--- title: Zak Phase Dislocations in Topological Lattices url: https://www.emergentmind.com/topics/zak-phase-dislocations type: topic --- # Zak Phase Dislocations in Topological Lattices 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 [2509.15894]. 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 [1506.08100], [1802.08099], [2306.12540]. ## 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 [2509.15894]. 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 [2509.15894]. The phrase “screw-type dislocation” refers to the helical behavior of \(Z_\mu\) around a degeneracy axis: as the cyclic parameter advances by \(2\pi\), the Zak phase changes by \(2\pi l_\mu\), 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 \(0\) or \(\pi\), but in the trimer problem the nontrivial topology resides in the winding of \(Z_\mu(\lambda)\) over a family of lattices, not in its value at a single point [2509.15894]. ## 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, \(A_n,B_n,C_n\), nearest-neighbor hopping only, and no on-site potentials. In real space the evolution equations are \[ \begin{aligned} i \frac{d}{dz}A_n + J_1 B_n + J_3 C_{n-1} &= 0,\\ i \frac{d}{dz}B_n + J_1 A_n + J_2 C_n &= 0,\\ i \frac{d}{dz}C_n + J_2 B_n + J_3 A_{n+1} &= 0, \end{aligned} \] with \(J_1\) the intra-cell \(A_n\leftrightarrow B_n\) hopping, \(J_2\) the intra-cell \(B_n\leftrightarrow C_n\) hopping, and \(J_3\) the inter-cell \(A_{n+1}\leftrightarrow C_n\) hopping [2509.15894]. In the photonic realization, the propagation coordinate \(z\) plays the role of time. The Bloch Hamiltonian is \[ H(q)=- \begin{pmatrix} 0 & J_1 & J_3 e^{-iq}\\ J_1 & 0 & J_2\\ J_3 e^{iq} & J_2 & 0 \end{pmatrix}, \] and the three bands satisfy \[ \omega(\omega^2-J^2)=-2p\cos q,\qquad J^2=J_1^2+J_2^2+J_3^2,\qquad p=J_1J_2J_3. \] Because the model is homogeneous in the overall scale, one can fix \(J=\sqrt{J_1^2+J_2^2+J_3^2}\) and regard the family of lattices as living on the sphere \(J=\mathrm{const}\). On that sphere, the dispersion depends only on the product \(p=J_1J_2J_3\), while motion along constant-\(p\) contours changes the eigenvectors but not the energies [2509.15894]. For the explicit eigenvectors of this model, the Berry connection becomes \[ A_{\mu q}=(J_2^2-J_1^2)\,J_3^2\,\Psi_\mu^2. \] Two consequences are immediate. First, \(A_{\mu q}=0\) when \(J_1=J_2\), i.e. on the inversion-symmetric trimer line. Second, the Zak phase itself vanishes there: \[ Z_\mu=0\qquad \text{for } J_1=J_2. \] This is a decisive departure from SSH intuition. In the trimer lattice, inversion symmetry forces \(Z_\mu=0\) even though edge states may still be present when \(J_3>J_1=J_2\) [2509.15894]. 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 \(\phi\) defined through rotated variables \[ \{j_1,j_2,j_3\}=\{r\cos\phi,r\sin\phi,\sqrt{1-r^2}\}, \] with \((J_1,J_2,J_3)\) expressed as linear combinations of the \(j_m\). Along a constant-\(p\) contour, \(\phi\) runs from \(0\) to \(2\pi\), and \(Z_\mu(\phi)\) develops the dislocation structure [2509.15894]. ## 3. Screw-type dislocations and their topological charges The organizing singularity in the trimer problem is the degeneracy axis \(J_1=J_2=J_3\), where the three bands collapse into a monatomic band and the gaps close. Near this axis, the Zak phases twist with quantized winding: \[ l_1=l_3=+1,\qquad l_2=-2. \] Equivalently, for a loop \(C\) encircling the degeneracy, \[ w_\mu(C)=\frac{1}{2\pi}\oint_C d\lambda\,\nabla_\lambda Z_\mu(\lambda)\in \mathbb{Z}, \] and \(w_\mu(C)\) is the dislocation charge of band \(\mu\) [2509.15894]. Geometrically, the dislocation is screw-like because \(Z_\mu\) 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 \(p=J_1J_2J_3\), so the parameter space contains both positive and negative dislocations [2509.15894]. 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 \(q=\pi\), band 3 has a cone at \(q=0\), 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 [2509.15894]. 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 \(\pi\), while same-branch differences vanish [1506.08100]. 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 \(\pi\) and acts as a localized phase jump [1802.08099]. 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 [2306.12540]. 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 \[ \vec{\tau}: t\in[0,T]\mapsto (J_1(t),J_2(t),J_3(t)). \] The associated band Chern number is \[ C_\mu=\frac{1}{2\pi}\int_{-\pi}^{\pi}dq\oint d\tau\,F_{\mu,q\tau}(q,\tau), \] with \[ F_{\mu,q\tau}=\partial_q A_{\mu\tau}-\partial_\tau A_{\mu q},\qquad A_{\mu\tau}=i\langle \psi_\mu\mid \partial_\tau \psi_\mu\rangle. \] For the explicit trimer eigenvectors, \[ A_{\mu\tau}=\frac{1}{2}\frac{\partial}{\partial q}\ln(\omega_\mu^2-J_3^2)\, \frac{d}{d\tau}\ln\frac{J_1}{J_2}, \] and the Brillouin-zone integral of \(\partial_q A_{\mu\tau}\) vanishes. One therefore obtains the central relation \[ C_\mu=-\frac{1}{2\pi}\oint d\tau\,\partial_\tau Z_\mu(\tau)=-w_\mu(\text{loop}), \] or, if several dislocations are enclosed, \[ C_\mu=-\sum_j w_{\mu,j}. \] The Chern number is the negative total winding number of the enclosed Zak-phase dislocations [2509.15894]. 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, \[ J_m(\phi)=\tilde{J}\left[1-\lambda\cos\left(\frac{2\pi m}{3}-\phi\right)\right],\qquad m=1,2,3, \] small \(\lambda\) gives a loop enclosing positive dislocations with \[ \{l_\mu\}=\{1,-2,1\},\qquad C_\mu=-l_\mu=\{-1,2,-1\}, \] whereas for \(\lambda>4\) the loop encloses additional negative dislocations and the Chern numbers become \[ C_\mu=\{2,-4,2\}. \] This explains the topological transition in the trimer AAH model as a change in which Zak-phase dislocations are enclosed by the pump cycle [2509.15894]. 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 \((q,\tau)\) 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: \[ \text{right-edge modes when } J_3>J_1,\qquad \text{left-edge modes when } J_3>J_2. \] If \(J_3>J_1,J_2\), both edges host modes; if \(J_3