[Paper Review] Theory of electric polarization induced by inhomogeneity in crystals

Di Xiao, Junren Shi+2 more|ArXiv.org|Nov 12, 2007

Electromagnetic Effects on Materials3 citations

TL;DR

This paper develops a general gauge-invariant theory of electric polarization in inhomogeneous crystals, showing that spatial gradients of an order parameter induce a new polarization contribution via the second Chern form of non-Abelian Berry curvatures. The theory enables quantitative band-structure calculations of fractional charge in ferroelectric domain walls, reproducing known continuum results while extending to lattice models with full microscopic consistency.

ABSTRACT

We develop a general theory of electric polarization induced by inhomogeneity in crystals. We show that contributions to polarization can be classified in powers of the gradient of the order parameter. The zeroth order contribution reduces to the well-known result obtained by King-Smith and Vanderbilt for uniform systems. The first order contribution, when expressed in a two-point formula, takes the Chern-Simons 3-form of the vector potentials derived from the Bloch wave functions. Using the relation between polarization and charge density, we demonstrate our formula by studying charge fractionalization in a two-dimensional dimer model recently proposed.

Motivation & Objective

• To develop a general theory of electric polarization in crystals with spatially varying order parameters, where translational symmetry is broken.
• To extend the Berry-phase formulation of polarization beyond uniform systems to inhomogeneous insulators, particularly relevant for multiferroics with long-wavelength magnetic order.
• To provide a microscopic, band-theoretic framework for calculating polarization and charge fractionalization without relying on continuum approximations.
• To identify general conditions under which inhomogeneity-induced polarization is non-zero, especially in systems with non-Abelian Berry curvatures.

Proposed method

• Formalism based on the relation between polarization change and bulk current density, using the semiclassical dynamics of Bloch electrons.
• Derivation of a two-point formula for polarization using the Chern-Simons 3-form of vector potentials from local Bloch wave functions.
• Expansion of polarization in powers of the order parameter gradient, with the first-order term involving the second Chern form of non-Abelian Berry curvatures.
• Use of the non-Abelian Berry curvature formalism to handle degenerate bands, particularly in systems with SU(2) symmetry.
• Application of the theory to a 2D dimer model with staggered sublattice potential and vortex-like dimerization, using both band-structure and continuum limits for comparison.
• Calculation of charge fractionalization via the relation ρ(r) = −∇·P, linking polarization to charge density.

Experimental results

Research questions

• RQ1 How does spatial inhomogeneity of an order parameter generate electric polarization in insulating crystals?
• RQ2 What is the microscopic origin of magnetically induced polarization in multiferroics, and how can it be quantitatively calculated?
• RQ3 Can fractional charge in dimerized systems be understood as a polarization charge associated with ferroelectric domain walls?
• RQ4 How does the band-structure calculation of polarization compare to the continuum limit in systems with Dirac-like excitations?

Key findings

• The polarization in inhomogeneous crystals contains a new contribution proportional to the gradient of the order parameter, expressed via the second Chern form of the non-Abelian Berry curvature.
• This first-order contribution can be written as a two-point formula depending only on initial and final states, with a topological quantum number identified as the second Chern number.
• In the 2D dimer model, a ferroelectric vortex domain wall carries a fractional charge Q = ne/2 (1 − Δ/√(Δ² + m²)), matching results from spectral analysis of the Dirac Hamiltonian.
• The band-structure calculation based on the new theory shows significant deviation from the continuum limit as the staggered potential Δ increases, highlighting the importance of lattice effects.
• The theory explains charge fractionalization as a direct consequence of polarization induced by ferroelectric domain walls, with sublattice symmetry breaking (Δ) enabling irrational fractional values.
• The zeroth-order polarization vanishes in systems with SU(2) symmetric non-Abelian Berry curvatures, making the inhomogeneity-induced term the dominant contribution.