., 2020).

3. Model-Specific Realizations

3.1. Tight-Binding Lattices with Long-Ranged Potentials

In lattice systems with non-Hermitian, PT-symmetric, long-ranged on-site potentials $V_{\alpha}(x)\sim i |x|^\alpha \mathrm{sign}(x)$:

• For $\alpha > -2$, there is a finite, positive PT threshold in the continuum limit: a finite value of the non-Hermitian strength at which the first pair of eigenvalues coalesce and become complex.
• For $\alpha \leq -2$, the steady-state threshold vanishes as the system size increases, corresponding to fragile PT-symmetry breaking (Joglekar et al., 2014).

A two-stage PT-breaking can further occur for $\alpha < 0$. At a higher (second) threshold $\gamma_c \gg \gamma_{PT}$, localized "bound-in-the-continuum" eigenstates with complex energies emerge, dramatically reducing the timescale associated with non-unitary growth in observable quantities such as total intensity (Joglekar et al., 2014).

3.2. Symmetry and Group-Theoretic Considerations

PT-symmetry breaking is intricately linked to the representation theory of point groups:

• If the PT-odd part of the Hamiltonian transforms in the same irrep as a degenerate block of the Hermitian part, immediate PT-symmetry breaking (zero threshold) occurs in that multiplet.
• Conversely, PT-odd components coupling only non-degenerate or symmetry-incompatible subspaces lead to a finite threshold for PT breaking (Fernández et al., 2013).

3.3. Random Matrix and Universality

Random-matrix theory classifies PT breaking into universality classes, reflecting the underlying symmetry (orthogonal for time-reversal symmetric and unitary in the presence of magnetic fields, etc.):

• In the orthogonal class, PT-breaking thresholds saturate at $\mu_{PT}\sim\sqrt{N}\Delta/(2\pi)$, independent of coupling in the strong-coupling regime, due to symmetry-induced level crossings.
• In the unitary class, level repulsion at avoided crossings yields a threshold dependent on both system size and coupling (Schomerus, 2010).

3.4. Scattering Systems and Order Parameters

In PT-symmetric scattering problems, the order parameter for PT-breaking is a nonlocal conserved current $Q$, derived from the invariance of the Helmholtz (or wave) equation under PT. This current $Q$ is exactly zero in the PT-symmetric phase (for all S-matrix eigenstates) and nonzero in the PT-broken phase, providing both a physical and computational tool for mapping the PT phase boundary. The phase diagram can be extended to general scattering states (not only S-matrix eigenstates), yielding a richer landscape of PT-related phenomena (Kalozoumis et al., 2014).

4. Physical Manifestations and Experimental Observables

PT-symmetry breaking is evidenced across physical platforms:

• Waveguides and Optical Cavities: In coupled photonic systems, the hallmark of PT-symmetry breaking is the transition from bounded, oscillatory intensity evolution (PT-symmetric phase) to an exponential, non-unitary growth (PT-broken phase), with pronounced changes in the behavior when localized PT-broken modes form (Joglekar et al., 2014, Riboli et al., 2023). Notably, the onset of broken symmetry can result in universal and quantized features in observables such as chirality or wavepacket handedness (Scott et al., 2012).
• Statistical Systems and the Sign Problem: In quantum many-body and classical spin models with generalized PT symmetry, the phase structure of correlations and the feasibility of numerical simulations is governed by PT-symmetry breaking. In the unbroken phase, the sign problem can always be resolved via a similarity transformation to a real (symmetric) basis, whereas the broken phase is accompanied by oscillatory or negative weights, precluding such remedies (Meisinger et al., 2010).
• Scattering, Transmission, and Resonances: In open quantum and wave systems, PT-breaking induces the coalescence or disappearance of resonance peaks, often classified as a "collapse of resonances," and can be further associated with catastrophe theory (e.g., cusp singularities) and dramatic changes in transmission behavior (Gorbatsevich et al., 2016, Gorbatsevich et al., 2016).
• Quantum Statistical Signatures: In few-photon or multiparticle interferometry, the PT-symmetry breaking point is sharply marked by the equality and subsequent reversal of photon coincidence statistics between bosons and fermions, exemplifying quantum-statistical signatures of exceptional points (Longhi, 2019).

5. Generalizations and Extensions

5.1. PT Breaking in Open Quantum Systems

Open quantum systems governed by Lindblad master equations exhibit PT-breaking transitions in their steady-state structure. As the ratio of dissipation rate to coherent coupling is tuned past a critical point, the stationary state changes non-analytically from maximally mixed to a pure "dark-state" product, with signatures also visible in purity, imbalance, and entanglement measures (Huber et al., 2020).

5.2. Non-Bloch PT Symmetry and Skin Effect

Recently, the interplay of PT-symmetry with non-Hermitian skin effect (NHSE) has been elucidated, leading to the concept of non-Bloch PT symmetry. Here, the PT-breaking threshold is governed by the formation of cusps in the generalized Brillouin zone (GBZ), not by band-edge exceptional points of the periodic (Bloch) Hamiltonian. This produces novel van Hove singularities in the density of states and enhanced sensitivity in physical response, with the transition captured by resultant algebraic conditions involving both $H(\beta)$ and its derivatives (Hu et al., 2022).

5.3. Catastrophe Theory and Topological Classification

The critical points of PT-symmetry breaking, especially in finite systems with tunable parameters, can be classified using catastrophe theory: merging of $N$ resonances at higher-order exceptional points leads to singularities associated with cuspoids $A_{2N-1}$, such as butterflies and swallowtails, which organize the response surface in the control space (Gorbatsevich et al., 2016).

6. Role of Degeneracy, Dimensionality, and Additional Symmetries

The scenario for PT-symmetry breaking is highly sensitive to the presence of degeneracies and additional discrete symmetries:

• In the absence of additional decoupling symmetries, degeneracy in $D>1$ immediately induces PT-broken complex eigenvalues for infinitesimal non-Hermiticity; the canonical PT transition with a finite threshold is recovered only when extra symmetries decouple the degenerate subspaces (Ge et al., 2014).
• Partial protection by symmetries or partial degeneracy may lead to mixed phases, where only a subset of states exhibit PT-broken behavior.
• Dark states, protected by symmetry or odd degeneracy, may remain real throughout parameter space, even after breakdown of PT symmetry for the rest of the spectrum.

7. Practical Engineering and Control

The sophistication of PT-breaking phenomena enables rational design of devices with tailored non-Hermitian responses:

• Robust phases in ring-like lattices are made possible by geometry-induced decoupling of criticality from impurity position, enabling universal chirality at the PT-breaking threshold (Scott et al., 2012).
• Broadband optical limiting and dynamic reflection control can be achieved by harnessing PT-symmetry breaking of reflectionless modes in multilayer resonators, with phase transitions induced by light-driven detuning (Riboli et al., 2023).
• Multi-cavity arrays and topological systems exhibit case-dependent PT-breaking that is sensitive to gain/loss localization, system size parity, and topological regime, facilitating controlled access to exotic spectral features (Xing et al., 2018).

---

PT-symmetry breaking is a unifying principle for a range of non-Hermitian phenomena, organizing spectral, dynamical, and statistical properties in quantum, classical, and open systems. It is fundamentally rooted in symmetry and topology, and is characterized by precise spectral signatures, exceptional-point behavior, and robustness or fragility dictated by both microscopic parameters and global symmetry structure. Its manifestations in lattice systems, wave physics, statistical mechanics, and quantum optics offer a complete stratified phenomenology, which continues to yield new physical effects and technological applications (Joglekar et al., 2014, Fernández et al., 2013, Schomerus, 2010, Qin et al., 2020, Zhang et al., 2019, Zhang et al., 2018, Hu et al., 2022).