Modern Theory of Polarization

• Modern theory of polarization is a Berry-phase based framework that provides a gauge-invariant definition of electric polarization in periodic crystals, overcoming the ambiguity of dipole moment calculations.
• The theory extends to inhomogeneous structures and supercells by employing Born effective charge and Wannier-center approaches for spatially resolved polarization analysis.
• First-principles methods like DFT and DFPT enable accurate ab initio computations of polarization textures, informing the design of ferroelectric and topologically advanced materials.

The modern theory of polarization provides a rigorous, gauge-invariant, and physically meaningful description of electric polarization in crystalline solids, both at the bulk and local (supercell) levels. Traditionally, polarization in finite systems is defined via the dipole moment per unit volume, but in periodic crystals, ambiguity arises due to the multi-valuedness implied by lattice periodicity. The modern theory resolves this by expressing polarization in terms of the geometric (Berry) phase accumulated by occupied electronic states in momentum space. Recent developments extend these concepts to inhomogeneous structures, supercells, and nontrivial band topologies, enabling ab initio calculation and analysis of spatially varying polarization textures and their interplay with crystal and electronic topology.

1. Fundamental Concepts in the Modern Theory

The classical definition of polarization as the dipole moment per unit volume fails in periodic solids due to the non-uniqueness of the unit cell, leading to a lattice of equally valid polarization values differing by a "quantum of polarization". The modern theory defines the change in macroscopic polarization as a geometric phase (Berry phase) acquired by occupied Bloch states as a parameter (e.g., atomic positions) is adiabatically varied. The Berry-phase polarization for the bulk in three dimensions is

$$P_i = -\frac{e\,f}{(2\pi)^3} \sum_n^{\mathrm{occ}} \int_{BZ} \left\langle u_{n\mathbf{k}} \big| \partial_{k_i} u_{n\mathbf{k}} \right\rangle d^3k,$$

where $e$ is the electron charge, $f$ the spin factor, $u_{n\mathbf{k}}$ the cell-periodic Bloch functions, and the integral runs over the Brillouin zone. Only differences in polarization, corresponding to Berry phase differences for different atomic configurations or structures, have physical meaning due to the multi-valuedness modulo the polarization quantum (Spaldin, 2012).

2. Gauge-Invariant Local Polarization in Supercells

In inhomogeneous or moiré superlattice systems, polarization becomes a spatially varying field. A gauge-invariant local polarization $P_i(r_j)$ can be defined as the change in total polarization of the supercell when atoms in cell $r_j$ are displaced from a reference non-polar configuration $x=0$ to their local configuration $x(r_j)$. Two main formulations are used:

• Born Effective Charge (BEC) Approach: The local polarization is given by

$$P_i(r_j) = \int_0^{x(r_j)} Z^*_{i\alpha}(x') dx'_\alpha = -\frac{2ie\,f}{(2\pi)^3} \sum_n^{\mathrm{occ}} \int_0^{x(r_j)} \oint_{\mathrm{scBZ}} \left\langle \partial_{x_\alpha} u_{n \mathbf{k}} \mid \partial_{k_i} u_{n \mathbf{k}} \right\rangle d^3k\, dx'_\alpha,$$

where $Z^*_{i\alpha}$ is the mixed Berry phase–derived effective charge tensor and $\mathrm{scBZ}$ is the supercell Brillouin zone.

• Wannier-Center Approach: Local polarization relates to shifts of maximally localized Wannier centers,

$$P_i(r_j) = -\frac{e f}{\Omega_0} \sum_n^{\mathrm{occ}} \int_0^{x(r_j)} \frac{\partial \bar{w}_n}{\partial x'_\alpha} dx'_\alpha,$$

where $\Omega_0$ is the reference cell volume and $\bar{w}_n$ are the Wannier centers (Bennett et al., 2023).

Both definitions reduce to the Berry-phase (King-Smith–Vanderbilt) formula for a primitive unit cell and are strictly gauge-invariant under $U(N)$ rotations of occupied states.

3. First-Principles Computational Methods

Gauge-invariant polarization in supercells is computed using electronic-structure techniques:

• Density Functional Theory (DFT) with appropriate pseudopotentials and exchange-correlation functionals, using plane-wave codes (e.g., ABINIT with PseudoDojo norm-conserving potentials) and converged reciprocal-space grids. Supercell geometries are relaxed and electronic ground states obtained.
• Berry-Phase Calculations are performed on the supercell bands for different stacking configurations, evaluating

$$P_i(x) = -\frac{e f}{(2\pi)^3} \sum_n^{\mathrm{occ}} \oint_{\mathrm{scBZ}} A_{n,i}(\mathbf{k})\, d^3k, \quad A_{n,i}(\mathbf{k}) = i\langle u_{n\mathbf{k}} | \partial_{k_i} u_{n\mathbf{k}} \rangle.$$

• Born Effective Charges are determined via Density Functional Perturbation Theory (DFPT), exploiting the equivalence between macroscopic field perturbations and atomic displacements.
• Wannier-Function Analysis is implemented using tools such as Wannier90, extracting the centers of maximally localized Wannier functions at each configuration.

Consistency among these approaches has been demonstrated in commensurate bilayer hexagonal boron nitride, confirming the robustness of the Berry-phase theory when extended to spatially resolved local polarization (Bennett et al., 2023).

4. Effective Models and Real-Space Visualization

Effective continuum models, such as the 1D "two-cosine" potential, validate the local polarization formalism. In such models, the local potential in a cell can be modulated to "depolarize" a specific region, and resulting shifts in local Wannier centers yield the local polarization profile. Summing these local contributions reproduces the global property, and gauge invariance holds for the total sum despite the gauge dependence of individual cluster sums. Real-space visualization, via local probes and perturbations, establishes that local polarization textures—stripes, vortices, merons—are rigorously connected to the Berry-phase characteristics of the full system (Bennett et al., 2023).

5. Polarization Textures, Topological Charge, and Band Topology

Local polarization fields can exhibit nontrivial textures characterized by topological invariants. For a normalized vector field $\hat{P}(\mathbf{r})$ in 2D, the real-space skyrmion (meron) number is

$$Q = \frac{1}{4\pi} \int \hat{P} \cdot (\partial_x \hat{P} \times \partial_y \hat{P})\, d^2r.$$

For moiré polar domains in twisted bilayer structures, $Q$ integrates to half-integer values (e.g., $\pm \frac{1}{2}$), corresponding to meron and antimeron textures. These topological characteristics are distinct from electronic band Chern numbers but can influence edge-state transport or serve as switching mechanisms in topological insulators.

While real-space polarization textures can be topologically nontrivial, their associated Chern invariants vanish unless the electronic bands themselves form a Chern insulator (i.e., break time-reversal symmetry). For systems with nontrivial band topology (Chern insulators), the modern theory of polarization must be further adapted using hybrid Wannier centers or Wilson loop techniques in the supercell Brillouin zone, tracking the evolution of Zak phases and their relationship to bulk polarization and dislocation-bound fractional charges (Gunawardana et al., 25 Feb 2025, Bennett et al., 2023).

6. Implications and Extensions of the Theory

A gauge-invariant, real-space field $P(r_j)$ allows the modern theory of polarization to be carried into generic inhomogeneous environments, overcoming limitations of configuration-space or bulk-charge approximations. This paves the way for:

• Direct ab initio characterization of nanoscale polarization textures, including domain walls, vortices, and moiré patterns.
• Investigation of the coupling between polarization texture topology and the topology of electronic bands, crucial for future materials where ferroelectric domains or heterostructure stacking engineer topological responses.
• Systematic extension to Chern insulators, capturing the effects of nontrivial band topology on polarization and defect-bound charges via hybrid Wannier center tracking and phase-space Berry curvature formulations.

A practical workflow involves: (1) full-supercell DFT calculations; (2) computation of BECs via DFPT (or electric field perturbation); (3) integration from a nonpolar reference to obtain local $P(r_j)$, or (3') extraction of Wannier centers and summation of their shifts for the same purpose.

The combination of Berry-phase theory, gauge-invariance, and real-space mapping tightly links polarization to underlying microscopic and topological properties, providing a cornerstone for the analysis of modern ferroelectric, multiferroic, and topological crystalline systems (Bennett et al., 2023).