Hinge Effect in Topological & Mechanical Systems

• The hinge effect is a phenomenon where the physical response concentrates at a hinge-like locus instead of the bulk, defining its unique boundary behavior.
• In topological systems, localized one-dimensional channels emerge at the intersections of gapped surfaces, facilitating robust conduction and waveguiding.
• In mechanical contexts, a hinge defect introduces localized rotational compliance that modifies force transmission and enables tunable elasticity.

Searching arXiv for recent and foundational papers on “hinge effect” across topological, non-Hermitian, acoustic, mechanical, and polymer contexts. In contemporary arXiv usage, the term hinge effect denotes a class of phenomena in which the dominant physics is concentrated at a hinge-like locus rather than in the bulk or on an extended surface. In higher-order topological matter, that locus is typically the one-dimensional intersection of two surfaces, where protected conducting or waveguiding channels can appear. In non-Hermitian systems, the same geometric locus can host skin accumulation driven by point-gap topology or gain/loss engineering. In superconducting hybrids, hinge channels can carry Josephson current and generate anomalous Shapiro-step responses. In mechanics and biophysics, by contrast, the hinge effect refers to a localized rotational degree of freedom or compliance defect that strongly modifies force transmission and deformation (Fu et al., 2021, Fang et al., 9 May 2025, Kazmin et al., 2024, Benetatos, 2017).

1. Terminological scope and recurring geometry

Across the literature, the unifying feature is not a single microscopic mechanism but the localization of response at a hinge. In three-dimensional higher-order topological systems, the hinge is the codimension-2 boundary where two gapped surfaces meet. In mechanical systems, it is a localized connector or defect that makes relative rotation energetically inexpensive. This common geometric language explains why the same term spans condensed matter, acoustics, metamaterials, and polymer physics.

Domain	Meaning of “hinge effect”	Representative papers
Higher-order topology	1D modes localized at the intersection of two surfaces	(Fu et al., 2021, Yue et al., 2018, Wei et al., 2021)
Non-Hermitian and hybrid wave physics	Skin or topological localization at intersections of gain/loss boundaries or surfaces	(Fang et al., 9 May 2025, Peters et al., 2024, Hu et al., 25 Dec 2025)
Superconducting probes	Josephson transport mediated by hinge channels	(Kazmin et al., 2024, Choi et al., 30 Jan 2025)
Mechanics and biophysics	Local rotational compliance from a defect, connector, or embedded reinforcement	(Benetatos, 2017, Meeussen et al., 2024, Khatua et al., 9 Aug 2025)

A recurrent misconception is that any hinge-localized mode is necessarily topological. That is not generally correct. Several works treat hinge states as explicit signatures of higher-order topology, but first-principles calculations for strained half-Heuslers show that hinge-localized boundary states can also be topologically trivial, strongly termination dependent, and partially obscured by surface-state hybridization (Das et al., 28 Apr 2025).

2. Higher-order topological hinge states

The canonical topological meaning of the hinge effect is the emergence of one-dimensional boundary channels in a three-dimensional system with gapped bulk and gapped surfaces. In this setting, the hinge mode is the boundary manifestation of higher-order bulk topology. A central formulation is the bulk-hinge correspondence developed for three-dimensional second-order topological insulators, where the flowing pattern of chiral hinge modes is encoded by a quadrupole index together with a slab Chern number (Fu et al., 2021). In that framework, the hinge current is related to spatial variation of an axion angle through

$$\mathbf{j}_{\theta}(\mathbf{r},t)=\frac{e^{2}}{2\pi h}\left[\partial_{t}\theta(\mathbf{r},t)\mathbf{B}-\nabla\theta(\mathbf{r},t)\times\mathbf{E}\right],$$

so a hinge channel is interpreted as the current-carrying singularity of the axion field.

A complementary and widely used microscopic picture is the surface-mass domain-wall mechanism. In ferromagnetic axion insulator $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$, mirror-related surfaces acquire opposite Dirac masses under appropriate in-plane magnetization, and a chiral mode is trapped where the mass changes sign (Yue et al., 2018). The same logic underlies effective descriptions of 3D second-order topological insulators with $R_4^zT$ symmetry, where adjacent side surfaces carry alternating mass signs and the hinge state appears as a Jackiw–Rebbi mode of the surface Dirac theory (Wang et al., 2022).

The hinge effect can also be field-tunable. In Cd$_3$As$_2$-like Dirac semimetals, a tilted magnetic field

$\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$

can convert the usual symmetric hinge pattern into one-sided higher-order topological hinge states. The mechanism combines Landau quantization, Zeeman mixing $\Delta_y$, and Zeeman splitting $\Delta_z$, which decouple the two time-reversed Weyl sectors enough to place hinge modes on only one side of the top surface and the opposite side of the bottom surface. The paper reports that a protective Landau-level gap requires roughly $B_y>10$ T and $\theta>5^\circ$, and that the conductance remains quantized at $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$0 up to disorder strengths of about $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$1 meV (Chen et al., 2021).

Higher-order hinge transport has been realized beyond electrons. A 3D acoustic higher-order topological insulator supports hinge states along three independent directions of the same sample, allowing sound to travel along a bottom $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$2 hinge, turn into a $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$3-directed hinge, and continue along a top $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$4 hinge. The effective hinge Hamiltonians

$\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$5

encode helical hinge propagation generated by mass-sign changes across interfaces (Wei et al., 2021).

Topological superconductors provide a further refinement: chiral Majorana hinge modes. In superconducting Dirac materials with $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$6 symmetry, a mixed $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$7 state with relative phase $\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$8 produces a second-order topological superconductor. The bulk diagnostic is the winding of the quadrupole moment,

$\mathrm{Bi}_{2-x}\mathrm{Sm}_x\mathrm{Se}_3$9

with $R_4^zT$0 in the nontrivial phase, while the low-energy hinge dispersion is

$R_4^zT$1

(Fu et al., 2020).

3. Non-Hermitian, Floquet, and hybrid hinge localization

A second major usage of the term concerns hinge accumulation induced by non-Hermiticity or periodic driving. Here the hinge effect is not merely higher-order boundary topology in the Hermitian sense; it arises from spectral winding, exceptional points, gain/loss-resolved boundary transport, or repeated boundary reflections.

In a hybrid skin-topological sonic crystal, the hinge effect is the appearance of hinge skin states at the intersections of a gain boundary and a loss boundary. The starting point is a 3D layer-stacked Haldane-type sonic crystal that reduces, at fixed $R_4^zT$2, to a 2D Haldane model with chiral boundary modes. When staggered gain/loss $R_4^zT$3 is introduced, the Hermitian chiral edge modes split into amplified and attenuated branches, with amplitude scaling

$R_4^zT$4

If adjacent zigzag boundaries carry opposite non-Hermitian character, the boundary modes accumulate at their common endpoint, giving a hinge skin state described as

$R_4^zT$5

Reversing the sign of $R_4^zT$6 swaps the gain and loss boundaries and moves the localized hinges to the opposite corners (Fang et al., 9 May 2025).

In a strongly correlated $R_4^zT$7-electron system, the hinge non-Hermitian skin effect arises from the effective single-particle Hamiltonian

$R_4^zT$8

which is non-Hermitian because the self-energy has an imaginary part. Surface exceptional points generate a nontrivial point-gap topology for one-dimensional momentum cuts crossing exactly one EP-pair branch cut. The corresponding winding number,

$R_4^zT$9

is nonzero only for those cuts, and under open boundary conditions right and left eigenstates accumulate on opposite sides of the surface, producing hinge-localized skin modes (Peters et al., 2024).

The non-Hermitian hinge effect can coexist with genuine higher-order semimetal topology. A higher-order Weyl exceptional ring semimetal in a lossy phononic crystal hosts both topological hinge states and a hinge-dependent skin effect. The Weyl exceptional rings carry a Chern number and a spectral winding number,

$_3$0

while the second-order index is the $_3$1-resolved bulk polarization. The reported topological hinge states remain nearly purely real in frequency despite loss, in contrast to trivial hinge states with larger imaginary parts (Hu et al., 25 Dec 2025).

Periodic driving generates a distinct hinge effect even without non-Hermiticity. In a six-step Floquet lattice, open boundaries in two directions reorganize the entire quasienergy spectrum into hinge-parallel quasi-one-dimensional modes. The central mechanism is repeated boundary reflection with Goos–Hänchen-like lateral shifts, which produce a net drift along the hinge. At the fine-tuned point $_3$2, the stroboscopic hinge velocity is

$_3$3

and the authors interpret the resulting localization as a second-order Floquet skin effect rather than a conventional in-gap higher-order boundary state (Huang et al., 2021).

4. Josephson and optical manifestations of hinge states

One of the most active experimental uses of the hinge effect is as a transport diagnosis of hinge-localized electronic channels. Superconducting proximity structures are especially important because the current-phase relation can distinguish ordinary surface or bulk transport from hinge-mediated Josephson dynamics.

In long In/NiTe$_3$4/In Josephson junctions with $_3$5m, the normal-state analysis gives $_3$6m and a diffusive bulk SNS coherence length

$_3$7

so $_3$8 and bulk Josephson transport is argued to be suppressed. Under $_3$9 GHz microwave irradiation, the junctions show integer Shapiro steps at $_2$0 together with half-integer steps $_2$1. The interpretation is a dominant $_2$2-periodic harmonic,

$_2$3

rather than a $_2$4-periodic relation. The persistence of odd integer steps is taken to rule out the canonical helical-surface-state scenario and to support interference of chiral topological hinge modes as the source of the fractional a.c. Josephson effect (Kazmin et al., 2024).

This distinction is important because a fractional Josephson signature is not unique. In NiTe$_2$5, half-integer Shapiro steps are interpreted as evidence for $_2$6-periodic hinge-mode interference, whereas in hinge-dominant Al-WTe$_2$7-Al junctions the salient feature is the absence of the first Shapiro step, attributed to a $_2$8-periodic component carried by topological hinge states. The WTe$_2$9 study compares a bulk-dominant junction of width $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$0 with a hinge-dominant junction of width $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$1, both with $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$2. Only the hinge-dominant device loses the first step at low microwave frequency, with $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$3 versus $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$4 in the bulk-dominant device. The data are modeled with

$\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$5

and the inferred $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$6-periodic current $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$7 agrees with the $\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$8 hinge current extracted independently from magnetic interference (Choi et al., 30 Jan 2025).

Optical probes target a different aspect of the hinge effect: the geometry of the hinge state wave function. In ferromagnetic MnBi$\mathbf{B}=(0,B_y,B_z=B_y\tan\theta)$9Te$\Delta_y$0, the hinge state can carry a non-Abelian optical Berry curvature that is detectable through a hinge circular photogalvanic effect. For a local illumination region satisfying $\Delta_y$1, the hinge CPGE coefficient is written as

$\Delta_y$2

and its frequency integral

$\Delta_y$3

measures the hinge-to-ground-state optical Berry curvature $\Delta_y$4 (Liu et al., 2022).

5. Boundary conditions, finite size, disorder, and non-topological hinge states

The visibility and interpretation of a hinge effect are often controlled by finite geometry and competing boundary channels. This is especially clear in analytic studies of finite-size hybridization. Starting from the effective surface theory of a 3D second-order topological insulator, one finds gapless hinge modes in the semi-infinite limit with

$\Delta_y$5

In a finite sample, however, the two hinge states on the same surface overlap and open a gap

$\Delta_y$6

For the parameter set $\Delta_y$7, $\Delta_y$8, $\Delta_y$9, $\Delta_z$0, the reported values are $\Delta_z$1 at $\Delta_z$2 and $\Delta_z$3 at $\Delta_z$4 (Wang et al., 2022).

An important caveat is that hinge localization need not certify higher-order topology. First-principles work on strained half-Heuslers LiSbZn and LiBiZn finds hinge-localized boundary states in both the trivial and band-inverted materials. The authors therefore identify the observed modes as topologically trivial hinge states. Their visibility depends strongly on prism termination and on whether surface states are present to hybridize with them. The sharpest hinge localization occurs in the topologically trivial LiSbZn case, where the bulk and surfaces are gapped; in LiBiZn, topological surface modes hybridize with the hinge states and obscure them (Das et al., 28 Apr 2025).

Disorder likewise distinguishes different hinge mechanisms. In a Dirac-semimetal realization of the 3D quantum Hall effect, one-sided hinge transport survives weak disorder with quantized $\Delta_z$5 up to $\Delta_z$6 meV (Chen et al., 2021). In disordered non-Hermitian 3DSOTIs, by contrast, hinge states remain robust but transport statistics deviate from the Hermitian paradigm: the mean transmission may cease to equal the number of hinge channels if the hinge states are non-Hermitian, and the transmission fluctuation

$\Delta_z$7

is always nonzero because of incoherent scattering from non-Hermitian potentials (Wang et al., 2022).

The hinge effect can also coexist with more conventional boundary modes. In a charge-density-modulated 3D quantum Hall stack of weakly coupled wires, the system can realize a second-order phase with chiral quasi-1D hinge modes, but for smaller CDW amplitude or larger interwire coupling it enters a hybrid higher-order topology in which 1D hinge modes coexist with 2D chiral surface QHE states. The hinge propagation direction depends on the CDW phase $\Delta_z$8 and can be reversed electrically without reversing the magnetic field (Szumniak et al., 2019).

6. Mechanical and biophysical hinge effects

Outside topological wave physics, the hinge effect refers to a localized rotational compliance that changes global mechanics. The most direct polymer example is a wormlike chain with a permanent hinge defect, modeled by the bending energy penalty

$\Delta_z$9

The hinge introduces an extra orientational degree of freedom and increases the differential tensile compliance

$B_y>10$0

In the small-force regime with free hinged-hinged ends, the paper finds

$B_y>10$1

whereas the intact chain gives

$B_y>10$2

The entropic elasticity is therefore enhanced by a factor of $B_y>10$3, and the position dependence is strongest near the ends over a scale set by the deflection length $B_y>10$4 (Benetatos, 2017).

Mechanical metamaterials use the same idea in engineered form. Textile hinges are designed to approximate ideal rotational joints in the mechanism limit, quantified by

$B_y>10$5

with $B_y>10$6 in the ideal limit. Textile hinges span nearly two orders of magnitude in bending stiffness,

$B_y>10$7

and for the main T1 hinge the reported values are $B_y>10$8, $B_y>10$9, $\theta>5^\circ$0, and $\theta>5^\circ$1. In rotating-square metamaterials, this enables nearly mechanism-like closure and deformation transmission that living hinges do not achieve (Meeussen et al., 2024).

A related but distinct hinge effect occurs in slender fiber-reinforced composites. Under compressive loading, a shell with carbon fiber showed about 20% less stiffness and 100% more strength than the unreinforced structure. Fiber-pullout tests ruled out poor axial bonding, leading to the hypothesis that the matrix rotates around the embedded fiber as around a hinge. The beam-scale analytical model treats the structure as two beam segments connected by a restrained hinge or rotational spring, and the experiments show that the stiffness reduction is strongest when the fiber is closer to the fixed end. The same mechanism is used to create hill and valley folds in deployable sheets (Khatua et al., 9 Aug 2025).

In this mechanical literature, the hinge effect is thus the inverse of its topological meaning. It does not denote a protected codimension-2 mode; it denotes a localized kinematic freedom that reduces the energetic cost of rotation and reorganizes deformation pathways.

7. Conceptual synthesis

The term hinge effect therefore names a geometric principle rather than a single theory. In higher-order topological systems, the hinge is where bulk topology, surface mass sign changes, or non-Hermitian point-gap winding become observable as one-dimensional channels. In superconducting hybrids, hinge channels are read out through Shapiro-step anomalies and related Josephson diagnostics. In non-Hermitian and Floquet media, the hinge can become the endpoint of directional pumping, gain/loss conversion, or reflection-induced drift. In mechanics, the hinge is a localized compliance element that converts distributed strain into concentrated rotation.

Two general lessons recur across these otherwise disparate literatures. First, hinge localization is highly sensitive to boundary conditions: surface termination, sample width, disorder, gain/loss patterning, and microwave drive all control whether the hinge dominates the observable response. Second, hinge localization is not automatically topological: some hinge states are enforced by bulk invariants, whereas others are trivial boundary resonances or mechanically engineered rotational modes (Das et al., 28 Apr 2025, Wang et al., 2022). The hinge effect is therefore best understood as a boundary-concentration phenomenon whose specific meaning must be fixed by the underlying Hamiltonian, constitutive law, and experimental observable.