Non-Hermitian physics has emerged as a transformative framework for understanding and engineering plasmonic systems, where the interplay between gain, loss, and near-field interactions leads to unconventional optical phenomena. Unlike Hermitian systems, which conserve energy and exhibit real eigenvalues, non-Hermitian systems embrace dissipative and amplifying processes, enabling unique functionalities such as exceptional points, parity-time (PT) symmetry, and tailored dispersion relations. Plasmonic platforms, with their strong field confinement and sensitivity to material losses, provide an ideal testbed for exploring these concepts.

At the heart of non-Hermitian plasmonics lies the concept of exceptional points (EPs), spectral singularities where eigenvalues and eigenvectors coalesce. EPs arise when a non-Hermitian system undergoes a parameter-dependent transition, leading to a square-root branch point in the eigenvalue spectrum. In plasmonic dimers or arrays, EPs can be engineered by balancing radiative and absorptive losses or by introducing asymmetric coupling between resonant modes. The presence of EPs enhances sensitivity to perturbations, a property leveraged for sensing applications, though practical implementations fall outside this discussion. Mathematically, a two-mode system near an EP is described by a Jordan block structure, with the Hamiltonian taking a defective form that cannot be diagonalized.

Parity-time symmetry, a cornerstone of non-Hermitian physics, requires that the complex potential satisfies V(r) = V*(-r). In plasmonic systems, PT symmetry is realized by judiciously arranging gain and loss regions such that the balanced condition γ_gain = γ_loss holds, where γ represents the imaginary part of the dielectric function. PT-symmetric plasmonic lattices exhibit a phase transition from a PT-exact regime, with real eigenvalues, to a PT-broken regime, where eigenvalues become complex conjugates. The transition point, known as the PT threshold, depends on the coupling strength κ between modes relative to the loss/gain rate γ, following the condition κ > γ/2. Below threshold, the system displays oscillatory dynamics, while above threshold, mode amplitudes grow or decay exponentially.

Loss engineering in plasmonic systems exploits non-Hermitian principles to manipulate scattering, absorption, and dispersion. By introducing controlled dissipation or gain, the linewidths of plasmon resonances can be tailored, leading to phenomena such as loss-induced transparency or coherence perfect absorption. The Fano resonance lineshape, prevalent in plasmonic systems, is particularly sensitive to non-Hermitian perturbations, with asymmetric coupling between bright and dark modes leading to sharp spectral features. The non-Hermitian perspective also clarifies the role of leakage radiation in plasmonic waveguides, where the interplay between intrinsic absorption and radiative losses dictates the quality factor of guided modes.

Coupled-mode theory (CMT) provides a powerful analytical tool for describing non-Hermitian plasmonic systems. For two interacting resonators with amplitudes a1 and a2, the dynamics are governed by the equations:
da1/dt = (-iω1 - γ1)a1 + iκa2
da2/dt = (-iω2 - γ2)a2 + iκa1
Here, ω1 and ω2 are resonant frequencies, γ1 and γ2 represent loss (or gain) rates, and κ is the coupling coefficient. The eigenvalues of this system exhibit avoided crossings in the Hermitian limit (γ1 = γ2 = 0) but coalesce at EPs when γ1 ≠ γ2 and detuning conditions are met. The non-Hermitian CMT framework naturally incorporates temporal dynamics, enabling studies of transient amplification and mode switching.

Scattering matrix (S-matrix) approaches offer complementary insights by relating incoming and outgoing waves in open plasmonic systems. The S-matrix for a non-Hermitian system is subunitary, with singularities corresponding to bound states in the continuum or lasing thresholds. The eigenvalues of the S-matrix, given by sn = exp(iφn), where φn are complex phase shifts, reveal information about transmission zeros and poles in the complex frequency plane. For PT-symmetric systems, the S-matrix satisfies the relation S(-ω*) = S^†(ω), imposing constraints on reflection and transmission coefficients.

Non-Hermitian effects also manifest in the topological properties of plasmonic systems. Complex Berry phases and non-Hermitian band structures arise in periodic plasmonic lattices, where gain-loss modulation creates non-reciprocal bulk modes. The non-Hermitian skin effect, wherein bulk modes localize at boundaries, has been theoretically predicted in plasmonic chains with asymmetric hopping. These phenomena are described by generalized Brillouin zones and complex-valued topological invariants, extending conventional band theory to dissipative systems.

The interplay between non-Hermiticity and nonlinearity in plasmonics introduces additional richness. Kerr nonlinearities in gain-compensated plasmonic waveguides can stabilize soliton-like modes despite inherent losses, while saturable absorption leads to bistability and self-pulsing. The Lugiato-Lefever equation, modified to include non-Hermitian terms, describes pattern formation in nonlinear plasmonic cavities, with solutions exhibiting complex spatiotemporal dynamics.

Analytical challenges in non-Hermitian plasmonics include the proper treatment of open boundary conditions and the non-orthogonality of eigenmodes. Petermann factors, which quantify mode non-orthogonality, become significant in systems with EPs, affecting spontaneous emission rates and noise properties. Green's function methods adapted for non-Hermitian systems reveal modified local density of states near plasmonic nanostructures, with implications for Purcell enhancement and near-field thermal radiation.

Theoretical advances continue to uncover fundamental limits imposed by non-Hermitian dynamics. Generalized uncertainty relations for non-Hermitian operators constrain simultaneous measurements of coupled plasmonic observables, while fluctuation-dissipation theorems require modification to account for gain media. Recent work has established bounds on exceptional point enhancement factors based on system dimensionality and mode overlap integrals.

Non-Hermitian physics provides a unified framework for understanding plasmonic systems where gain, loss, and coupling cannot be treated as perturbations. From exceptional point-enhanced sensing to PT-symmetric plasmonic circuits, the theoretical insights derived from this perspective continue to reshape our understanding of nanoscale light-matter interactions. Future directions include the development of ab initio non-Hermitian density functional theory for plasmonic materials and the integration of machine learning methods for inverse design of non-Hermitian plasmonic architectures.