Zak Phase Dislocations in Topological Lattices

• Zak phase dislocations are singular, vortex-like structures in the Zak phase field across multidimensional lattice parameter spaces that exhibit quantized winding near degeneracies.
• In the off-diagonal trimer lattice, these screw-type dislocations manifest through quantized winding numbers that directly determine Chern numbers and govern edge-state behavior.
• This framework offers a computational tool for mapping topological invariants onto geometric singularities, elucidating Thouless pumping and experimental photonic transport.

Zak phase dislocations are singular, vortex-like structures of the Zak phase $Z_\mu(\lambda)$ when the Zak phase is regarded not as a number attached to a single one-dimensional Bloch band at a fixed parameter point, but as a field over a multidimensional parameter space of lattices. In this formulation, band degeneracies act as singular anchors, and the Zak phase develops quantized winding around them. The clearest explicit realization is the off-diagonal trimer lattice, where the Zak phase carries quantized screw-type dislocations in hopping-parameter space, and the Chern number of an adiabatic Thouless pump equals the negative total winding number of the enclosed Zak-phase dislocations (Uakhitov et al., 19 Sep 2025). Related work treats analogous singularities at Dirac points in discrete-time quantum walks, phase-jump singular points in dimerized metamaterials, and Berry-connection holonomy on multidimensional Zak-phase landscapes (Puentes et al., 2015, Zhu et al., 2018, Puentes, 2023).

1. Definition and general meaning

For a one-dimensional Bloch band $\mu$, the Zak phase is the Berry phase accumulated across the Brillouin zone,

$$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$

where $\lambda$ denotes a point in parameter space, such as a set of hopping amplitudes (Uakhitov et al., 19 Sep 2025). In the usual fixed-parameter viewpoint, $Z_\mu$ is a band quantity. In the dislocation viewpoint, $Z_\mu$ is promoted to a function on parameter space, and its singular behavior near degeneracies becomes the relevant topological object.

In the trimer-lattice formulation, a closed loop $C$ in parameter space can produce a net change

$$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$

with integer $l_\mu$. This integer is the winding number, or topological charge, of the Zak-phase dislocation (Uakhitov et al., 19 Sep 2025). The phrase “screw-type dislocation” refers to the helical behavior of $Z_\mu$ around a degeneracy axis: as the cyclic parameter advances by $\mu$0, the Zak phase changes by $\mu$1, so the phase field is globally multivalued and must be described with branch cuts.

This usage differs from the more familiar one-dimensional setting in which the Zak phase is quantized at a fixed parameter point by inversion or chiral symmetry. In SSH-type systems the fixed-point Zak phase can be $\mu$2 or $\mu$3, but in the trimer problem the nontrivial topology resides in the winding of $\mu$4 over a family of lattices, not in its value at a single point (Uakhitov et al., 19 Sep 2025).

2. Off-diagonal trimer lattice as the canonical realization

The explicit construction in the trimer case starts from a one-dimensional lattice with three equivalent sites per unit cell, $\mu$5, nearest-neighbor hopping only, and no on-site potentials. In real space the evolution equations are

$\mu$6

with $\mu$7 the intra-cell $\mu$8 hopping, $\mu$9 the intra-cell $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$0 hopping, and $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$1 the inter-cell $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$2 hopping (Uakhitov et al., 19 Sep 2025). In the photonic realization, the propagation coordinate $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$3 plays the role of time.

The Bloch Hamiltonian is

$$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$4

and the three bands satisfy

$$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$5

Because the model is homogeneous in the overall scale, one can fix $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$6 and regard the family of lattices as living on the sphere $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$7. On that sphere, the dispersion depends only on the product $$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$8, while motion along constant-$$Z_\mu(\lambda)=\int_{-\pi}^{\pi}A_{\mu q}(q,\lambda)\,dq,\qquad A_{\mu q}=i\langle \psi_\mu(q,\lambda)\mid \partial_q\psi_\mu(q,\lambda)\rangle,$$9 contours changes the eigenvectors but not the energies (Uakhitov et al., 19 Sep 2025).

For the explicit eigenvectors of this model, the Berry connection becomes

$\lambda$0

Two consequences are immediate. First, $\lambda$1 when $\lambda$2, i.e. on the inversion-symmetric trimer line. Second, the Zak phase itself vanishes there: $\lambda$3 This is a decisive departure from SSH intuition. In the trimer lattice, inversion symmetry forces $\lambda$4 even though edge states may still be present when $\lambda$5 (Uakhitov et al., 19 Sep 2025). A common misconception is therefore that a static Zak phase at a fixed parameter point must remain the primary topological marker. In this model it does not.

The convenient cyclic coordinate is an angle $\lambda$6 defined through rotated variables

$\lambda$7

with $\lambda$8 expressed as linear combinations of the $\lambda$9. Along a constant-$Z_\mu$0 contour, $Z_\mu$1 runs from $Z_\mu$2 to $Z_\mu$3, and $Z_\mu$4 develops the dislocation structure (Uakhitov et al., 19 Sep 2025).

3. Screw-type dislocations and their topological charges

The organizing singularity in the trimer problem is the degeneracy axis $Z_\mu$5, where the three bands collapse into a monatomic band and the gaps close. Near this axis, the Zak phases twist with quantized winding: $Z_\mu$6 Equivalently, for a loop $Z_\mu$7 encircling the degeneracy,

$Z_\mu$8

and $Z_\mu$9 is the dislocation charge of band $Z_\mu$0 (Uakhitov et al., 19 Sep 2025).

Geometrically, the dislocation is screw-like because $Z_\mu$1 behaves as a helical phase around the degeneracy axis. The lower and upper bands each carry unit positive winding, while the middle band carries double winding with opposite sign. The sign changes for opposite sign of $Z_\mu$2, so the parameter space contains both positive and negative dislocations (Uakhitov et al., 19 Sep 2025).

This structure is tied to Dirac-like cones in the dispersion. In the trimer lattice, the degeneracy line corresponds to Dirac-like cones at specific momenta: band 1 has a cone at $Z_\mu$3, band 3 has a cone at $Z_\mu$4, and band 2 touches both cones and is threaded by a “Berry flux loop.” The doubled winding of the middle band reflects how Berry curvature is routed through these cones (Uakhitov et al., 19 Sep 2025).

Related formulations in other one-dimensional systems treat the same underlying phenomenon in different language. In discrete-time quantum walks, the Zak phase is undefined at Dirac points, and phase differences between trajectories ending on opposite quasi-momentum branches are quantized to $Z_\mu$5, while same-branch differences vanish (Puentes et al., 2015). In dimerized locally resonant metamaterials, the Berry phase is accumulated almost entirely at singular points in the bulk band where the reflection vanishes; each such singular point contributes $Z_\mu$6 and acts as a localized phase jump (Zhu et al., 2018). In two-dimensional parameter landscapes of photonic quantum walks, Berry curvature can vanish while the Berry connection still produces nontrivial Zak-phase holonomy around singular points, in close analogy with the Aharonov–Bohm effect (Puentes, 2023). These constructions differ in implementation, but they all localize topological information in singular structures of the Zak-phase field rather than in a fixed-point band phase.

4. Thouless pumping and the conversion of dislocation winding into Chern number

When the trimer lattice is driven adiabatically and periodically, the hopping parameters trace a closed loop

$Z_\mu$7

The associated band Chern number is

$Z_\mu$8

with

$Z_\mu$9

For the explicit trimer eigenvectors,

$C$0

and the Brillouin-zone integral of $C$1 vanishes. One therefore obtains the central relation

$C$2

or, if several dislocations are enclosed,

$C$3

The Chern number is the negative total winding number of the enclosed Zak-phase dislocations (Uakhitov et al., 19 Sep 2025).

This formula turns the dislocation picture into a direct computational tool. An adiabatic pump is a loop around phase screws in parameter space; the net turn of the screw is the pumped topological charge. In the trimerized commensurate off-diagonal Aubry–André–Harper modulation,

$C$4

small $C$5 gives a loop enclosing positive dislocations with

$C$6

whereas for $C$7 the loop encloses additional negative dislocations and the Chern numbers become

$C$8

This explains the topological transition in the trimer AAH model as a change in which Zak-phase dislocations are enclosed by the pump cycle (Uakhitov et al., 19 Sep 2025).

A plausible implication is that this viewpoint reorganizes Thouless pumping into a parameter-space singularity counting problem. Instead of computing the Berry curvature over the full $C$9 manifold, one can map the dislocation charges and integrate only their winding content.

5. Bulk–edge correspondence, edge-state evolution, and experimental manifestations

For open boundaries, the trimer lattice supports edge states in its band gaps. Their existence regions are sharply partitioned by the hopping inequalities: $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$0 If $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$1, both edges host modes; if $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$2, no edge states exist (Uakhitov et al., 19 Sep 2025). During an adiabatic pump, the loop in parameter space crosses these sectors and generates a characteristic edge-state sequence.

For the bottom gap, the relevant Chern number is $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$3. The paper introduces “Thouless pairs” to encode the edge-state dynamics: a right-to-left sequence corresponds to $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$4, while a left-to-right sequence corresponds to $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$5. The examples are explicit. A simple contour at $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$6 gives the bottom-gap sequence right $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$7 both $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$8 left and yields one right-to-left pair, hence $$\Delta Z_\mu=\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)=2\pi l_\mu,$$9. An elliptical loop encircling two negative dislocations yields two left-to-right pairs and $l_\mu$0. A figure-eight loop gives one positive and one negative pair, so $l_\mu$1. A zig-zag loop encircling many dislocations yields four negative pairs and maximal Chern numbers $l_\mu$2 for the three bands (Uakhitov et al., 19 Sep 2025).

The correspondence is therefore threefold. The band Chern number equals the net number of signed edge-state pairs crossing the gap during one cycle. The same Chern number equals the negative total Zak-phase winding enclosed by the pump loop. The same integer also gives the quantized pumped transport. In this system, geometric singularities in parameter space, bulk topological invariants, and edge-state dynamics are directly identified with one another (Uakhitov et al., 19 Sep 2025).

The primary experimental setting is a photonic waveguide array, where the couplings $l_\mu$3 are tuned by changing waveguide separations along the propagation direction. Observable consequences include quantized transport per cycle, measurable as the center-of-mass displacement of light, and direct imaging of edge-state evolution along the pump (Uakhitov et al., 19 Sep 2025). More broadly, photonic work on Bloch oscillations in dimerized waveguide lattices, interface-state spectroscopy in metasurface/photonic-crystal structures, and far-field diffraction in leaky one-dimensional photonic systems shows that Zak-phase differences, interface manifestations, and topological phase mismatches can be accessed experimentally without direct Berry-curvature reconstruction (Longhi, 2013, Wang et al., 2016, Liu et al., 2023).

6. Broader theoretical setting, limitations, and related formulations

The broader literature makes clear that Zak-phase dislocations are not a single mechanism but a family of singular Berry-phase phenomena whose interpretation depends on symmetry, gauge, and boundary structure. Several clarifications are especially important.

First, absolute Zak phases are subtle. In one-dimensional crystals the Zak phase can depend on the choice of origin or unit cell, motivating decompositions into origin-dependent and origin-independent pieces, such as the intra-cellular versus inter-cellular Zak phase, or the global versus internal phase in Fourier-space formulations (Rhim et al., 2016, Martí-Sabaté et al., 2021). Related work argues that the conventional expression can also be gauge dependent and introduces the Pancharatnam–Zak phase to obtain a gauge- and origin-independent geometric phase (Vyas et al., 2019). This suggests that the most robust content of a “Zak phase dislocation” is often not an absolute phase value but a phase difference, parity, winding number, or boundary-sensitive mismatch.

Second, single-band Zak phases are not universally reliable topological markers. In a generalized Dirac–Kronig–Penney continuum model, the Zak phase is quantized in classes AIII and BDI but takes non-quantized values in class D, and the bulk–boundary correspondence becomes highly sensitive to truncation position and boundary conditions (Angelone et al., 3 Feb 2026). In a related symmetry analysis of one-dimensional Altland–Zirnbauer–Cartan classes, a $l_\mu$4-valued invariant can be extracted from the abelian Zak phase in some classes, but quaternionic structures from anti-unitary symmetries squaring to $l_\mu$5 force that invariant to vanish (Manzoni et al., 16 Mar 2026). These results place a limit on any attempt to universalize single-band Zak dislocations across all continuum or symmetry settings.

Third, real-space approaches reinforce the boundary sensitivity of the problem. A Weyl-$l_\mu$6-function formulation expresses the Zak phase for periodic Jacobi operators directly in terms of boundary spectral data and recovers quantization under inversion symmetry (Ammari et al., 21 Jan 2026). This supports the view that dislocations, interfaces, and truncations are not merely illustrative boundary phenomena but can enter the phase itself through the spectral boundary terms.

Within those limitations, the trimer-lattice construction remains distinctive because it supplies an explicit parameter-space singularity theory: dislocations are localized at degeneracies, their winding numbers are integer-valued, and their sum is directly convertible into Chern numbers and edge-state counts (Uakhitov et al., 19 Sep 2025). In that sense, Zak phase dislocations elevate the Zak phase from a static one-dimensional Berry phase to a geometric field over parameter space whose singularities encode the topology of Thouless pumps and the evolution of edge states.