Berry Connection Polarizability Tensor

• Berry Connection Polarizability (BCP) tensor is an intrinsic band geometric quantity that measures the electric-field induced polarization of the Berry connection.
• It appears in extended semiclassical theory to govern third-order Hall conductivity, linking the nonlinear response to underlying band geometry in both Hermitian and non-Hermitian systems.
• Model calculations and experiments in materials like FeSe and TaIrTe4 demonstrate its role in modulating nonlinear Hall effects, suggesting applications in advanced electronic devices.

The Berry Connection Polarizability (BCP) tensor is an intrinsic, gauge-invariant band geometric quantity that quantifies the linear susceptibility of the band-resolved Berry connection to an external electric field. It plays a central role in controlling third-order (cubic in field) nonlinear Hall effects in systems with appropriate symmetry constraints, extending the geometric framework that connects Berry curvature and its dipole to lower-order (linear and second-order) Hall responses. The BCP tensor emerges in the extended semiclassical formalism as the key band-structure parameter dictating the magnitude and symmetry of the third-order Hall conductivity in both Hermitian and non-Hermitian quantum materials.

1. Formal Definitions and Tensor Structure

For a crystal described by Bloch eigenstates $\vert u_n(\mathbf{k})\rangle$ and energies $\varepsilon_n(\mathbf{k})$, the Berry connection of band $n$ is

$$\mathcal{A}_{n,a}(\mathbf{k}) = \langle u_n(\mathbf{k}) | i\partial_{k_a} | u_n(\mathbf{k}) \rangle.$$

In a static, spatially uniform field $\mathbf{E}$, this connection acquires a first-order correction: $$\mathcal{A}^{(1)}_{n,a}(\mathbf{k}) = G_{n,ab}(\mathbf{k}) E_b,\quad G_{n,ab}(\mathbf{k}) = \frac{\partial\mathcal{A}^{(1)}_{n,a}}{\partial E_b}.$$ The BCP tensor $G_{n,ab}(\mathbf{k})$ (Hermitian systems) is

$$G_{ab}(\mathbf{k}) = 2\, \mathrm{Re} \sum_{m\neq n} \frac{(\mathcal{A}_a)_{nm} (\mathcal{A}_b)_{mn}}{\varepsilon_n - \varepsilon_m},\quad (\mathcal{A}_a)_{nm} = \langle u_n | i \partial_{k_a} | u_m \rangle.$$

In the context of modern semiclassical theory and experimental work, a more general form (including non-Hermitian band structures) uses the tensor $\alpha_{bc,a}$, defined as the electric-field derivative of the Berry connection dipole: $$\alpha_{bc,a} = \frac{\partial}{\partial E_d} \Gamma_{bc,a}\big|_{E=0},\quad \Gamma_{bc,a} \equiv e^2\sum_n \int [dk]\, f_0(\varepsilon_n(k))\, \partial_{k_a} [A_n^b(k) A_n^c(k)]$$ where $$A_n^a = i\langle u_n(k)|\partial_{k_a} u_n(k)\rangle$$ and $f_0$ is the Fermi–Dirac distribution (Liu et al., 2021, Yang et al., 12 Jun 2025, Qin et al., 10 Nov 2024).

2. Physical Interpretation and Origin in Semiclassical Transport

The BCP tensor describes the field-induced "polarization" of the Berry connection in analogy to a dielectric's field-induced dipole moment. In extended semiclassical dynamics, $G_{ab}$ produces both:

• A second-order Stark energy shift

$$\tilde\varepsilon_n = \varepsilon_n - \frac{1}{2} E_a G_{ab} E_b,$$

• A field-induced correction to the Berry curvature

$$\tilde\Omega_{n,a} = \Omega_{n,a} + \Omega^{(1)}_{n,a},\quad \Omega^{(1)}_{n,a} = \epsilon_{abc} \partial_b G_{cd} E_d,$$

which directly enters the anomalous velocity and thus the third-order Hall current (Liu et al., 2021).

3. Role in Third-Order Hall and Nonlinear Hall Conductivity

The third-order Hall current $j^{(3)}_a$ is given by

$j^{(3)}_a = \chi_{abcd} E_b E_c E_d.$

Here, the symmetric ($\tau$-linear, geometric) part of the third-order Hall conductivity in clean systems is

$$\chi^{I}_{abcd} = \tau\int [d\mathbf{k}]\, \left[ - \partial_a\partial_b G_{cd} + \partial_a\partial_d G_{bc} - \partial_b\partial_d G_{ac} \right] f_0 + \frac{\tau}{2} \int [d\mathbf{k}]\, v_a v_b G_{cd} f_0'',$$

while the $\tau^3$ Drude-like term derives purely from the band dispersion (Liu et al., 2021, Yang et al., 12 Jun 2025). In experimental scaling,

$$S_3 \equiv V_{3\omega}/(V_\omega)^3 = n\, \sigma + \mathcal{C}\, \sigma^3,$$

where $n$ and $\mathcal{C}$ parameterize the BCP and Drude contributions, respectively (Yang et al., 12 Jun 2025).

In non-Hermitian systems, the intrinsic BCP-mediated nonlinear Hall effect is controlled by the BCP density $P_{abc}^{\rm BCP}(\mathbf{k})$ and shows distinct scaling behavior near band singularities (exceptional rings) (Qin et al., 10 Nov 2024).

4. Model Calculations and Material Realizations

In two-dimensional Dirac models, the BCP tensor takes the form

$$G_{ab}(\mathbf{k}) = -\frac{v_x^2 v_y^2}{4 \Lambda^5} \begin{pmatrix} k_y^2 + \Delta^2/v_y^2 & -k_x k_y\ - k_x k_y & k_x^2 + \Delta^2/v_x^2 \end{pmatrix}_{ab},$$

with $$\Lambda(\mathbf{k}) = \sqrt{v_x^2 k_x^2 + v_y^2 k_y^2 + \Delta^2}$$ (Liu et al., 2021). First-principles calculations for monolayer FeSe (Quantum ESPRESSO + Wannier90) reveal $G_{xx}, G_{yy}$ concentrated near $\Gamma$ and $M$, $G_{xy}$ quadrupolar, and a significant magnitude for the BCP-driven third-order conductivity at charge neutrality ($$\chi_\perp/\tau \approx -0.048\,\text{cm}^2\,\text{V}^{-2}\Omega^{-1}\text{s}^{-1}$$).

In Weyl semimetals such as TaIrTe$_4$, the BCP tensor components allowed by the space group (Pmn2$_1$, point group C$_{2v}$) are constrained by symmetry: $$\alpha_{xz,y},\; \alpha_{zx,y},\; \alpha_{xy,z},\; \alpha_{yx,z},\; \alpha_{yz,x},\; \alpha_{zy,x},$$ and the measured third-order Hall signal in experiment directly probes these components (Yang et al., 12 Jun 2025).

Non-Hermitian dissipative Dirac models with an exceptional ring display BCP densities divergent at the singular ring; the scaling of the BCP (intrinsic nonlinear Hall) response falls off as $\gamma^{-1/2}$ with increasing non-Hermiticity $\gamma$, and is completely suppressed in the strongly non-Hermitian limit (Qin et al., 10 Nov 2024).

5. Symmetry Considerations

The presence and allowed components of the BCP tensor are dictated by crystalline and anti-unitary symmetries.

• In materials with time-reversal symmetry but without inversion (or with only a two-fold rotation), the first- and second-order Hall effects vanish, and the BCP-controlled third-order Hall effect becomes the leading transverse signal.
• In C$_{2v}$ symmetry (as in TaIrTe$_4$), only those $\alpha_{bc,a}$ tensors with one index along the mirror axis and two along the orthogonal axes are nonzero (Yang et al., 12 Jun 2025).
• In non-Hermitian models with mirror symmetry, only specific BCP tensor components survive by parity (antisymmetry), such as $\bar{v}_{xyy}^{\mathrm{BCP}} \neq 0$ while $\bar{v}_{xxy} = 0$ (Qin et al., 10 Nov 2024).

6. Experimental Manifestations and Tuning

The third-order nonlinear Hall effect (NLHE), $j^{(3\omega)} \propto E^3$, provides an experimental probe of the BCP tensor. In TaIrTe$_4$, temperature and electric field studies reveal:

• A sign-reversal of the third-order NLHE at $T \approx 23\,\text{K}$, with high-$T$ dominated by BCP and low-$T$ by impurity scattering (Yang et al., 12 Jun 2025).
• Application of an in-plane DC electric field modulates both the BCP tensor and the third-order NLHE, with modulation strength up to 65.3% observed at $4\,\text{K}$ for $E_{dc} = 0.3$ kV/cm.
• The scaling law $S_3 = n\,\sigma + \mathcal{C}\, \sigma^3$ allows clean separation and quantification of the geometric (BCP) versus extrinsic (impurity) contributions.

In non-Hermitian systems, the presence of exceptional rings induces singularities in the BCP density, markedly affecting higher harmonic generation and nonlinear Hall responses (Qin et al., 10 Nov 2024).

7. Implications and Outlook

The BCP tensor encodes a higher-order geometric response intrinsically tied to the quantum metric and interband coherence structure of the electronic bands. Its experimental tunability via electric gating and its distinctive symmetry filtering properties position it as a tool for band-geometry-based device engineering, notably in nonmagnetic systems where lower-order Hall effects are symmetry-forbidden. Demonstrations in monolayer FeSe and TaIrTe$_4$ suggest broad applicability for studying new classes of nonlinear Hall phenomena, and future work in materials with large intrinsic BCP or engineered non-Hermiticity may yield enhanced third-order responses suitable for device applications such as vectorial sensors and multi-state functional logic (Liu et al., 2021, Yang et al., 12 Jun 2025, Qin et al., 10 Nov 2024).