Geometric Interface Phase: Theory & Applications

• Geometric interface phase is a phase contribution originating from geometric configurations at material interfaces, distinct from dynamical effects.
• It appears in diverse fields such as GPA in microscopy, Berry/Zak phase analysis in photonics, and twist-induced effects in superconducting junctions.
• Researchers leverage methods like Fourier analysis, DFT models, and interferometry to distinguish genuine strain effects from basis-induced artifacts.

Searching arXiv for recent and foundational papers on geometric/interface phase across microscopy, superconducting junctions, photonics, mechanics, and transport. Geometric interface phase denotes an interface-associated phase contribution whose origin is geometric rather than purely dynamical. In the literature represented here, the concept appears in several technically distinct forms: as a basis-dependent Fourier phase in geometric phase analysis (GPA) of compound materials that can mimic lattice strain across chemically distinct interfaces (Petersa et al., 2015); as an electronic phase shift induced by twist or coherent reflection in superconducting and valleytronic junctions (Song et al., 21 Nov 2025, Zeng, 2024); as a Berry- or Zak-phase mismatch that controls localized interface states in photonic, plasmonic-photonic, and elastic periodic media (Gao et al., 2016, Ge et al., 2017, Kumar et al., 2024); and as a geometric quantity accessible through modal conversion or closure-phase geometry in optics and interferometry (Li et al., 2023, Thyagarajan et al., 2020). The same corpus also shows a separate usage in which “phase” denotes a material phase and “geometric interface” denotes a reconstruction or evolution method rather than a phase angle (Du et al., 2019).

1. Fundamental definitions and formal structure

Across the phase-angle literature, the common structure is an observable phase term that is added to the usual dynamical or displacement phase because an internal basis, eigenstate, or modal geometry changes at or across an interface. In atomically resolved GPA, the measured phase is

$$\phi_{\mathrm{meas}}(g,r)= -2\pi\, g\cdot u(r) + \phi_{\mathrm{basis}}(g,r),$$

so the physically desired lattice-displacement term and an additional basis-dependent term are superposed in the same quantity (Petersa et al., 2015).

In Berry-type formulations, the geometric phase is written either as a discrete loop product,

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$

or in continuum form,

$$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$

with the parameter $R$ chosen, for example, as twist angle or crystal momentum (Song et al., 21 Nov 2025).

In one-dimensional periodic media, the relevant bulk invariant is often the Zak phase,

$$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$

which, in inversion-symmetric systems, is restricted to $0$ or $\pi$, and whose mismatch across an interface controls whether a bound state is guaranteed in a common gap (Gao et al., 2016).

In interferometry, closure phase is the phase of a closed-loop product of correlations,

$$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$

and its invariance to element-based phase corruption and translation is recast geometrically through the conserved shape, orientation, and size of the principal fringe triangle (Thyagarajan et al., 2020).

2. Basis-dependent interface phase in geometric phase analysis

In aberration-corrected STEM or phase-contrast HRTEM, the recorded image can be written as the convolution of a perfect Bravais lattice $\Lambda(r)=\sum_j \delta(r-R_j)$ with a basis image $f(r)$. For a compound lattice with more than one atom per unit cell, the Fourier component at reciprocal vector $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$0 carries an additional structure-factor phase,

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$1

where $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$2 is the sublattice displacement within the unit cell and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$3 are relative scattering weights. After GPA masking and inverse Fourier transformation, the measured phase becomes

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$4

so any spatial variation in the basis term is mathematically indistinguishable from a displacement phase unless it is treated explicitly (Petersa et al., 2015).

This point is decisive at interfaces. If the basis phase jumps by $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$5 across a sharp interface, differentiation produces a strain-like singularity: $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$6 In the bi-atomic case, $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$7 often equals $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$8 for reflections with $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$9, so conventional GPA can report an apparent $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$0 displacement jump. In practice, finite mask width broadens the interface, and the apparent strain is measured as a $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$1–$$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$2 signal spread over a few unit cells rather than as an exact delta-like spike.

The InGaAs/AlAsSb quantum-cascade example demonstrates the mechanism directly. In the [110] ADF-STEM image, the group-III and group-V dumbbells have sublattice shift $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$3. Across the thin $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$4 layer the A–B contrast inverts, so $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$5 for $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$6 with $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$7 jumps by approximately $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$8. GPA using $$\phi_g=i\oint_C \langle u(R)|\nabla_R u(R)\rangle\cdot dR,$$9 reports $R$0 at the interface, although visual inspection shows virtually no lattice strain. Choosing $R$1 and $R$2, for which $R$3, removes the basis phase and recovers $R$4. Strain-free simulations reproduce both the spurious strain for $R$5 and its disappearance for integer $R$6.

The $R$7 perovskite interface shows the same artifact in a different lattice. In [001] ADF-STEM the Sr sublattice persists, while the B-site columns invert as Ti and Ru are exchanged; the sublattice shift is $R$8. GPA with $R$9 or $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$0, again satisfying $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$1, yields huge false $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$2 and $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$3 values greater than $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$4 at the interface. Switching to $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$5, for which $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$6, eliminates the basis phase and yields negligible strain below $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$7.

The practical consequence is methodological rather than merely interpretive. The recommended strategy is to use only spot pairs satisfying $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$8, inspect $$\Phi_n^{\mathrm{Zak}} = i\int_{\mathrm{BZ}} \langle u_{n,k}|\partial_k u_{n,k}\rangle\,dk,$$9 before differentiation for sudden jumps at known interfaces, compare GPA results from different $0$0-sets, and run the same GPA workflow on an ideal strain-free structure. In this context, the geometric interface phase is a chemically induced imaging phase that can masquerade as mechanics.

3. Electronic interface phases in twisted and coherent junctions

In twisted bilayer $0$1, the geometric interface phase is defined in direct analogy with the Pancharatnam–Berry phase. Large-scale DFT near $0$2 twist reveals flat interface bands from Bi-O layers with bandwidth $0$3 that become spin-split by approximately $0$4 below approximately $0$5, producing spontaneous spin polarization and local ferromagnetic order at the interface, while the Cu-$0$6 bands remain dispersive and nearly decoupled. The spin texture $0$7 of the spin-polarized flat bands generates a Berry connection

$0$8

and the DFT fit yields an interlayer coupling $0$9–$\pi$0 for twist angles $\pi$1–$\pi$2 (Song et al., 21 Nov 2025).

Within a single chiral component of the $\pi$3-wave order, twisting one layer by $\pi$4 gives a global phase

$\pi$5

For cuprates, $\pi$6 gives $\pi$7, while the opposite chirality gives $\pi$8. This geometric contribution shifts the Josephson current-phase relation from

$\pi$9

to

$$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$0

Because opposite chiralities experience opposite shifts, the same twist can selectively enhance or suppress one chirality. If the two layers host opposite chirality states, the two phase shifts cancel and the net Josephson charge current vanishes, leaving only a neutral orbital current. When time-reversal symmetry is weakly broken by interface ferromagnetism or an applied field, the cancellation is incomplete and a net Josephson current emerges with strong chirality polarization. The DFT-based model shows $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$1 decreasing from approximately $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$2 at $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$3 to approximately $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$4 at $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$5, free-energy minima shifted by $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$6, and a $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$7 that reproduces the experimental $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$8 peak when the geometric phase shift is included. Proposed measurements include SQUID or tri-junction interferometry, low-temperature twist-angle sweeps near $$\widetilde\phi_N=\arg\!\prod_{\rm loop}\widetilde V_{ij},$$9, Josephson diode measurements in small in-plane fields, Shapiro-step spectroscopy, and tunneling detection of

$\Lambda(r)=\sum_j \delta(r-R_j)$0

A second electronic realization appears in coherent tunneling through two combined barriers in the $\Lambda(r)=\sum_j \delta(r-R_j)$1-$\Lambda(r)=\sum_j \delta(r-R_j)$2 lattice. At a single barrier interface, the reflection phase can be decomposed as

$\Lambda(r)=\sum_j \delta(r-R_j)$3

where $\Lambda(r)=\sum_j \delta(r-R_j)$4 is the dynamical phase and $\Lambda(r)=\sum_j \delta(r-R_j)$5 is the geometric Pancharatnam phase determined by a geodesic triangle on the Bloch sphere. The explicit result,

$\Lambda(r)=\sum_j \delta(r-R_j)$6

shows that the interface phase is valley dependent. In a double-barrier Fabry–Pérot geometry, the coherent sum of the two interface phases generates a total phase $\Lambda(r)=\sum_j \delta(r-R_j)$7, and the transmission becomes valley skewed. This skew tunneling produces a transverse valley current with zero net charge. The effect is electrically controlled by the barrier heights and disappears exactly when the two barriers are equal, because then $\Lambda(r)=\sum_j \delta(r-R_j)$8 and $\Lambda(r)=\sum_j \delta(r-R_j)$9 (Zeng, 2024).

4. Bulk geometric phases and interface-localized states

In one-dimensional photonic crystals, the interface phase is governed by bulk Zak phases rather than by an explicitly inserted defect. For the $f(r)$0th band,

$f(r)$1

and in inversion-symmetric systems this quantity is $f(r)$2 or $f(r)$3. The reflection phase inside the $f(r)$4th bandgap,

$f(r)$5

is determined modulo $f(r)$6 by the cumulative Zak phase of all bands below that gap: $f(r)$7 If two semi-infinite photonic crystals meet at an abrupt interface, an interface mode exists when their reflection phases differ by $f(r)$8. By choosing the unit-cell inversion center and tuning the layer-thickness ratio, Gao et al. showed that interface states can be guaranteed in odd gaps, even gaps, or all gaps without any extrinsic defect layer. The designs were verified experimentally by fabricating $f(r)$9 and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$00 multilayers, measuring the reflection phase with a $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$01–$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$02 Fabry–Pérot etalon formed with a flat glass plate, and correcting numerical-aperture artifacts by repeating the measurement after coating a thin Ti film (Gao et al., 2016).

A closely related topological mechanism occurs in the one-dimensional hybrid plasmonic-photonic crystal formed by a simple lattice of graphene sheets. For TM modes, plasmonic and photonic branches cross at an accidental degeneracy point at $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$03. This crossing is a diabolic point accompanied by a topological phase transition. A closed loop around the degeneracy carries Berry phase $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$04, and the Zak phase of the lower band jumps from $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$05 to $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$06 at $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$07. The semi-infinite system has an analytic surface impedance,

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$08

and interface states satisfy $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$09 with the ambient impedance $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$10. In the projected spectrum, the corresponding interface-state branches either start from or terminate at the diabolic point (Ge et al., 2017).

Elastic waveguides extend the same logic to guided mechanical modes on a parameter manifold. For a slowly varying parameter vector $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$11, the geometric phase is

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$12

When the interface joins two periodic waveguides with Zak phases $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$13 and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$14, a difference

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$15

guarantees one protected mode in the common bandgap. The review summarizes several explicit cases: a triangular-cross-section waveguide in which loops enclosing a degeneracy pick up a $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$16 topological phase, a stepped-beam phononic crystal in which varying a unit-cell parameter across a critical value closes and reopens a gap, and helical waveguides in which the polarization phase is geometric but not topological because it varies continuously with enclosed solid angle (Kumar et al., 2024).

These systems collectively realize a bulk–interface correspondence in which the interface phase is not an independent local parameter but a manifestation of a global phase structure already encoded in the adjoining media.

5. Modal and interferometric realizations

Planar optics with liquid-crystal geometric phase provides a modal interface in which the phase is written directly into the spatial pattern of a half-wave retarder. Under the circular-polarization basis $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$17, a local liquid-crystal director angle $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$18 imposes

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$19

Using a fractional Fourier transform design with independent astigmatic phases $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$20 and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$21, the planar device produces a net retardance

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$22

which acts as a modal waveplate between Hermite–Gaussian and Laguerre–Gaussian states. The device is capable of reciprocal conversion between all possible OAM states on the same modal sphere. At $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$23, the reported conversion efficiency is $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$24, the state fidelity is $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$25, crosstalk is below $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$26, insertion loss is $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$27, and the operational bandwidth is approximately $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$28. A cyclic trajectory on the HLG modal sphere encloses a solid angle $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$29 and yields a higher-order geometric phase

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$30

which was measured to within $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$31 over $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$32 (Li et al., 2023).

Interferometric closure phase supplies a different geometric realization. For corrupted visibilities $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$33, the closure phase on a loop is

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$34

and the antenna-based phase errors cancel identically. The geometric interpretation is the “shape–orientation–size” conservation principle: the three null-phase curves generated by a closed triad define a principal triangle in the image plane, and antenna-based phase errors shift each fringe only parallel to itself, preserving the triangle’s shape, orientation, and size. This invariance yields two direct image-domain measurements. In the height method,

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$35

where $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$36 is the perpendicular distance from a chosen vertex to the third null-phase curve. In the area-product method,

$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$37

where $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$38 and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$39 are the triad areas in the aperture and image planes. The framework was validated on 3C 286, Cygnus A, and M87 data from the VLA and EHT, with height- and area-based measurements agreeing with the standard visibility-sum closure phase (Thyagarajan et al., 2020).

These modal and interferometric cases broaden the notion of geometric interface phase beyond material interfaces in the strict condensed-matter sense. Here the “interface” is the operational boundary between modal bases, polarization states, or fringe families, while the phase remains a genuinely geometric observable.

6. Interpretive boundaries, diagnostics, and distinct uses of “phase”

The literature also makes clear that geometric interface phases are not interchangeable. In GPA, an interface phase may be entirely non-mechanical and may generate a false strain map. In elastic waveguides, a geometric phase may be topological or non-topological depending on whether it is tied to a degeneracy and a quantized invariant. In the $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$40-$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$41 double barrier, the valley Hall current disappears when the two barriers are of equal height. In twisted cuprate junctions, opposite chirality states can cancel the net Josephson charge current even though each chirality experiences a nonzero phase shift. These cases show that the experimental signature depends not only on the presence of an interface phase but also on whether the phase survives symmetry, gauge, or basis cancellation.

A separate body of work uses “phase” in the thermodynamic or material sense and “geometric interface” in the numerical sense of sharp or diffuse interface tracking. Du and Feng survey the phase-field method for geometric moving interfaces through Allen–Cahn and Cahn–Hilliard dynamics, sharp-interface limits, and adaptive numerical approximations (Du et al., 2019). Sato et al. present a geometric VOF method with PLIC, conservative thermal-energy advection, and a novel two-step VOF advection scheme for sharp-interface phase change; the Stefan problem errors are $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$42, $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$43, and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$44 on $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$45 cells, and the 3D bubble final-radius errors are $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$46, $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$47, and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$48 on $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$49 grids (Malan et al., 2020). SimPLIC combines PLIC and Simpson’s rule on arbitrary polyhedral meshes, with mass conservation at machine zero $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$50 and near-second-order convergence in several reconstruction and advection tests (Dai et al., 2024). The unstructured sharp-interface VOF method of 2026 combines algebraic VOF, geometric reconstruction, interfacial heat-flux evaluation, and interface-modified least squares; on polyhedral Scriven benchmarks the relative radius error drops from approximately $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$51 to approximately $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$52 between $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$53 and $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$54 meshes, while Cartesian meshes exhibit coherent four-fold anisotropy (Kren et al., 16 Apr 2026). In fully Eulerian FSI, the interface-and-geometry preserving method uses a gradient-minimizing velocity to reduce curvature flow, keeps interface thickness constant to within $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$55–$$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$56, and maintains volume conservation below $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$57 (Mao et al., 2022). The nonlocal diffusion-bonding phase-field model replaces Allen–Cahn descent with a geometric conservation law and a coalescence switch $$\phi_g=\mathrm{Arg}\,[\langle \psi_1|\psi_2\rangle \langle \psi_2|\psi_3\rangle \cdots \langle \psi_n|\psi_1\rangle],$$58 built from higher-derivative curvature invariants, thereby arresting interface merging under calibrated conditions (Khodadad et al., 18 Feb 2026).

This suggests two distinct meanings must be separated in advanced usage. One meaning concerns a phase angle produced by geometry at an interface; the other concerns the geometry of an interface separating material phases. The former is central to GPA artifacts, Berry/Zak physics, Josephson transport, valley skew tunneling, modal conversion, and closure phase. The latter is central to geometric reconstruction, interface preservation, and phase-change computation. The two literatures share the words “geometric,” “interface,” and “phase,” but they do not share the same observable.