Berry phases and finite electric fields

Berry phase expression for the macroscopic polarization

Calculating the change in dipole moment per unit cell under PBC's, is a nontrivial task. In general one cannot define it as the first moment of the induced change in charge density δ(r), through

$\Delta \mathbf{P}= \frac{1}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta \left( \mathbf{r} \right) d^{3}r$

without introducing a dependency on the shape of Ω₀, the chosen unit cell.

Recently King-Smith and Vanderbilt, building on the work of Resta, showed that the electronic contribution to the difference in polarization ΔP_e, due to a finite adiabatic change in the Hamiltonian of a system, can be identified as a geometric quantum phase or Berry phase of the valence wave functions. We will briefly summarize the essential results (for a review of geometric quantum phases in polarization theory see the papers of Resta).

Central to the modern theory of polarization is the proposition of Resta to write the electronic contribution to the change in polarization due to a finite adiabatic change in the Kohn-Sham Hamiltonian of the crystalline solid, as

$\Delta \mathbf{P}_{e}= \int^{\lambda_{2}}_{\lambda_{1}}{\partial \mathbf{P}_{e} \over \partial \lambda} d\lambda$

with

${\partial \mathbf{P}_{e} \over \partial \lambda}= {i |e| \hbar \over N \Omega_{0} m_{e}} \sum_{\mathbf{k}} \sum^{M}_{n=1} \sum^{\infty}_{m=M+1} {\langle \psi^{\left(\lambda\right)}_{n\mathbf{k}} | \mathbf{\hat{p}} | \psi^{\left(\lambda\right)}_{m\mathbf{k}}\rangle \langle \psi^{\left(\lambda\right)}_{m\mathbf{k}} | \partial V^{\left(\lambda\right)}/\partial \lambda | \psi^{\left(\lambda\right)}_{n\mathbf{k}}\rangle \over \left( \epsilon^{\left(\lambda\right)}_{n\mathbf{k}}- \epsilon^{\left(\lambda\right)}_{m\mathbf{k}} \right)^{2}}+ \mathrm{c.c.}$

where m_e and e are the electronic mass and charge, N is the number of unit cells in the crystal, Ω₀ is the unit cell volume, M is the number of occupied bands, p is the momentum operator, and the functions ψ^(λ)_nk are the usual Bloch solutions to the crystalline Hamiltonian. Within Kohn-Sham density-functional theory, the potential V^(λ) is to be interpreted as the Kohn-Sham potential V^(λ)_KS, where λ parameterizes some change in this potential, for instance due to the displacement of an atom in the unit cell.

King-Smith and Vanderbilt have cast this expression in a form in which the conduction band states ψ^(λ)_mk no longer explicitly appear, and they show that the change in polarization along an arbitrary path, can be found from only a knowledge of the system at the end points

$\label{PolarizationChange2} \Delta \mathbf{P}_{e}= \mathbf{P}^{\left(\lambda_{2} \right)}_{e} - {\bf P}^{\left(\lambda_{1}\right)}_{e}$

\noindent with

$\label{Polarization} \mathbf{P}^{\left(\lambda\right)}_{e}=-{if|e|\over 8\pi^{3}} \sum^{M}_{n=1} \int_{BZ} d^{3}k \langle u^{\left(\lambda\right)}_{n\mathbf{k}} |\> \nabla_{\mathbf{k}} \>| u^{\left(\lambda\right)}_{n\mathbf{k}} \rangle$

\noindent where $f$ is the occupation number of the states in the valence bands, $u^{\left(\lambda\right)}_{n\mathbf{k}}$ is the cell-periodic part of the Bloch function $\psi^{\left(\lambda\right)}_{n\mathbf{k}}$, and the sum $n$ runs over all $M$ occupied bands.

The physics behind Eq.\ (\ref{Polarization}) becomes more transparent when this expression is written in terms of the Wannier functions of the occupied bands,

$\label{Wannier} \mathbf{P}^{\left(\lambda\right)}_{e}=-{f |e| \over \Omega_{0}} \sum^{M}_{n=1} \langle W^{\left(\lambda\right)}_{n} | \mathbf{r} |W^{\left(\lambda\right)}_{n} \rangle$

\noindent where $W_{n}$ is the Wannier function corresponding to valence band $n$.

Eq.\ (\ref{Wannier}) shows the change in polarization of a solid, induced by an adiabatic change in the Hamiltonian, to be proportional to the displacement of the charge centers $\mathbf{r}_{n} = \langle W^{\left(\lambda\right)}_{n} | \mathbf{r} | W^{\left(\lambda\right)}_{n} \rangle$, of the Wannier functions corresponding to the valence bands.

It is important to realize that the polarization as given by Eq.\ (\ref{Polarization}) or (\ref{Wannier}), and consequently the change in polarization as given by Eq.\ (\ref{PolarizationChange2}) is only well-defined modulo $fe\mathbf{R}/\Omega_{0}$, where $\mathbf{R}$ is a lattice vector. This indeterminacy stems from the fact that the charge center of a Wannier function is only invariant modulo $\mathbf{R}$, with respect to the choice of phase of the Bloch functions.

In practice one is usually interested in polarization changes $|\Delta{\bf P}_{e}| \ll |fe\mathbf{R}_{1}/\Omega_{0}|$, where $\mathbf{R}_{1}$ is the shortest nonzero lattice vector. An arbitrary term $fe\mathbf{R}/\Omega_{0}$ can therefore often be removed by simple inspection of the results. In cases where $|\Delta\mathbf{P}_{e}|$ is of the same order of magnitude as $|fe{\bf R}_{1}/\Omega_{0}|$ any uncertainty can always be removed by dividing the total change in the Hamiltonian $\lambda_{1} \rightarrow \lambda_{2}$ into a number of intervals.

Related Tags and Sections

LBERRY, IGPAR, NPPSTR, LPEAD, IPEAD, LCALCPOL, LCALCEPS, EFIELD_PEAD, DIPOL

References

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