Quantized Edge Polarization in Topological Systems

• Quantized Edge Polarization is a phenomenon where robust, discrete electric or spin polarizations appear at boundaries, driven by Berry phase mechanisms and symmetry constraints.
• 1D models like the SSH and Rice–Mele chains, along with 2D higher-order topological insulators, illustrate how bulk–boundary correspondences enforce quantized interface charges.
• This concept underpins applications in topological qubits, edge state detection, and is validated through experimental platforms ranging from atomic chains to photonic lattices.

Quantized edge polarization refers to the robust, discrete values of electric or spin polarization (dipole moments) that appear localized at boundaries, interfaces, or edges in diverse crystalline, electronic, spin, or superconducting systems. These quantized polarizations are topological boundary signatures, often protected or enforced by bulk–boundary correspondences, crystalline symmetries, or Berry phase (Zak phase) topological invariants, and can arise even where conventional symmetry-based bulk indices vanish or are forbidden. This phenomenon controls observable edge (or domain-wall) charges, qubit polarizations, and corner charges in higher-order topological insulators.

1. Quantized Edge Polarization in 1D: Domain Walls and Berry-Phase Mechanisms

The Su–Schrieffer–Heeger (SSH) and generalized Rice–Mele models epitomize the 1D origin of quantized edge and domain-wall polarization. In the generalized SSH chain with hopping alternation and staggered on-site mass: $$H_{\mathrm{AI}}(k) = 2t_0\cos k \, \sigma_x - \Delta\sin k \, \sigma_y + m_z \sigma_z,$$ the polarization in an insulating ground state is given by the Berry (Zak) phase of the occupied band(s): $$P = (e/2\pi) \int_{\mathrm{BZ}} A(k) \, \mathrm{d}k,\quad A(k) = i \langle u_k | \partial_k u_k \rangle,$$ where $|u_k\rangle$ is the Bloch eigenstate.

When spatially smooth (kink-like) interpolations of $\Delta(x)$ and/or $m_z(x)$ generate a domain wall between two degenerate but topologically distinct bulk regions, the difference in Berry/Zak phases ensures a quantized dipole moment at the interface: $$Q_{\mathrm{DW}} = P_R - P_L = (e/2\pi)\Delta\varphi,$$ with $\Delta\varphi$ the difference in Zak phases between right and left domains. In the SSH-type case, $\Delta\varphi = \pi$ yields $Q_{\mathrm{DW}} = e/2$; remarkably, this value persists for more general interpolations—even when bulk polarizations are unquantized—whenever $\Delta\varphi$ is pinned to an integer multiple of $\pi$ by underlying (even hidden) symmetries (Han et al., 2023). This result is robust to disorder, as supported by tight-binding exact diagonalization and field-theoretic (Jackiw–Rebbi) continuum arguments.

Notably, quantized domain-wall polarization can appear in 1D Altland–Zirnbauer classes forbidden from hosting conventional edge zero modes; thus, quantized interface charge emerges when the joining bulks differ by a quantized Berry phase, exemplifying a "generalized bulk–boundary principle" that goes beyond the tenfold classification (Han et al., 2023).

2. Bulk–Edge Polarization Decomposition and Quantization in 1D Chains

In generalized Rice–Mele chains at arbitrary filling, the many-body polarization (Resta formula) can be decomposed under open boundary conditions into bulk and edge contributions: $$\langle \hat{P}_x \rangle \simeq P_{\mathrm{bulk}} + P_{\mathrm{edge}},$$ where $P_{\mathrm{bulk}}$ comes from the local charge configuration deep in the chain, and $P_{\mathrm{edge}}$ encodes net edge excess charge. Under appropriate combined inversion–translation symmetries, the polarization is symmetry-quantized: $2 P_x = k\nu \pmod{1},$ where $\nu$ is the filling factor and $k$ is symmetry-dependent. Edge charges $\pm 1/2$ (modulo 1) accumulate at phase boundaries or at specific fillings (filling anomaly), enforcing edge charge quantization even when bulk polarization is not symmetry-quantized (Tada, 2024).

The polarization crossover from edge- to bulk-dominated regimes is controlled by the system gap. As the modulation amplitude increases, $P_{\mathrm{bulk}}$ rises smoothly to quantized values, while $P_{\mathrm{edge}}$ decreases, vanishing in the deep gapped (atomic) limit. At bulk–boundary phase transitions or in the presence of symmetry-protected zero modes, the quantized edge (or end) polarization becomes an observable charge signature (Tada, 2024).

3. Quantized Edge Polarization in 2D: Higher-Order Topological Insulators

In 2D systems with vanishing bulk dipole polarization, edges can still exhibit robust quantized polarizations. For topological quadrupole insulators (e.g., the Benalcazar–Bernevig–Hughes (BBH) model), the hierarchy is as follows:

• The bulk possesses a quadrupole density moment but vanishing net dipole.
• Each edge hosts quantized "entanglement polarization" (the "hidden" dipole polarization of edge-restricted entanglement Hamiltonians), which, under mirror symmetry, is quantized to $0$ or $1/2$ (lattice units) (Fukui et al., 2018).
• This quantized edge polarization guarantees, via a 1D bulk–boundary correspondence, the appearance of protected zero-energy corner modes when open boundaries cut both directions.

Mathematically, for an edge along, e.g., $x$, the entanglement edge polarization is defined as

$$P_{\mathrm{edge},y}^{(13)} = (1/2\pi)\int_0^{2\pi} A_y^{\mathrm{edge}}(k_y)\,dk_y,$$

where $A_y^\mathrm{edge}$ is the Berry connection of the mid-gap entanglement edge state for partition $(1,3)$. Reflection or rotation symmetry enforces quantized values, and the sum of edge polarizations directly determines the corner charge when both edges meet (Fukui et al., 2018, Ren et al., 2020).

4. Wannier Formalism and Bulk–Edge–Corner Correspondence

The Wannier function framework systematically formalizes bulk, edge, and corner contributions in both 1D and 2D. Edge polarization in a ribbon geometry is defined from the dipole moment of maximally localized Wannier functions at the edge: $P_x^{\mathrm{edge}} = \frac{d_x}{a},$ where $d_x$ is the dipole per edge tile.

For rectangular 2D insulators, the corner charge $q_c$ can be expressed as

$q_c = Q^{xy} + {\cal P}_x + {\cal P}_y,$

with $Q^{xy}$ the bulk quadrupole density and ${\cal P}_{x(y)}$ the edge polarization on the respective edges. Crucially, while $Q^{xy}$ and each ${\cal P}_\alpha$ are individually gauge-dependent, their sum (the physical corner charge) is gauge-invariant (Ren et al., 2020).

The "Wannier-cut" prescription, as formalized in (Trifunovic, 2020), enables explicit computation of the edge polarization in terms of both a projected bulk quadrupole piece and an edge (Wannier) contribution: $$\mathbf{P}_\alpha^{\mathrm{edge}} = \frac{L_\alpha}{2}\hat{q}\,\mathbf{n}_\alpha + {}^{\mathrm{edge}}_\alpha,$$ where $\hat{q}$ is the bulk quadrupole tensor and $L_\alpha$ the edge length. Crystalline point-group symmetries (e.g., $C_n$) enforce quantization: with $C_4$, $p_\alpha = e/2$ mod $e$, so the edge polarization along each edge is quantized, explaining the robust fractional corner charges in H.O.T.I.s. (Trifunovic, 2020).

5. Quantized Edge Polarization in Topological Spin and Superconducting Systems

Quantized edge polarization is not restricted to charge; it arises in spin and superconducting systems:

• Spin-1 Haldane chains (as realized with Rydberg excitons in $\mathrm{Cu}_2\mathrm{O}$) display fractionalized spin-$1/2$ edge states in finite chains. Optical selection rules render the boundary emission circularly polarized, with degree $P_{\mathrm{edge}} \approx \pm 1$, sharply distinct from the bulk ($\approx 0$) (Poddubny et al., 2020). The edge value of $\langle L_j^z \rangle$ is analytically $L_0 \approx 0.8$, effectively "half-integer quantized."
• In topological superconducting systems (e.g., the Kitaev chain), domain walls separating regions with distinct phase windings can bind Majorana zero modes and induce quantized polarization analogous to the SSH model (Han et al., 2023).

In the Moore–Read $\nu = 5/2$ fractional quantum Hall state, quantized edge polarization applies to topological qubits (encoded by Ising anyons). Controlled coupling between edge Majorana modes and localized bulk quasiparticles can lock the polarization $\langle \sigma_z \rangle$ into quantized plateaus ($0$, $\pm 2/\pi$, $\pm 1$), set by symmetry and braiding statistics of Ising anyons (Clarke et al., 2011).

6. Chern Insulators and Gauge-Fixed Edge Polarization

In 2D Chern insulators, defining absolute edge polarization is complicated by the absence of exponentially localized, gauge-invariant Wannier centers. The bulk polarization, formulated via the Zak phase and incorporating gauge-vortex singularities, controls both fractional dislocation charge and, in a suitable gauge, the edge dipole jump. When the Berry phase $\gamma_x(k_y)$ jumps by $2\pi C$ at a vortex, the edge polarization jumps by $eC$, enforcing quantization per the Chern number (Gunawardana et al., 25 Feb 2025).

7. Symmetry Constraints and Experimental Realizations

Crystalline symmetries dictate the allowed quantized values of edge polarization via rigorous symmetry constraints:

• $C_n$ rotation forces $p_\alpha = j e/n$ mod $e$.
• Reflection or inversion reduces this to $p_\alpha = 0$ or $e/2$.
• The quantization of edge polarization is reflected directly in observable quantities such as fractional edge/corner charges, midgap boundary modes, and, in the spin case, quantized polarized emission (Trifunovic, 2020, Ren et al., 2020, Tada, 2024, Han et al., 2023, Fukui et al., 2018).

Experimental platforms include atomic and molecular chains on surfaces (STM), artificial lattices (CO-on-Cu), graphene nanoribbons, ultracold atomic optical lattices, photonic waveguide arrays, and quantum Hall interferometers, validating the universality of quantized edge polarization as a topological boundary invariant (Han et al., 2023, Clarke et al., 2011, Poddubny et al., 2020, Tada, 2024).

---

References:

• Topological domain-wall quantized polarization and Majorana zero modes (Han et al., 2023)
• Quantized bulk and edge polarization in charge-ordered 1D insulators (Tada, 2024)
• Entanglement polarization and higher-order topology (Fukui et al., 2018)
• Microscopic theory of Chern and edge polarization (Gunawardana et al., 25 Feb 2025)
• Wannier-based edge/corner charge correspondence (Ren et al., 2020)
• General bulk-and-edge to corner correspondence (Trifunovic, 2020)
• Quantized spin edge polarization in Rydberg exciton chains (Poddubny et al., 2020)
• Quantized edge-induced qubit polarization and Ising anyons (Clarke et al., 2011)