Vectored Zak Phases in Topological Systems

• Vectored Zak phases are geometric phase vectors derived from Berry connection integrals across the Brillouin zone in higher-dimensional systems.
• They exhibit distinct topological winding and phase singularities, such as quantized vortex charges, which directly correlate with robust edge states.
• Experimental implementations in photonic lattices and discrete-time quantum walks validate their role in engineering topologically protected states even with vanishing Berry curvature.

Vectored Zak phases generalize the concept of the Zak phase—originally defined for one-dimensional periodic systems as a geometric phase acquired along the Brillouin zone—to higher-dimensional and multi-parameter systems. In these systems, the Zak phase acquires a vectorial character, encoding multicomponent geometric phase information with distinct topological and physical consequences for quantum and classical wave dynamics. This framework is fundamental in describing the band topology of photonic, electronic, and cold-atom systems, particularly when characterized by Berry connection structures that remain nontrivial even in the absence of Berry curvature.

1. Mathematical Definition and Physical Context

In a two-dimensional periodic system, such as a crystal or a discrete-time quantum walk (DTQW) protocol, the vectored Zak phase is defined as a two-component vector

${\boldsymbol{\gamma}} = (\gamma_x, \gamma_y),$

where each component is given by integration of the Berry connection $\mathbf{A}(\mathbf{k})$ along a direction in reciprocal space:

$$\gamma_i = i \int_{BZ} \langle u_\mathbf{k} | \partial_{k_i} u_\mathbf{k} \rangle\, dk_j,$$

with $i,j \in \{x, y\}$ and the path taken along $k_i$ at fixed $k_j$. Equivalently,

$${\boldsymbol{\gamma}} = \left( i \int dk_y \langle u_\mathbf{k} | \partial_{k_x} u_\mathbf{k} \rangle,\quad i \int dk_x \langle u_\mathbf{k} | \partial_{k_y} u_\mathbf{k} \rangle \right).$$

Here, $|u_\mathbf{k}\rangle$ are the cell-periodic Bloch eigenstates, and the integrals extend over the Brillouin zone (Puentes, 2023).

In systems with multiple tunable parameters (e.g., trimer lattices with two independent hopping amplitudes), the Zak phase forms a scalar field $\gamma_\mu(\lambda_1, \lambda_2)$ over the parameter plane, but topological features such as dislocations and vortices in the phase field endow the notion of a vectorial winding structure (Uakhitov et al., 19 Sep 2025).

2. Topological Structure and Phase Singularities

The parameter-space landscape of the Zak phase is marked by quantized screw-type dislocations or vortices, which occur at points of band degeneracy (gap-closing points). In the off-diagonal trimer lattice, the Zak phase $\gamma_\mu(\lambda_1, \lambda_2)$ for band $\mu = 1,2,3$ exhibits such singularities at the monatomic degeneracy $\lambda_1 = \lambda_2 = 1$, corresponding to Dirac-cone points ($q=0, \pi$). These points act as sources of phase winding:

$$l_\mu = \frac{1}{2\pi} \oint_C d\gamma_\mu \in \mathbb{Z},$$

where $C$ is a loop in parameter space around the degeneracy. In concrete calculations:

• $l_1=+1$
• $l_2=-2$
• $l_3=+1$

This quantized winding directly underpins the topological properties and the robustness of edge states (Uakhitov et al., 19 Sep 2025).

3. Vectored Zak Phases in Discrete-Time Quantum Walks

Photonic DTQW protocols provide a versatile platform to realize and probe vectored Zak phases. Three classes—Hadamard quantum walk (HQW), non-commuting rotations quantum walk (NCRQW), and split-step quantum walk (SSQW)—realize nontrivial Zak phase structures under spatial inversion symmetry (SIS) and time-reversal symmetry (TRS) (Puentes, 2023).

In these models, the step operator $U$ is constructed from coin rotations $R$ and spin-dependent shifts $T$, leading to an effective Hamiltonian $$H(\mathbf{k}) = E(\mathbf{k}) \,\mathbf{n}(\mathbf{k}) \cdot \boldsymbol{\sigma}$$ such that $U = \exp(-iH)$. Separable step operators $U_x \otimes U_y$ ensure SIS and TRS, enforcing identically vanishing Berry curvature $F = \nabla_\mathbf{k} \times \mathbf{A} = 0$, but leaving nonzero Berry connection and allowing intricate vectored Zak phase structure.

Key results include:

• HQW: $\gamma_x$ and $\gamma_y$ are functions of the coin angle $\theta$ and coincide when SIS and TRS hold.
• NCRQW: $\gamma_x(\theta, \phi)$ and $\gamma_y(\theta, \phi)$ display multi-lobed landscapes, with singularities at Dirac points (gap closings).
• SSQW: For special parameter choices ($n_3 = 0$, e.g., $\theta_2=0$), the analytic result $$\gamma_x = \gamma_y = \tan \theta_2 / \tan \theta_1$$ is obtained.

The table below summarizes the main analytic or numerical results for these protocols:

Protocol	Analytic Zak Phase	Structure
HQW	Function of $\theta$	$\gamma_x = \gamma_y$ under SIS+TRS
NCRQW	Numerical $(\theta,\phi)$	Singular at Dirac points
SSQW ($n_3=0$)	$\tan \theta_2/\tan \theta_1$	$\gamma_x = \gamma_y$

4. Manipulation of Symmetry and Applications

Breaking time-reversal symmetry while retaining vanishing Berry curvature produces antisymmetric vectored Zak phases $(\gamma_x, \gamma_y) = (+\gamma, -\gamma)$. For example, in the SSQW, imposing $\theta_1^x = -\theta_1^y$ flips the sign of the Zak phase in one direction. This kind of manipulation enables experimental control over geometric phases without inducing gauge-invariant field strengths, revealing a direct Aharonov–Bohm-type analogy: in a field-free (curvature-free) but multiply-connected parameter region, global phase evolution is determined by the Berry connection (vector potential), not just the local fields (Puentes, 2023).

A plausible implication is that such symmetry breaking schemes offer a way to engineer topological transport and localization effects even in systems with trivial Chern numbers, through the phase structure encoded in the Zak vector.

5. Bulk–Edge Correspondence via Zak Vortices

The connection between bulk vectored Zak phase winding and edge-state structure is explicit in trimer chains. As the parameters $(\lambda_1, \lambda_2)$ are varied along a closed pump trajectory, the total Zak-phase winding (sum of vortex charges) enclosed by the pump cycle in parameter space determines the band Chern number $C_\mu$ via

$$C_\mu = -\frac{1}{2\pi} \oint_{\text{cycle}} d\gamma_\mu(\lambda_1(\tau), \lambda_2(\tau))$$

(Uakhitov et al., 19 Sep 2025). In a finite system, the sequence of edge-state bifurcations (where edge states peel off or merge into the bulk) across parameter-space "bifurcation lines" matches the bulk winding: each "left-right" followed by "right-left" transition sequence accounts for $\pm1$ in $C_1$ of the bottom band. This demonstrates a bulk–edge correspondence strictly mediated by the topology of the vectored Zak phase field and its singularities.

6. Experimental Realizations and Observable Signatures

Numerical and analytic computations of Zak-phase landscapes, both in trimer lattices and 2D DTQWs, have yielded contour plots, 3D phase maps, and explicit formulas for experimental design and verification. In photonic systems, the direct measurement of nonvanishing vectored Zak phases despite zero Berry curvature provides clear signatures of geometric phase control and topological protection in field-free regimes (Puentes, 2023). Edge-state observations in finite lattices track the predicted sequence of bifurcations dictated by bulk Zak-vortex structure (Uakhitov et al., 19 Sep 2025).

7. Theoretical Significance and Future Directions

Vectored Zak phases clarify the structure of geometric phases in higher dimensions and multi-parameter spaces, bridging band topology, quantum walks, and modern geometric phase theory. The identification of topological invariants via winding of Zak phase dislocations, their equivalence to Chern numbers under adiabatic pumping cycles, and the observed exact correspondence to edge-state phenomena collectively establish vectored Zak phases as central topological markers, especially in systems where the Berry curvature vanishes but the Berry connection remains nontrivial.

A plausible implication is that future work will further explore the use of vectored Zak phase engineering for robust, tunable topological states in synthetic and real materials, including quantum simulation platforms and photonic lattices, guided by the analytic and numerical frameworks constructed in these studies (Uakhitov et al., 19 Sep 2025, Puentes, 2023).