PT01 — PT-Symmetric Non-Hermitian System
| Field | Value |
|---|---|
| Domain | Non-Hermitian Physics |
| System | Gain-loss balanced optical waveguide pair |
| Group | PT (parity-time symmetry group) |
| H^k tier | H¹ |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · LABEL |
| Papers | Paper 543, Paper 460 |
Physical system
A PT-symmetric system has a non-Hermitian Hamiltonian H satisfying [H, PT] = 0, where P is parity (x → −x) and T is time-reversal (i → −i). Despite H ≠ H†, the spectrum is real in the PT-unbroken phase — an apparently impossible result that follows from the PT symmetry constraining eigenvalues to come in complex conjugate pairs, which are forced real when PT symmetry is unbroken.
The canonical experimental realisation (Rüter et al. 2010): a pair of coupled optical waveguides, one with gain (amplification, Im(n) < 0) and one with loss (absorption, Im(n) > 0), with equal magnitudes. The coupling κ between waveguides and the gain/loss rate γ set the two control parameters.
The exceptional point (EP) at γ = κ is the β* snap: for γ < κ (PT-unbroken), eigenvalues are real; at γ = κ they coalesce into a degenerate pair; for γ > κ (PT-broken) they split into a complex conjugate pair. The EP is a branch point of the eigenvalue surface — not a crossing but a coalescence where two eigenvalues and their eigenvectors become identical.
Target category
PT-Hilb — the category of finite-dimensional Hilbert spaces with a PT-inner product ⟨φ|ψ⟩_{PT} = ⟨φ|PT|ψ⟩ (indefinite metric, not positive definite). Objects: PT-symmetric Hamiltonians H on ℂⁿ. Morphisms: PT-preserving similarities S with SHS⁻¹ Hermitian. In the PT-unbroken phase, the PT-inner product is positive definite and PT-Hilb ≅ standard Hilbert space. At the EP the metric becomes degenerate — this is the TWIST failure event.
Interpretation functor
F: C → PT-Hilb defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | PT-symmetric time evolution: U(t) = e^{−iHt}; in the unbroken phase this is unitary with respect to the PT-inner product; eigenstates propagate with real eigenfrequencies |
| TWIST | Berry phase around the exceptional point: encircling the EP in parameter space (γ, κ) applies a geometric phase that swaps the two coalescing eigenvalues; one full loop returns eigenvalue 1 to eigenvalue 2 and vice versa — a half-integer TWIST requiring two loops to return to identity |
| LABEL | Eigenvalue pair (E₊, E₋): real in PT-unbroken phase (E± = ±√(κ²−γ²)); purely imaginary in PT-broken phase (E± = ±i√(γ²−κ²)); coalesce at EP (E± = 0, γ=κ) |
ISA programme
PARAMS: LABEL[kappa, gamma | coupling and gain/loss rates]
PHASE?: LABEL[gamma < kappa? | PT-unbroken vs broken]
EIGEN: LABEL[E_pm = pm sqrt(kappa^2 - gamma^2) | real eigenvalues]
EVOLVE: ORBIT[U(t) = exp(-i H t) | PT-symmetric evolution]
WIND: TWIST[encircle EP in (gamma,kappa) plane]
SWAP: TWIST[E_+ <-> E_- after one loop | half-integer Berry phase]
SNAP: LABEL[gamma = kappa | exceptional point, EP, beta* snap]
BROKEN: LABEL[E_pm = pm i sqrt(gamma^2 - kappa^2) | complex eigenvalues]
Computable output
- Eigenvalue coalescence at γ = κ: E± = 0. At the EP, both eigenvalues and both eigenvectors become identical — the Hamiltonian is not diagonalisable but only Jordanisable: H = λI + N where N² = 0 (Jordan block). This is the LABEL degeneracy; the response function near an EP diverges as 1/(γ−κ)^{1/2} rather than the Lorentzian 1/(γ−κ) of a normal degeneracy.
- Eigenvalue swap under encirclement: encircling the EP once in the (γ,κ) parameter plane causes E₊ → E₋ and E₋ → E₊. Two encirclements restore the original labelling. This is a TWIST with winding number 1/2 — a spinorial Berry phase. Measured experimentally in: microwave cavities (Doppler et al. 2016, Nature), optical systems (Rüter et al. 2010), and acoustic resonators.
- PT-broken mode growth: for γ > κ, one eigenvalue has Im(E) < 0 giving exponential growth in the cavity mode. The amplified output power grows as e^{2|Im(E)|t}. Measured in gain-loss waveguide pairs to match theory.
- Unidirectional invisibility: near the EP, the scattering matrix becomes non-reciprocal — a wave incident from the left is partially reflected while the same wave from the right passes through invisibly. This is an ORBIT asymmetry (the forward/backward propagation ORBITs are not time-reversal images of each other), arising from the PT-broken gauge connection.
The β-plane interpretation (Paper 543 §1)
The β-plane gives an exact home for PT-symmetric systems. Writing β = σ + it:
| β-plane region | Physical regime | Spectrum |
|---|---|---|
| σ = 0 (imaginary axis) | Standard Hermitian QM | Real (always) |
| σ > 0 small (right half-plane, small) | PT-unbroken phase | Real (PT symmetry protects) |
| σ = σ* (EP locus) | Exceptional point | Degenerate real |
| σ > σ* (right half-plane, large) | PT-broken phase | Complex conjugate pairs |
| σ < 0 (left half-plane) | Gain-dominated | Growing modes |
The PT phase transition — EP at γ = κ — is a phase boundary on the β-plane at fixed |β|, varying arg(β). It is the arg(β) analogue of the BKT transition (which is at fixed arg(β), varying |β|). Both are β* snap events, just on different axes.
The TWIST half-integer winding around the EP is the β-plane version of the TWIST opcode: the eigenvalue surface is a Riemann sheet with a branch point at the EP, and the branch cut is the PT phase boundary. The half-integer winding (two loops needed to return) is the β-plane TWIST at a branch point of order 2 — exactly the structure of a √z Riemann surface.
Why H¹ and not H²
The EP is a degeneracy of eigenvalues, not a topological charge. The Berry phase from encircling an EP is geometric (depends on the path) but not quantised to an integer — it is π (half-integer), not 2π (integer). The H¹ assignment reflects:
- The TWIST winding number is ½ (spinorial), consistent with H¹ (half-integer windings are H¹ in the ℤ₂ cohomology of the parameter space)
- No H² BIND is needed: the EP has no associated second Chern class; it is a branch point (codimension-2 feature) of the H¹ eigenvalue bundle
- The exceptional point is a TWIST failure event (the H¹ bundle becomes non-trivial), not a BIND event (which would require a non-Abelian holonomy)
Compare to Yang-Mills instantons (G01), which are genuinely H² — the Chern number Q ∈ ℤ requires a full 4-manifold integral. The EP is H¹ because the relevant cohomology group is H¹(ℂ{EP}, ℤ) = ℤ, with the single generator being the winding around the branch point.
Connections to other entries
- Paper 543 §1 (β-plane): PT-symmetric QM lives at β = σ + it with σ > 0; EP is the PT phase boundary in the β-plane; explicitly in the §1 table
- G03 Higgs mechanism: the electroweak phase transition is an analogous snap — the order parameter (Higgs VEV) goes through a real→complex transition at T_c, just as PT eigenvalues go real→complex at γ = κ
- SC02 Abrikosov vortex: the vortex core is a localised region where the superconducting order parameter passes through zero — the same H¹ TWIST structure as the EP encirclement
- CA02 Residue theorem: the EP is a branch point of the eigenvalue surface E(γ,κ), analogous to a pole of a meromorphic function; the half-integer winding is the residue-theorem H² content of the branch cut
Validation
- Bender & Boettcher (1998), PRL 80, 5243: first proof that PT-symmetric Hamiltonians can have real spectra. The foundational theoretical paper.
- Rüter et al. (2010), Nature Physics 6, 192: first experimental observation of PT symmetry and its spontaneous breaking in coupled optical waveguides. EP observed at balanced gain/loss.
- Doppler et al. (2016), Nature 537, 76: encirclement of EP in microwave cavity; eigenvalue swap (TWIST half-integer winding) directly observed.
- Asymmetric state switching: Xu et al. (2016), Nature 537, 80: EP encirclement gives non-reciprocal mode switching in optomechanical system.
- Review: Özdemir et al. (2019), Nature Materials 18, 783.