Theory of electric, magnetic, and toroidal polarizations in crystalline solids
with applications to hexagonal lonsdaleite and cubic diamond

It is well-known that a proper definition of electric dipolarization as a bulk property is nontrivial [res94, res10]. The naive electromagnetic definition of electric dipolarization as the dipole moment of a unit cell is unsatisfactory as this quantity generally depends on the arbitrary choice of a unit cell [mar74]. Thus a proper description of the electromagnetic properties of solids requires tools beyond those supplied by classical electrodynamics.

Important progress has been made by introducing the modern theories of electric dipolarization and magnetization where geometric phases are used to quantify these dipole densities (multipole order $\ell=1$) independently of the choice of unit cell [kin93, res94, res10, res20]. Within the modern theory, the electric dipolarization has a clear physical interpretation relative to a reference state. However, for systems showing a spontaneous electric dipolarization, the interpretation and observability of this quantity have remained ambiguous. Also, it is a significant challenge to extend the modern-theory approaches to multipole densities of higher multipole order $\ell>1$ [gli22].

Even before the advent of the modern theories, some early studies did not make any reference to electromagnetism in their investigation of dipolarizations in materials, as they recognized how crystal symmetry allows one to identify crystal structures that permit a bulk electric dipolarization (so-called polar crystals include pyroelectric and ferroelectric media [voi10, nye85, lit86]) or a bulk magnetization (ferromagnetic crystals [tav56, tav58, cor58a]). According to Neumann’s principle (see Refs. [voi10, nye85] for seminal discussions), the crystal classes can be rigorously divided into those that permit a macroscopic electric dipolarization or magnetization, and those for which these phenomena are forbidden. Magnetic crystal classes that do not permit a magnetization have been generically associated with antiferromagnetism [cor58a].

The approach pursued in the present work overcomes the unsatisfactory electromagnetic definition of electric and magnetic multipole densities that is inadequate for crystalline solids; we rely entirely on symmetry to extend the notion of bulk dipolarization and magnetization to electric and magnetic multipole densities of higher orders $\ell>1$. To this end, we treat the black-white symmetries space inversion symmetry (SIS) and time inversion symmetry (TIS) on the same footing [lud96]. Moreover, we treat electric and magnetic order on the same footing. Our systematic theory provides a broader framework for recent efforts to study electric and magnetic multipolar order in solids [wat18, hay18c, yat21, bho22, bho22a] and lends itself for wider application in the context of complex materials [san09, kur09, fu15, nor15, hay19a, vol21]. Throughout this work, we focus on systems that are in thermal equilibrium, thus leaving aside the interesting topic of current-induced multipolar order [tah22].

In the following, the term polarization refers to a general realization of bulk multipolar order with $\ell\geq 0$. Four types of polarizations — electric, magnetic, electrotoroidal, and magnetotoroidal — are presented in Table 1. The signature $ss^{\prime}$ indicates how a polarization behaves under space inversion (even/odd if $s=+/-$) and time inversion (even/odd if $s^{\prime}=+/-$). The electric (magnetic) polarization of order $\ell=1$ corresponds to the electric dipolarization (magnetization), having signature $-+$ ($+-$). This general group-theoretical definition of electric and magnetic multipolar order is independent of the arbitrary choice of a unit cell. It is also independent of a material’s electrodynamic properties and, therefore, applies to both insulators and metals.

A comprehensive classification of ways to combine polarizations is based on their transformation behavior under SIS, TIS, and the combined inversion symmetry (CIS) represented by the operations $i$, $\theta$, and $i\theta$, respectively. Five distinct inversion groups can be formed from these symmetry operations, as defined in Table 2, where we also indicate which polarizations are permitted under these groups. The only types of polarizations permitted under the full inversion group $C_{i\times\theta}\equiv C_{i}\times C_{\theta}$ are even/̄$\ell$ electric polarizations called parapolarizations, and we label the associated matter category parapolar. On the opposite extreme, the trivial inversion group $C_{1}$ containing only the identity $e$ as a symmetry element allows all polarization types, and we label the category of polarized matter associated with $C_{1}$ multipolar. Each of the remaining inversion groups contains strictly one inversion operation $i$, $\theta$, or $i\theta$ as a symmetry element. As a result, only a single type of electric or magnetic polarization is symmetry-allowed: odd/̄$\ell$ electropolarizations for the time inversion group $C_{\theta}$, odd/̄$\ell$ magnetopolarizations for the space inversion group $C_{i}$, and even/̄$\ell$ antimagnetopolarizations for the combined inversion group $C_{i\theta}$. Our unified treatment reveals a far-reaching correspondence between electric and magnetic order in crystalline solids.

Our theory enables us to identify measurable indicators that signal the presence of electric and magnetic order in the electronic band structure. Some of these indicators are quite familiar, though their relation to electric and magnetic order was not established previously.

A classical example for a bulk crystal with a spontaneous electric dipolarization is wurtzite [pos90, ber97a]. Wurtzite is also the classical example for a bulk crystal showing the Rashba effect [ras59a]. Generally, the Rashba effect is characterized by a term ${\bm{\mathrm{\alpha}}}\cdot({\bm{\mathrm{k}}}\times{\bm{\mathrm{\sigma}}})$, where $\hbar{\bm{\mathrm{k}}}$ is crystal momentum and the vector ${\bm{\mathrm{\sigma}}}$ of Pauli spin matrices represents the spin degree of freedom of the Bloch electrons [ras06]. By definition, the Rashba effect is proportional to a polar vector ${\bm{\mathrm{\alpha}}}$. In confined geometries, the vector ${\bm{\mathrm{\alpha}}}$ is commonly associated with a built-in, or external, electric field that controls the magnitude of the Rashba effect [las85, win03]. In bulk materials like wurtzite, no such intuitive picture exists for the vector ${\bm{\mathrm{\alpha}}}$, and its physical meaning has remained unclear [ras59a, voo96]. The Rashba effect exists in all bulk crystal structures that belong to one of the ten polar crystal classes [nye85]. We argue that, in these structures, the vector ${\bm{\mathrm{\alpha}}}$ represents the bulk dipolarization ($\ell=1$). The Rashba effect is thus a measure of the spontaneous electric dipolarization in bulk wurtzite structures and other polar crystals. Similarly, the Dresselhaus term [dre55] in bulk zincblende structures represents an electric octupolarization ($\ell=3$), and Dresselhaus spin splitting provides a measure of the spontaneous electric octupolarization in bulk zincblende structures.