Berry phase polarization and orbital magnetization responses of insulators: Formulas for generalized polarizabilities and their application

TL;DR

The paper develops a projector-based perturbative framework to compute generalized polarizabilities in insulating crystals, linking Berry phase polarization, orbital magnetization, and Hall-vector polarizability under static perturbations. It provides compact formulas for two- and four-band models and extends to general $N$-band systems, enabling transparent expressions in terms of the Hamiltonian and perturbations. Through explicit 1D and 2D magnetoelectric applications—such as antiferromagnets, bilayer systems, Dirac fermions, and altermagnets—it reveals geometric and interband contributions, curvature-dipole interpretations, and Maxwell relations between conjugate polarizabilities. The results offer a versatile toolkit for predicting magnetoelectric and strain-engineered responses in topologically nontrivial insulators, with potential guidance for material realizations and experimental probes.

Abstract

Condensed matter physics is often concerned with determining the response of a solid to an external stimulus. This paper revisits and extends the microscopic formalism for calculating response coefficients -- here referred to as (generalized) polarizabilities -- in crystalline electronic insulators. The main focus is on the Berry phase polarization and orbital magnetization, for which we obtain general formulas describing the linear response to an arbitrary (but static and uniform) perturbation. The response of an arbitrary lattice-periodic observable (e.g. spin, layer pseudospin) to electric and magnetic fields is also examined, and serves as a basis for mircoscopically establishing Maxwell relations between conjugate generalized polarizabilities. We furthermore introduce and examine the notion of Berry curvature or Hall vector polarizability, i.e., the response of the Berry curvature to a general perturbation, and show how it relates to Berry phase polarization and orbital magnetization responses. For all polarizabilities considered, we obtain simplified formulas applicable to two- and four-band models, expressed directly in terms of the Hamiltonian and the perturbation. Three specific applications of these formulas are discussed: (i) a computation of the magnetoelectric polarizabilities of model antiferromagnets in one and two dimensions; (ii) a general proof of (quasi)topological signatures in the polarizabilities of Dirac fermions in two dimensions; (iii) a calculation of the strain-induced Berry curvature polarizability in an altermagnet.

Figures (6)

• Figure 1: Schematic sketches of the energy bands of the unperturbed Hamiltonian $H_0$. (a) All valence bands and all conduction bands have the same energy ${\varepsilon}^v_{\bf k}$ and ${\varepsilon}^c_{\bf k}$ ("degeneracy"), and ${\varepsilon}^v_{\bf k}+{\varepsilon}^c_{\bf k} = \text{constant}$ ("reflection"). (b) A spectrum which has the property of degeneracy but not reflection.
• Figure 2: (a) Depiction of the Néel antiferromagnetic zigzag chain in 1D. Red arrows indicate the ordered moments, which are chosen to point in the $\hat{y}$ direction. (b) Calculation of the Berry phase polarization $P_x$ as a function of Zeeman field $B_x$ for the 1D antiferromagnetic chain model, as defined in Eqs. \ref{['eq:H_0-1D-A']} and \ref{['eq:W-1D-A']}. Red and blue curves correspond to $N_y/t_1 = 0.05, 0.15$ and $N_y/t_1 = -0.05, -0.15$, respectively. (We have set $t_{\text{SO}} = 0.8t_1$.) The dashed straight lines correspond to the linear polarizabilities $\partial P_x /\partial B_x$ computed using Eq. \ref{['eq:dPx/dBx']}. (c) Calculation of the Berry phase polarization $P_x$ as a function of Néel order parameter $N_y$ in the presence of a finite Zeeman field $B_x$. Here red and blue curves correspond to $b_x/t_1 = 0.1, 0.3$ and $b_x/t_1 = -0.1, -0.3$, respectively. The dashed straight lines correspond to the linear polarizabilities $\partial P_x /\partial N_y$ computed using Eq. \ref{['eq:dPx/dNy']}. As $N_y\rightarrow 0$ the polarization $P_x/e$ tends to the value $-\frac{1}{2} \text{sgn}(b_x)$, which reflects the topological half-quantized polarization of the inversion symmetric insulating state at $N_y=0$.
• Figure 3: (a) Schematic representation of two linear Dirac band crossings described by Eq. \ref{['eq:H_0-Dirac']}. The two Dirac nodes have opposite helicity (indicated as red and blue) and can be tilted when they occur at points of low symmetry in the Brillouin zone. (b) In the presence of a perturbation the Dirac points can be shifted in momentum and in energy, as described by Eq. \ref{['eq:W-Dirac']}.
• Figure 4: (a) The unit cell of the two-dimensional Lieb lattice has two magnetic sites, shown in black and labeled $A$ and $B$, and one non-magnetic site, shown in white. The ordered moments of the altermagnetic collinear Néel state are indicated as red and blue arrows. (b) Energy spectrum of the Lieb lattice model defined by Eq. \ref{['eq:Lieb-def']}. We have used the parameters $(t_0,t_d,t_{\text{SO}},N_z) = (0.5t_1,2.0t_1,0.75t_1,4.0t_1)$; blue and red bands correspond to $\sigma=\uparrow$ and $\sigma=\downarrow$, respectively.
• Figure 5: Berry curvature and Berry curvature polarizability of the Lieb lattice model. (a) and (b) show the Berry curvature $\Omega_{xy}$ of the valence band in the $\sigma=\uparrow$ and $\sigma=\downarrow$ sector, respectively. (c) and (d) show the Berry curvature (or Hall vector) polarizability $\partial_\lambda \Omega_{xy}$ of the valence band in the $\sigma=\uparrow$ and $\sigma=\downarrow$ sector, respectively. The Hall vector polarizability is calculated using the formula of Eq. \ref{['eq:dOmega-2band']}, with model parameters set to $(t_0,t_d,t_{\text{SO}},N_z) = (0.5t_1,2.0t_1,0.75t_1,4.0t_1)$ [See Eq. \ref{['eq:Lieb-def']}].