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# Clarifying the non-uniqueness in Polarization

For ionic and dielectric materials, the polarization (dipole moment per unit cell) is plagued with non-uniqueness. As a 1D example, consider a 1D line of alternating positive and negative charges. If we agree that the dipole is a vector from the negative to the positive charge, then we have two choices for the dipole. This is because every negative charge has two different positive charges in its immediate vicinity. The situation gets more complicated in higher dimensions. This definition of polarization, based on classical electrostatics was deemed incorrect and alternative ideas were researched.

The current approach to determining polarization (popular in the physics community) goes by the name of "The Modern Theory of Polarization". The claim is that while the polarization is non-unique, the change in polarization is unique modulo a constant (called a quantum of polarization). The change is obtained using an adiabatic assumption which gives rise to the berry phase of the changing wave function.

Alternatively, one can consider an energetic/thermodynamic definition of polarization. The thermodynamic equilibrium of a body is determined by the minimization of the Gibbs free energy. The electronic property of the material shows up in the Gibbs free energy via the polarization. Thus the derivative of the Gibbs free energy with respect to an external Electric field is the polarization. This definition seems to have no non-uniqueness associated with it and seemingly takes the polarization for granted.

My paper with Yang Wang, Timothy Breitzman and Kaushik Dayal has come out in JMPS. The paper deals with connecting and clarifying the different definitions of polarization. We first claim (the rigorous proof is kept aside for another paper) that the electrostatic definition of polarization is perfectly fine and the non-uniqueness is an outcome of using infinite bodies. When we have finite bodies, different choices for polarizations will have corresponding surface charge densities; not all the negative charges will have a corresponding positive charge and some choices of dipoles will leave uncompensated charges near the boundaries. Accounting for both of them provides a route to computing the fields, and energy correctly.

In this paper, we go on to show that this definition encompasses the "Modern theory of polarization definition" and the energetic definition of polarization clarifying for each of the cases how the non-uniqueness is dealt with.

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