Berry phase & the modern theory of polarization

The bulk electric polarization $\mathbf{P}$ of a crystalline insulator is not a function of the charge density in the unit cell — the naive dipole-per-volume $\frac{1}{\Omega }{\int }_{\text{cell}}\mathbf{r}\rho (\mathbf{r})d\mathbf{r}$ depends on where you cut the cell and is therefore ill-defined. The resolution (King-Smith, Vanderbilt; Resta) is that the physical observable is the change $\Delta \mathbf{P}$ along an adiabatic path, and that $\mathbf{P}$ itself is a geometric (Berry) phase of the occupied Bloch manifold, well-defined only modulo a quantum. This note states the geometry, the polarization quantum, the Wannier-center picture, and then closes the LO–TO loop by recovering the Born effective charge as a polarization derivative — the same $Z^{\ast }$ that DFPT reaches from linear response. It is also the gateway to band topology.

Prerequisites

tight-binding & Bloch · Wannier functions

1. Berry phase and Berry curvature

Let $|u_{n\mathbf{k}}⟩$ be the cell-periodic part of a Bloch state, an eigenstate of $H(\mathbf{k})$ that depends smoothly on a parameter $\mathbf{k}$ (here crystal momentum, but the construction is generic in any parameter manifold). The Berry connection of band $n$ is the expectation of $i{\nabla }_{\mathbf{k}}$,

$${\mathbf{A}}_n(\mathbf{k})=i⟨u_{n\mathbf{k}}|{\nabla }_{\mathbf{k}}|u_{n\mathbf{k}}⟩,$$

a real (up to the $i$) gauge-dependent vector field: under a gauge change $|u_{n\mathbf{k}}⟩\to e^{i{\phi }_n(\mathbf{k})}|u_{n\mathbf{k}}⟩$ it shifts by ${\mathbf{A}}_n\to {\mathbf{A}}_n-{\nabla }_{\mathbf{k}}{\phi }_n$, exactly like a $U(1)$ vector potential. Its curl is the gauge-invariant Berry curvature

$${\mathbf{\Omega }}_n(\mathbf{k})={\nabla }_{\mathbf{k}}\times {\mathbf{A}}_n(\mathbf{k})=i⟨{\nabla }_{\mathbf{k}}{u}_{n\mathbf{k}}|\times |{\nabla }_{\mathbf{k}}{u}_{n\mathbf{k}}⟩,$$

equivalently ${\Omega }_n^{\gamma }=-{ϵ}^{\alpha \beta \gamma }\mathrm{Im}{\sum }_{m\ne n}\frac{⟨u_n|{\partial }_{\alpha }H|u_m⟩⟨u_m|{\partial }_{\beta }H|u_n⟩}{(E_n-E_m{)}^2}$, the sum-over-states form that makes the role of band touchings (where $E_n-E_m\to 0$ and curvature concentrates) explicit. The Berry phase accrued on a closed loop $\mathcal{C}$ in $\mathbf{k}$-space is ${\gamma }_n={\oint }_{\mathcal{C}}{\mathbf{A}}_n\cdot d\mathbf{k}$, gauge-invariant modulo $2\pi$, and the curvature is its local density via Stokes. For a degenerate $J$-band manifold the connection and curvature become $J\times J$ matrices (non-Abelian, Wilczek–Zee), and only manifold-traced quantities $\mathrm{tr}\mathbf{A}$, $\mathrm{tr}\mathbf{\Omega }$ are gauge-invariant under the full $U(J)$ — which is exactly the gauge freedom of the Wannier construction.

2. Polarization as a Berry phase (King-Smith–Vanderbilt)

The modern theory identifies the electronic polarization with the Berry phase of the occupied manifold along each reciprocal-lattice direction. For direction $\alpha$ (reciprocal vector ${\mathbf{G}}_{\alpha }$, real-space period ${\mathbf{a}}_{\alpha }$), parametrize the BZ as the loop in $k_{\alpha }$ at fixed transverse ${\mathbf{k}}_{\perp }$ and integrate:

$$\boxed{P_{\alpha }^{\text{el}}=-\frac{e}{(2\pi {)}^3}{\int }_{\mathrm{BZ}}d\mathbf{k}\sum _{n\in \text{occ}}i⟨u_{n\mathbf{k}}|{\partial }_{k_{\alpha }}|u_{n\mathbf{k}}⟩}$$

i.e. ${\mathbf{P}}^{\text{el}}=-\frac{e}{(2\pi {)}^3}{\int }_{\mathrm{BZ}}\mathrm{tr}\mathbf{A}(\mathbf{k})d\mathbf{k}$ over occupied bands, plus the nuclear point-charge contribution ${\mathbf{P}}^{\text{ion}}=\frac{e}{\Omega }{\sum }_{\kappa }{Z}_{\kappa }^{\text{ion}}{\mathbf{\tau }}_{\kappa }$. Operationally one discretizes the $k_{\alpha }$ string and computes the phase of the product of overlap determinants between neighboring $\mathbf{k}$ points,

$${\gamma }_{\alpha }({\mathbf{k}}_{\perp })=-\mathrm{Im}\ln \prod _{j=0}^{N_k-1}\det [⟨u_{m{\mathbf{k}}_j}|u_{n{\mathbf{k}}_{j+1}}⟩{]}_{mn},$$

the discrete (Zak) Berry phase — manifestly gauge-invariant because each interior gauge phase appears once with each sign, and the global $U(J)$ at the string endpoints cancels under the determinant. This is the form codes actually evaluate (the “Berry-phase polarization” of QE/berryphase, ABINIT, VASP LCALCPOL).

3. The polarization quantum — the multivaluedness is physical

The string Berry phase is defined only modulo $2\pi$, so $\mathbf{P}$ is defined only modulo the polarization quantum

$$\boxed{\Delta {\mathbf{P}}_{\text{quantum}}=\frac{e\mathbf{R}}{\Omega }}(\mathbf{R}\text{any lattice vector}).$$

This is not a numerical nuisance to be gauged away — it is the statement that $\mathbf{P}$ is a lattice-valued quantity, an element of ${\mathbb{R}}^3/(e\mathbf{R}/\Omega )$, because shifting one electron by a lattice vector is a legal relabeling of the same crystal that changes the single-cell dipole by exactly one quantum. Two consequences a careful calculation must respect:

• Only differences are observable. A measurable polarization change (a switching path, a piezoelectric response, a Born charge) is $\Delta \mathbf{P}$ along a continuous adiabatic path on which the gap stays open; one must track the same branch of $\mathbf{P}$ along the path, never subtracting two values reduced into the home Brillouin zone of $\mathbf{P}$. Reducing endpoints independently into $(-\frac{e\mathbf{R}}{2\Omega },\frac{e\mathbf{R}}{2\Omega }]$ and subtracting is the classic way to get a piezo or Born-charge sign/magnitude wrong by a quantum.
• Symmetry quantizes $\mathbf{P}$. In a centrosymmetric crystal $\mathbf{P}$ must equal $-\mathbf{P}$ modulo the quantum, forcing $\mathbf{P}\in \{0,\frac{1}{2}e\mathbf{R}/\Omega \}$ — the ${\mathbb{Z}}_2$ Berry-phase invariant ($\gamma =0$ or $\pi$) behind the Su–Schrieffer–Heeger / Zak-phase classification of 1D insulators.

4. The Wannier-center picture

The Berry-phase formula has a transparent real-space reading. Build occupied Wannier functions $|w_{n0}⟩$ with centers ${\bar{\mathbf{r}}}_n=⟨w_{n0}|\mathbf{r}|w_{n0}⟩$. The Wannier center is exactly the band-projected Berry connection (the diagonal of the position operator in the Bloch basis), so the electronic polarization is the dipole of a lattice of point charges $-e$ sitting at the Wannier centers:

$${\mathbf{P}}^{\text{el}}=-\frac{e}{\Omega }\sum _{n\in \text{occ}}{\bar{\mathbf{r}}}_n.$$

The quantum reappears as the freedom to assign each Wannier function to any home cell: moving a center by $\mathbf{R}$ shifts $\mathbf{P}$ by $e\mathbf{R}/\Omega$. This picture also makes the topological obstruction visible: if the occupied manifold carries a nonzero Chern number, no choice of smooth gauge yields exponentially localized Wannier centers (the localization failure is the obstruction), the Wannier-center “strings” wind across the BZ as ${\mathbf{k}}_{\perp }$ is varied, and the polarization branch cannot be defined globally — the same physics, viewed downstream, that makes a Chern insulator topologically nontrivial.

5. Born effective charge — closing the LO–TO loop from the other side

The dynamical (Born) effective charge of sublattice $\kappa$ is, by definition in the modern theory, the macroscopic polarization linearly induced by a sublattice displacement at zero macroscopic field:

$$\boxed{Z_{\kappa ,\alpha \beta }^{\ast }=\Omega \frac{\partial P_{\alpha }}{\partial u_{\kappa \beta }}{|}_{\mathbf{E}=0}}$$

Because each $\partial P_{\alpha }$ is a Berry phase, $Z^{\ast }$ is computed by finite differences of the Berry-phase polarization between the equilibrium and slightly displaced sublattice — the original King-Smith–Vanderbilt application. This is the same tensor that DFPT obtains as the mixed second derivative $\Omega \partial P_{\alpha }/\partial u_{\kappa \beta }=\partial F_{\kappa \beta }/\partial {\mathcal{E}}_{\alpha }$ from a single linear-response electric-field calculation: two routes — geometric phase vs. Sternheimer response — to one physical quantity, a useful cross-check. $Z^{\ast }$ is the dynamical charge that sources the long-range dipole field, hence the non-analytic LO–TO term of the dynamical matrix; the modern-theory route is “the other side” of the loop the DFPT note already closes. The same exact constraints apply: charge neutrality ${\sum }_{\kappa }{Z}_{\kappa ,\alpha \beta }^{\ast }=0$ (a rigid translation produces no polarization — the Berry phase is invariant under a uniform shift) and the site-symmetry constraints on each $Z_{\kappa }^{\ast }$.

6. Gateway to band topology

The geometry above is the entry point to topological band theory. The integral of the Berry curvature over the (2D) Brillouin zone is quantized,

$$C=\frac{1}{2\pi }{\int }_{\mathrm{BZ}}{\Omega }^z(\mathbf{k})d^{2}k\in \mathbb{Z},$$

the Chern number — the first Chern class of the occupied bundle, the obstruction to a global smooth gauge, and (via the TKNN formula) the quantized Hall conductance ${\sigma }_{xy}=C{e}^2/h$. The same Wannier-center winding that obstructs polarization defines the Berry-phase / Wilson-loop invariants (${\mathbb{Z}}_2$ for time-reversal-invariant insulators). Polarization, Born charges, orbital magnetization, and the anomalous Hall effect are all moments of the same Berry curvature and connection of the occupied manifold; the modern theory of polarization is the first and simplest member of that family.

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Builds toward: Chern insulators / ${\mathbb{Z}}_2$ topological invariants · orbital magnetization · the anomalous Hall effect · [[physics/dynamical-matrix-asr|LO–TO splitting via $Z^{\ast }$]]

Key references

• Modern theory of polarization — R. D. King-Smith & D. Vanderbilt, Phys. Rev. B 47, 1651 (1993); R. Resta, Rev. Mod. Phys. 66, 899 (1994); D. Vanderbilt & R. D. King-Smith, Phys. Rev. B 48, 4442 (1993).
• Berry phase & curvature in solids — M. V. Berry, Proc. R. Soc. A 392, 45 (1984); D. Xiao, M.-C. Chang & Q. Niu, Rev. Mod. Phys. 82, 1959 (2010).
• Wannier-center / topology — R. Resta & D. Vanderbilt, in Physics of Ferroelectrics (Springer, 2007); D. Vanderbilt, Berry Phases in Electronic Structure Theory (Cambridge, 2018).
• Chern number / quantized Hall — D. J. Thouless, M. Kohmoto, M. P. Nightingale & M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).