Pancharatnam-Zak Phase in 1D Systems

• Pancharatnam-Zak phase is defined as the geometric phase in 1D band transport, incorporating an endpoint overlap term to correct gauge and boundary sensitivities.
• Its formulation combines the Zak integral with Pancharatnam’s endpoint contribution, leading to quantized values in symmetric systems and improved topological classification.
• Applied in photonic, acoustic, and condensed-matter systems, this phase aids in experimental design by revealing transport properties and boundary-dependent effects.

Searching arXiv for recent and foundational papers on the Pancharatnam-Zak phase. arXiv search query: "Pancharatnam-Zak phase Zak phase geometric phase 1D periodic systems" The Pancharatnam-Zak phase is the geometric phase associated with one-dimensional band transport when the physically relevant cycle is treated in the Pancharatnam sense—through the return of physical observables or rays in projective Hilbert space—rather than through strict recurrence of the state vector. In this formulation, the familiar Zak integral over the Brillouin zone is supplemented by an endpoint overlap term, yielding a phase that Vyas and Roy identify as free of the gauge and origin flaws of the original Zak expression (Vyas et al., 2019). In parallel, recent work has shown that the same band phase admits a real-space representation in terms of Weyl $m$-functions for periodic Jacobi operators, making its boundary sensitivity explicit and avoiding Floquet-Bloch theory (Ammari et al., 21 Jan 2026).

1. Conceptual lineage

Pancharatnam’s phase is the geometric phase between two non-orthogonal quantum states, given in polarization optics by

$\Phi_P=\arg\langle i|f\rangle .$

Loredo et al. studied this phase for arbitrary $SU(2)$ transformations on polarization states of light and implemented both interferometric and polarimetric measurements over the full Poincaré sphere (Loredo et al., 2015). In that setting, the phase is inherently geometric: it depends on the path of the polarization state on the Poincaré sphere rather than on propagation time.

The Zak phase is the one-dimensional Bloch-band specialization of Berry’s phase. In its standard form, for a band eigenstate $|u_n(k)\rangle$,

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$

This quantity is central to the modern theory of polarization and to the classification of topological phases in one-dimensional crystalline systems (Sergeev, 2018). It is also the phase that underlies the SSH distinction between trivial and non-trivial dimerizations in photonic, acoustic, and condensed-matter realizations (Longhi, 2013).

The Pancharatnam-Zak phase emerges from the observation that adiabatic transport across the Brillouin zone is not cyclic in the naive sense assumed in the original Zak construction. Vyas and Roy argue that the relevant cyclicity is instead a generalized cyclicity in which physical observables return although the state may return only up to a large gauge transformation (Vyas et al., 2019). This places the one-dimensional band phase in the same geometric family as open-path Pancharatnam phases and clarifies why a boundary term is required.

2. Mathematical formulations

The corrected single-band Pancharatnam-Zak phase introduced by Vyas and Roy is

$$\gamma_g(n)=\arg\!\left[\langle u_n(0)|u_n(2\pi/a)\rangle\right] +i\int_0^{2\pi/a}\langle u_n(\kappa)|\partial_\kappa u_n(\kappa)\rangle\,d\kappa .$$

The first term is a Pancharatnam endpoint contribution; the second is the usual Berry-connection integral. The construction can also be written as a discretized Pancharatnam chain,

$$\gamma_g(n)=\lim_{N\to\infty}\arg\!\Big( \langle u_{n,0}|u_{n,N}\rangle \langle u_{n,N}|u_{n,N-1}\rangle\cdots \langle u_{n,1}|u_{n,0}\rangle \Big),$$

which makes its gauge invariance manifest (Vyas et al., 2019).

A geometrically related viewpoint is provided by the theory of projected connections. There the Zak phase is interpreted as the holonomy of a projected connection on the Bloch bundle, and the projected Berry potential depends on the basis used to represent Bloch states. For two bases related by a unitary $U$,

$$\langle\psi|\partial_k^{(2)}\psi\rangle = \langle\psi|\partial_k^{(1)}\psi\rangle + \langle\psi|U^{-1}(\partial_k U)|\psi\rangle ,$$

so basis changes contribute explicitly to the connection and therefore to polarization-related observables (Sergeev, 2018).

A distinct formulation was established for one-dimensional periodic Jacobi operators by expressing the Zak phase through the Weyl $m_+$-function of a half-line operator: $\Phi_P=\arg\langle i|f\rangle .$0 The associated Berry connection is

$\Phi_P=\arg\langle i|f\rangle .$1

This formula dispenses with $\Phi_P=\arg\langle i|f\rangle .$2-space machinery and links the phase directly to boundary values of the resolvent, i.e. to a boundary Green function (Ammari et al., 21 Jan 2026).

3. Gauge, basis, unit cell, and boundary dependence

A central issue in the subject is that the conventional Zak phase is not an absolute invariant. In photonic SSH lattices, the value shifts under a translation of the lattice origin, while the Zak-phase difference between topologically distinct dimerizations remains physically meaningful (Longhi, 2013). In the projected-connection framework, this sensitivity is not an accident but reflects the dependence of the Berry potential on the chosen Bloch basis and on the orbital arrangement inside the unit cell (Sergeev, 2018).

The Pancharatnam-Zak construction addresses this by closing the path in ray space rather than forcing a periodic gauge. The endpoint overlap

$\Phi_P=\arg\langle i|f\rangle .$3

is exactly the term required to compensate the gauge and origin dependence of the bare integral (Vyas et al., 2019). In this sense, the Pancharatnam-Zak phase is not merely a renamed Zak phase; it is a corrected geometric phase for Bloch transport.

Boundary sensitivity remains fundamental even in the corrected picture. In the Weyl-$\Phi_P=\arg\langle i|f\rangle .$4 formulation, the definition of $\Phi_P=\arg\langle i|f\rangle .$5 depends on the boundary site $\Phi_P=\arg\langle i|f\rangle .$6, and different choices shift the Zak phase by integer multiples of $\Phi_P=\arg\langle i|f\rangle .$7 (Ammari et al., 21 Jan 2026). Likewise, in the generalized Dirac-Kronig-Penney model, translating the unit-cell edge by $\Phi_P=\arg\langle i|f\rangle .$8 changes the band Zak phase according to

$\Phi_P=\arg\langle i|f\rangle .$9

while the relative Zak phase between two phases is independent of this choice (Angelone et al., 3 Feb 2026). A common misconception is therefore that the Zak or Pancharatnam-Zak phase is a strictly bulk quantity with no boundary convention dependence. The literature instead indicates that bulk geometry and boundary convention are inseparable in one dimension.

4. Quantization, symmetry, and topological status

In inversion-symmetric or mirror-symmetric settings, the phase can become quantized. For periodic Jacobi operators with an inversion symmetric fundamental cell, the Weyl function satisfies

$SU(2)$0

along a band, so that writing $SU(2)$1 gives

$SU(2)$2

Since $SU(2)$3 becomes real at band edges, $SU(2)$4, and the phase is quantized in integer multiples of $SU(2)$5 (Ammari et al., 21 Jan 2026). In photonic crystals with mirror-symmetric unit cells, the same $SU(2)$6 structure can be inferred from band-edge parity or from the presence or absence of interface states in the two adjacent gaps (Wang et al., 2016).

The topological content of the phase depends strongly on symmetry class. In one-dimensional translation-invariant topological insulators with Altland-Zirnbauer-Cartan symmetries, a $SU(2)$7-valued invariant can be extracted from the abelian Zak phase using symmetric Bloch bases,

$SU(2)$8

In classes such as AIII, BDI, and D, this parity can be basis-independent under the appropriate symmetry constraints, but if the occupied space carries a quaternionic structure generated by an anti-unitary symmetry squaring to $SU(2)$9, the same work shows that the corresponding $|u_n(k)\rangle$0 invariant necessarily vanishes (Manzoni et al., 16 Mar 2026).

Continuum models sharpen the limitations of the phase as a topological marker. In the generalized Dirac-Kronig-Penney model, the Zak phase is quantized in classes AIII and BDI but becomes non-quantized in class D, even though class D admits non-trivial one-dimensional topology in K-theoretic classification (Angelone et al., 3 Feb 2026). This challenges the traditional assumption that a single-band Zak phase uniformly diagnoses one-dimensional topology. The evidence suggests a narrower statement: the Pancharatnam-Zak phase is a useful topological indicator when the symmetry and geometric structure support its quantization, but not in every one-dimensional setting.

5. Measurement and physical realizations

Direct experimental access to the phase has been demonstrated in several platforms. In an electroacoustic cavity system, Loredo-style interferometric logic is adapted to adiabatic bulk evolution: momentum $|u_n(k)\rangle$1 is mapped to a time-dependent parameter $|u_n(k)\rangle$2, the SSH Hamiltonian is represented by a real-valued four-state Hamiltonian, and the geometric phase is extracted by comparing final states evolved along two symmetric paths whose dynamical phases cancel (He et al., 27 May 2025). In that experiment, the accumulated phase is explicitly described as a Pancharatnam-type geometric phase, termed the Pancharatnam-Zak phase when evaluated over an adiabatic cycle in the Brillouin zone. The reported phase accumulation is near $|u_n(k)\rangle$3, $|u_n(k)\rangle$4, and $|u_n(k)\rangle$5 for winding numbers $|u_n(k)\rangle$6, $|u_n(k)\rangle$7, and $|u_n(k)\rangle$8, respectively (He et al., 27 May 2025).

In optical waveguide lattices, the phase difference between topologically distinct SSH dimerizations is converted into a measurable $|u_n(k)\rangle$9 output shift after one Bloch oscillation. The proposed detection scheme uses two vertically displaced arrays and a three-port coupler: if the two outputs are in phase the central port is bright, whereas a $$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$0 phase difference keeps it dark (Longhi, 2013). This realizes the geometric phase as an interferometric observable of photonic band transport.

A different strategy uses bulk-boundary correspondence rather than direct phase interferometry. In a metasurface/photonic-crystal system, the Zak phase of a band is inferred from whether interface states occur in the two gaps adjacent to that band. The interface-state condition is

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$1

and sharp dips in reflection spectra determine whether the condition is met in a given gap (Wang et al., 2016). This method exploits the relation between gap topology, reflection phase, and interface-state existence.

Locally resonant metamaterials exhibit yet another route. In dimerized one-dimensional metamaterials, the contribution to the Zak phase is concentrated at singular points in the bulk band where the phase difference jumps by $$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$2. The singular points arise either from dimerization-independent anti-resonance,

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$3

or from dimerization-dependent destructive interference,

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$4

Changing the dimerization parameter shifts these singular points, induces band inversion, changes the Zak phase, and produces experimentally observable interface states (Zhu et al., 2018).

6. Extensions, open-system variants, and current limits

The phase has been generalized beyond Hermitian, static, and strictly continuous settings. Zhang and Song studied non-Hermitian bipartite lattices using biorthogonal left and right Bloch states and defined a complex Zak phase

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$5

For staggered imaginary potentials in a non-Hermitian SSH model, they found that the real part of the Zak phase remains unchanged while the imaginary part encodes amplification or attenuation (Zhang et al., 2018). In this usage, the Pancharatnam-Zak phase is effectively the complex geometric phase accumulated by cyclic adiabatic transport of biorthogonal states.

A further extension appears in measurement-driven and Floquet many-body dynamics. In generalized radical Floquet dynamics, the Pancharatnam phase is defined for a discrete sequence of states

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$6

and is proposed as a natural invariant for monitored trajectories (Roberts et al., 2023). For the $$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$7 Floquet code of Hastings and Haah, the associated family of unitary evolutions is identified with the radical chiral Floquet phase, and the quantized Pancharatnam phase can be extracted by a computationally-assisted interferometry protocol (Roberts et al., 2023). This suggests a discrete-sequence generalization of the same geometric logic that underlies the band-theoretic Pancharatnam-Zak phase.

The phase has also been tied to transport and device functionality. In room-temperature atomic SSH superradiance lattices, opposite probe directions perceive different SSH topological phases, and the interplay between the Zak phase and thermal motion produces direction-dependent absorption. The Zak-phase-induced energy shift,

$$\gamma_Z(n)= i\int_{\mathrm{BZ}} \langle u_n(k)|\partial_k u_n(k)\rangle\,dk .$$8

contributes directly to optical nonreciprocity (Liu et al., 2024). This suggests that in some driven or momentum-space platforms the Pancharatnam-Zak geometry is not only diagnostic but operational.

Two broad cautions recur across the literature. First, the phase is not universally gauge-immune unless the Pancharatnam closure or an equivalent geometric correction is included (Vyas et al., 2019). Second, even the corrected phase is not a universal classifier of one-dimensional topology: continuum class-D models and quaternionic symmetry classes exhibit explicit obstructions to such a reading (Angelone et al., 3 Feb 2026, Manzoni et al., 16 Mar 2026). What remains robust is the conceptual role of the Pancharatnam-Zak phase as the geometric phase appropriate to one-dimensional band transport once open-path closure, basis dependence, and boundary convention are handled explicitly.