Extended Zak Phase in 1D Lattices

• Extended Zak phase is a generalized geometric phase in one-dimensional periodic lattices that integrates bulk-boundary interplay and symmetry constraints.
• It employs multiple formulations, including real-space (Weyl m-function) and inter-cellular approaches to quantify phase contributions and predict edge phenomena.
• The theory extends to non-Hermitian, dynamically driven, and many-body regimes, enabling precise control over polarization and topological invariants.

The extended Zak phase generalizes the geometric phase acquired by a quantum state in a one-dimensional periodic lattice, systematically embedding bulk-boundary interplay, symmetry constraints, real-space representations, and extensions to non-Hermitian, interacting, or dynamically driven settings. Originally defined as the Berry phase over the Brillouin zone of a Bloch band, the Zak phase underlies polarization, the existence and character of boundary modes, and topological classification of 1D crystalline systems. Recent theoretical and experimental work has developed multiple formulations—real-space, gauge-invariant, inter-cellular, partial, and many-body—that reveal the full subtleties of unit-cell dependence, symmetry quantization, non-adiabaticity, and multi-band/topological structure.

1. Real-Space Formulation and the Weyl m-Function

A central development is the explicit real-space representation of the Zak phase for periodic Jacobi operators via the Weyl $m_+$-function, circumventing Floquet-Bloch theory (Ammari et al., 21 Jan 2026). For a $p$-periodic Jacobi operator on $\ell^2(\mathbb{Z})$

$$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$

the half-line Weyl function

$$m_+(\lambda) = \left\langle (\mathcal J_+ - \lambda)^{-1} e_1,\,e_1 \right\rangle$$

admits a real-space integral formula for the Zak phase of the $n$th isolated band: $$\gamma_n = \int_{\lambda_{n,\min}}^{\lambda_{n,\max}} \frac{\Re[m_+'(\lambda+i0)]}{\Im[m_+(\lambda+i0)]} d\lambda - \pi.$$ Here, $\lambda_{n,\min}$, $\lambda_{n,\max}$ are the band edges. The integrand quantifies a spectral phase-twist, while the $-\pi$ term encodes explicit boundary (unit-cell) dependence. This real-space formula elucidates the boundary-localized contributions that are only implicit in momentum-space Berry phase descriptions and directly ties the extended Zak phase to half-line impedance/resolvent data, providing computational and conceptual advantages for systems where Bloch theory is fragile or absent.

2. Bulk-Boundary Structure: Boundary Terms and Origin Dependence

Unlike topological invariants in higher dimensions, the Zak phase is not purely a bulk invariant; its value depends explicitly on the choice of real-space origin and unit cell. In the $p$0-function framework, shifting the fundamental cell alters the boundary contribution ($p$1 shift) without affecting the bulk spectral integral. The distinction between unit-cell–dependent and origin-independent components is made precise in the decomposition (Martí-Sabaté et al., 2021, Rhim et al., 2016): $p$2 where $p$3 accounts for the real-space origin and is quantized under inversion symmetry, while $p$4, derived as an internal “phase delay” between Fourier components, vanishes for inversion-symmetric systems and can be non-zero and continuous otherwise. The inter-cellular (origin-independent) Zak phase governs bulk-boundary correspondence robustly, predicting accumulated boundary charges and the parity of in-gap boundary modes (Rhim et al., 2016). Only differences in Zak phase between two regions or two parameter points have unambiguous physical implications for edge-state formation and fractionalization.

3. Symmetry, Quantization, and Topological Classification

In systems possessing discrete symmetries—particularly inversion, chiral, or combined time-reversal—quantization of the extended Zak phase (to $p$5 or $p$6 modulo $p$7) emerges naturally. For inversion-symmetric Jacobi operators, the modulus of $p$8 is constant on each band, and the Zak phase reduces to the difference of the boundary phases at band edges, strictly quantized (Ammari et al., 21 Jan 2026). This quantization is the origin of the $p$9 invariant that distinguishes topologically trivial and nontrivial 1D insulators (Manzoni et al., 16 Mar 2026, Rhim et al., 2016). In multi-band settings and within the Altland-Zirnbauer symmetry classification, the extended (mod $\ell^2(\mathbb{Z})$0) Zak invariant coincides with the target $\ell^2(\mathbb{Z})$1-theoretic invariant, but vanishes in classes with quaternionic (anti-unitary, $\ell^2(\mathbb{Z})$2) structure due to paired holonomy eigenvalues (Manzoni et al., 16 Mar 2026).

Formally, for a symmetric Bloch frame $\ell^2(\mathbb{Z})$3 spanning the negative-energy (occupied) bands,

$\ell^2(\mathbb{Z})$4

defines the $\ell^2(\mathbb{Z})$5 Zak invariant protected by symmetry class.

4. Origin-Independent and Partial/Inter-Cellular Zak Phase

The extended, origin-independent (inter-cellular) Zak phase, $\ell^2(\mathbb{Z})$6, is constructed by extracting from the conventional Berry-phase integral the contribution that is invariant under unit-cell shifts (Rhim et al., 2016, Martí-Sabaté et al., 2021). In tight-binding notation, this component is expressed in terms of the $\ell^2(\mathbb{Z})$7-derivatives of the tight-binding coefficients, isolated from the position-dependent intra-cellular dipole moment: $\ell^2(\mathbb{Z})$8 The surface (boundary) charge accumulation and the number of in-gap edge states are directly given by

$\ell^2(\mathbb{Z})$9

ensuring that the physical manifestation of topological invariance is codified in $$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$0, not in the entirety of the convention-dependent Zak phase. In non-Hermitian bipartite systems, the real part of the Zak phase coincides with the Hermitian limit and thus retains its topological role under adiabatic flux threading, while the imaginary part governs dynamical gain/loss phenomena (Zhang et al., 2018).

5. Extensions: Non-Adiabatic, Non-Hermitian, and Many-Body Regimes

For time-dependent or driven systems where adiabaticity is broken, the extended Zak phase admits non-adiabatic corrections associated with Landau-Zener transitions and Stokes phases, as established in the Rice-Mele model under finite-frequency modulation (Kuno, 2018). The non-adiabatic constructive formula replaces the occupation factor in the Berry connection integral with the instantaneously-evolved lower-band population,

$$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$1

where $$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$2 encapsulates Landau-Zener population leakage. In the adiabatic limit, this protocol recovers the usual quantized charge pumping; deviations signal the breakdown of topological quantization.

In interacting systems, a many-body extension arises via twisted boundary conditions and computation of Berry phases in the ground-state as a function of imposed boundary flux. The expectation value of a “twist operator” yields a quantized (mod $$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$3) many-body Zak phase per spin sector, confirmed even in strongly correlated Mott-insulating phases where spin-polarized edge excitations are observed (Misawa et al., 2023).

6. Gauge and Origin-Invariant Reformulations: Pancharatnam-Zak Phase

The foundational issue of gauge and origin dependence in the geometric phase is resolved in the Pancharatnam–Zak phase formalism (Vyas et al., 2019). Here, the phase accumulates not through a line integral of the Berry connection but as the argument of a cyclic product of non-orthogonal inner products across infinitesimal Brillouin zone increments,

$$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$4

This formulation is rigorously invariant under smooth gauge transformations and arbitrary unit-cell shifts, endowing the extended Zak phase with the status of a true bulk geometric invariant. The filled-band extension directly controls electrical polarization and topological indices, with quantization protected by inversion symmetry.

7. Experimental Access and Applications

Direct measurements of the extended Zak phase have been realized in ultracold atom interferometry (Atala et al., 2012), electroacoustic cavity systems (He et al., 27 May 2025), and photonic lattices (Jiao et al., 2021). Experimental protocols exploit adiabatic (or nearly adiabatic) traversals of parameter cycles coupled with Ramsey or interferometric readout to extract phase accumulation differing between topological sectors. Notably, in electroacoustic cavities, higher winding numbers and extended model parameters (e.g., next-nearest-neighbor couplings) are accessible by engineered Hamiltonian evolution and phase-difference measurement, enabling quantized and robust retrieval of the Zak phase and its generalizations.

The extended Zak phase further underlies bulk-boundary phenomena in planar magnonic crystals (Mieszczak et al., 2021), non-chiral yet inversion-symmetric photonic systems (Jiao et al., 2021), and dynamical confinement in non-Hermitian devices (Zhang et al., 2018). The boundary signature is encoded in the difference of extended Zak phases across interfaces, with precise prediction (and tuning) of boundary or interface mode count and localization.

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Summary Table: Key Features Across Selected Extended Zak Phase Formulations

Formulation	Core Invariant	Physical Role
Real-space (Weyl $$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$5-function) (Ammari et al., 21 Jan 2026)	Spectral integral + explicit boundary term	Bulk-boundary decomposition, real-space computation
Inter-cellular/partial (Rhim et al., 2016, Martí-Sabaté et al., 2021, Zhang et al., 2018)	Origin-independent Berry/wavefunction term	Boundary charge, number of edge modes
Pancharatnam–Zak (Vyas et al., 2019)	Cyclic product of inner products	Gauge and origin independence, true geometric phase
Non-adiabatic extension (Kuno, 2018)	Weighted Berry phase (Landau-Zener)	Breakdown of quantized pumping, dynamical corrections
Many-body extension (Misawa et al., 2023)	Berry phase under boundary twist	Topology in correlated insulators
Multi-band symmetry-adapted (Manzoni et al., 16 Mar 2026)	$$(\mathcal J\phi)(n) = a_{n-1}\phi(n-1) + b_n\phi(n) + a_n\phi(n+1), \qquad a_{n+p}=a_n,\,b_{n+p}=b_n,$$6 parity of extended Zak phase	AZC-class classification, vanishing with quaternionic structure

The extended Zak phase thus unifies real-space and bulk geometric approaches, enables robust classification and prediction of topological and edge phenomena across model generalizations, and provides a versatile experimental and theoretical tool for driven, non-Hermitian, interacting, and multi-band systems.