Nonreciprocal & Non-Hermitian Metamaterials

• Nonreciprocal and non-Hermitian metamaterials are artificially engineered structures that break time-reversal symmetry, enabling asymmetric wave propagation and novel transport phenomena.
• They leverage diverse mechanisms like asymmetric coupling, synthetic gauge fields, and time-Floquet modulation to induce effects such as the non-Hermitian skin effect and topologically protected edge states.
• These systems offer practical applications in ultracompact isolators, reconfigurable routers, and robust sensors across photonics, acoustics, and electronics.

Nonreciprocal and non-Hermitian metamaterials are artificial media whose electromagnetic, acoustic, or elastic response is governed by non-Hermitian Hamiltonians and broken reciprocity, leading to a suite of phenomena including asymmetric transmission, non-Hermitian skin effects, topologically protected edge and corner states, anomalous amplification/attenuation, and novel transport signatures such as reflectionless conduction, perfect absorption, and giant nonreciprocal phase shifts. Their microscopic origins span spatial asymmetry in hopping or coupling, time-dependent (Floquet) modulation, parity-time (PT) symmetry breaking by distributed gain/loss, synthetic gauge fields, and dissipative-reservoir engineering. These systems have rapidly transitioned from theoretical constructs to implementable platforms in optics, electronics, acoustics, and mechanics, enabling ultracompact isolators, reconfigurable routers, frequency converters, and robust sensors.

1. Physical Origins and Classes of Nonreciprocal, Non-Hermitian Response

Nonreciprocity in metamaterials arises when the linear response fails Lorentz reciprocity—usually encoded by the permittivity tensor $\varepsilon$ lacking transpose symmetry, $\varepsilon^T \neq \varepsilon$—and/or when the effective Hamiltonian is non-Hermitian, $H \neq H^\dagger$, admitting amplification or dissipation. Traditional magneto-optic effects, time-modulation, or chiral symmetry breaking realize such responses (e.g., Faraday isolators). Recent paradigms include:

• Spatially asymmetric hopping or coupling (Hatano–Nelson model (Jana et al., 20 Dec 2025, Xiao et al., 28 Mar 2024, Ezawa, 2018)): yields non-Hermitian skin effect (NHSE), with bulk eigenstates localized at a boundary depending on direction and complex hopping parameter.
• Parity-Time (PT)-symmetry: balanced gain/loss distributions, such as alternate metal-dielectric stacks with $n(x) = n_0 \pm i\kappa$, with $\kappa$ tuned to the exceptional-point threshold, create PT-symmetric bandstructures and enable broadband nonreciprocity (III et al., 2017).
• Synthetic gauge fields: reciprocal couplings combined with a tunable Peierls phase induce spin-resolved skin effects and nonreciprocal transport even in the absence of explicit gain/loss (Li et al., 25 Apr 2025).
• Time-Floquet modulation: time-periodic coupling (e.g., via capacitive modulation), especially when the modulation is non-Hermitian, enables one-way frequency conversion and parametric gain (Koutserimpas et al., 2017, Wen et al., 2020, Chen et al., 2021).
• Dissipative coupling and interface with reservoirs: even when both subsystems are Hermitian, the self-energy of a conductor coupled to a topological system induces dissipative nonreciprocal transport along the interface (Maopeng et al., 29 Aug 2024).
• Nonlinear bias-induced nonreciprocity: static electric fields linearized around a biased operating point in nonlinear dielectrics yield real but non-symmetric permittivity tensors, breaking both reciprocity and Hermiticity (Lannebère et al., 2022).
• Multi-valued band structure and exceptional points: near-zero-index media with engineered exceptional points show path-dependent mode switching—the nonlocal "arrow of time" principle with giant nonreciprocal phase and loss (Li et al., 7 Sep 2025).

2. Non-Hermitian Skin Effect and Energy-dependent Topological Transitions

A distinctive feature of nonreciprocal non-Hermitian metamaterials is the NHSE, in which all bulk right-eigenstates pile up at one edge under open boundary conditions. This effect is analytically captured by the complex dispersion $E(k)$, where the spectral winding number $$W(E) = \frac{1}{2\pi i}\int_{-\pi}^{\pi} dk\, \partial_k \log \det[H(k)-E]$$ partitions the spectrum into energy regions localized at opposite ends—skin edges (Xiao et al., 28 Mar 2024, Jana et al., 20 Dec 2025). NHSE can be reversed or dissolved entirely by tuning onsite modulations between real and imaginary regimes.

The coexistence of skin and extended states, and the interplay between nearest- and next-nearest-neighbor nonreciprocal hopping, enables programmable spatial filters, robust transport channels, and energy routing. Extended states reside on point-gapped segments in the complex $E$-plane, immune to boundary effects, while skin states exhibit exponential profile set by localization length $\xi = 1/|\kappa|$, with $\kappa$ determined by the complex momentum shift.

In higher-dimensions, nonreciprocal skin effects persist, leading to higher-order topological corner states (Ezawa, 2018), and are classified by non-Bloch winding numbers $\Gamma$ computed in the complexified Brillouin zone.

3. PT-Symmetric and Saturable-Gain Nonreciprocal Multilayers

Parity-time symmetric metamaterials composed of metal-dielectric stacks with alternate gain/loss yield broadband and wide-angle nonreciprocity (III et al., 2017). Bandgap closes at the PT-breaking threshold $\kappa_{\rm th}$, giving a complex band structure with exceptional points, followed by gap reopening and field localization that depends strongly on illumination direction.

Upon inclusion of nonlinear saturation—modeled as intensity-dependent susceptibility $$\chi_{\rm PT}(|E|^2) = \chi_{\rm PT}^\infty/[1 + |E|^2/|E_{\rm sat}|^2]$$—the permittivity becomes spatially non-PT-symmetric, leading to pronounced nonreciprocal transmission contrasts over large bandwidth and incident angular ranges. Isolation ratio and transmission contrast, e.g., $T_F/T_B$ values exceeding 6.5, over 50 nm and $\pm60^\circ$ are achievable in submicron slabs.

Design guidelines emphasize tuning non-Hermitian strength near the exceptional point, maximizing the PT-induced band splitting, and adjusting gain saturation intensity for desired functional performance.

4. Synthetic Gauge Fields, Spin Helicity, and Skin Effects without Explicit Gain/Loss

Metamaterials exploiting gauge-field-induced topology can realize helical spin skin effects (SSE) even with purely reciprocal, dissipative coupling and no explicit gain or loss (Li et al., 25 Apr 2025). In bilayer lattice models with gauge flux $\theta$, tuning from 0 to $\pi$ creates complex energy loops with nontrivial spectral winding, manifesting first-order (edge) and second-order (corner) SSE with opposite spin accumulation directions.

Circuit implementations use capacitors, resistors, and phase-shifting elements to realize the model's Laplacian; S-parameter measurements then resolve spin-dependent nonreciprocal transmission channels $T_{\uparrow \to \downarrow}$ and $T_{\downarrow \to \uparrow}$. Control of the flux $\theta$ provides reconfigurable transfer, with direct applications in topological routing and spin-selective isolation.

5. Floquet-driven Nonreciprocity: Frequency Conversion and Topological Edge Channels

Time-periodic modulation of coupling or impedance, particularly when engineered as a non-Hermitian Floquet protocol, enables nonreciprocal behavior and parametric gain/amplification (Koutserimpas et al., 2017, Li et al., 2018, Wen et al., 2020, Chen et al., 2021). In optics/microwave, modulating the coupling between two resonators at the frequency difference $\Omega$ ensures upward-only frequency conversion; the Floquet-S-matrix becomes non-Hermitian, supporting one-way amplification and perfect isolation ($S_{21} \gg 0, S_{12} = 0$).

Space-time modulated acoustic metamaterials, in which programmable impedance profiles are applied along deeply subwavelength membranes, allow for directional evanescent conversion and nonreciprocal focusing by controlling the phase and amplitude of the modulation. Floquet Chern numbers, calculated over the $(k_x, t)$ torus, predict unidirectional edge channels in the non-Hermitian Floquet Hamiltonian, paralleling topological effects in standard Hermitian systems (Chen et al., 2021).

Nonreciprocal frequency conversion efficiency exceeding 100% (active gain), strong suppression of undesired bands via PT-balanced protocols, and programmable, software-defined interference control are demonstrated experimentally.

6. Dissipation-free Non-Hermiticity and Geometry-driven Strong Arrow of Time

Interfaces between Hermitian topological and conducting subsystems can exhibit effective non-Hermitian, nonreciprocal behavior driven solely by the conductor self-energy (reservoir-induced dissipation) (Maopeng et al., 29 Aug 2024). The resulting edge Hamiltonian, of Hatano–Nelson form, underlines a dissipative “non-Bloch” transport, spatial decay rates, and directional amplification, all without explicit gain or loss.

Zero-index magneto-optical metawaveguides with embedded exceptional points and nonzero residues realize multi-valued complex band structures—each junction between the sheets (Riemann surfaces) corresponds to mode switching and enormous nonreciprocal differences in momentum or loss (Li et al., 7 Sep 2025). Even tiny magneto-optical bias produces giant nonreciprocal phase shifts ($\sim 48$ rad/mm) and attenuation ($\sim 54$ dB/mm), orders of magnitude above conventional devices. This geometry-based arrow of time is fundamentally path-dependent and nonlocal, with broad implications for isolators, circulators, and sensor architectures.

7. Design Principles, Topological Transport, and Experimental Platforms

Universal design features across platforms include:

• Unit cell engineering: asymmetric hopping (diode-resistor networks, circulator-aided phase links), gain/loss modal profiles (op-amp circuits, active optical cavities), synthetic gauge fields (phase-shifted couplers).
• Topological invariants and skin effect control: calculation of spectral winding numbers ($W$), non-Bloch winding indices ($\Gamma$), spectral conditions for reflectionless/invisible/lasing/CPA transport (Ghaemi-Dizicheh et al., 2021).
• Experimental realization: microstrip/waveguide circuits (Maopeng et al., 29 Aug 2024), electronically configurable meta-atoms (Wen et al., 2020), stacked atomically-thin van der Waals heterostructures (Buddhiraju et al., 2020), acoustic membranes (Chen et al., 2021), photon-magnon planar hybrids with field-tunable negative refraction (Kim et al., 27 Jun 2024).
• Metrics and figures of merit: isolation ratio ($T_F/T_B$ or $|S_{21}|/|S_{12}|$), frequency bandwidth, operational angular range, negative refractive index (magnitude and fractional band), insertion loss/gain, robustness to disorder.

Integration of higher-order topology (corner states, multi-dimensional invariants), reflectionless edge modes, and programmable amplification/attenuation enables next-generation metamaterial devices for communications, signal processing, and quantum information, with magnet-free, CMOS-compatible, and ultra-compact architectures.

Table: Mechanisms and Effects in Nonreciprocal Non-Hermitian Metamaterials

Mechanism	Key Physical Effect	Representative Platform
Asymmetric hopping (Hatano–Nelson)	Skin effect, NH jump, amplification	Electronic/photonic lattice
PT symmetry (gain/loss layer)	Bandgap closing, directional transmission	Metal-dielectric multilayers
Synthetic gauge field	Spin-helical skin effect	Circuit/metamaterial bilayer
Floquet time-modulation	Parametric amplification, unidirectional conversion	Microwave/optical/acoustic
Reservoir-induced dissipation	Non-Bloch transport	Edge of TI/conductor junction
EP-induced multi-valued bands	Giant nonreciprocity, path dependence	Zero-index MO metawaveguide

In sum, nonreciprocal and non-Hermitian metamaterials constitute a rapidly maturing domain combining advanced symmetry principles, active and passive engineering, and quantum-inspired topological design toward robust, highly asymmetric wave transport across the electromagnetic, acoustic, and elastic spectrum.