Quantum Metric Effects in PT-Symmetric Antiferromagnets

• The paper reveals that intrinsic nonlinear thermal noise is governed solely by the quantum metric due to the suppression of Berry curvature in PT-symmetric antiferromagnets.
• It employs dc electric field-induced corrections to the Berry connection polarizability tensor and band energy, highlighting temperature scaling and relaxation-time independence.
• The study establishes CuMnAs as a key material platform where noise spectroscopy effectively probes band geometry and quantum phase characteristics.

Quantum metric induced nonlinear thermal noise in PT-symmetric antiferromagnets refers to an intrinsic contribution to electrical current fluctuations that arises from the nontrivial geometry of Bloch wavefunctions—specifically, the quantum metric—in materials with combined space-time inversion symmetry (𝒫𝒯 symmetry). In these systems, Berry curvature-induced effects vanish identically, elevating the quantum metric as the dominant geometric entity governing transport noise at nonlinear (second-order) response.

1. Quantum Metric and Electric Field-Induced Corrections

Application of a dc electric field modifies both the energy bands and the wavefunction geometry of Bloch electrons. The main geometric response is captured by the Berry connection polarizability (BCP) tensor: $$G_{ab}^{nn}(\mathbf{k}) = 2 \operatorname{Re}\left[ \sum_{\ell \neq n} \frac{A_a^{n\ell}(\mathbf{k}) A_b^{\ell n}(\mathbf{k})}{\epsilon_n(\mathbf{k}) - \epsilon_\ell(\mathbf{k})} \right]$$ where $$A_a^{n\ell}(\mathbf{k}) = \langle n(\mathbf{k})|i \partial_{k_a}|\ell(\mathbf{k})\rangle$$ is the interband Berry connection. This BCP is directly related to the quantum metric (real part of the quantum geometric tensor). Under an applied field, the Berry connection is renormalized: $$A_a^{nn}(\mathbf{k}) \rightarrow A_a^{nn}(\mathbf{k}) + e G_{ab}^{nn}(\mathbf{k}) E_b$$ Band energies also acquire a correction: $$\epsilon_n(\mathbf{k}) \rightarrow \epsilon_n(\mathbf{k}) - \frac{e^2}{2} G_{ab}^{nn}(\mathbf{k}) E_a E_b$$

2. Nonlinear Thermal Noise: Field Expansion and Intrinsic Quantum Metric Effects

Thermal noise—current fluctuations due to thermal agitation—can be expanded as a series in the electric field: $$\eta_{ab}^{\text{th}} = \Gamma_{ab}^{(1)} + \Gamma_{ab}^{(2)} + \mathcal{O}(E^3)$$ The second-order term ($\Gamma^{(2)}$), in PT-symmetric antiferromagnets, separates into extrinsic (relaxation-time dependent) and intrinsic (band-geometric, scattering-time independent) contributions. The intrinsic part, entirely governed by quantum metric, is given by: $$\eta_{ab}^{\text{th,int,BCP}} = \frac{e^4 k_B T}{2} E_c E_d \sum_{n,\mathbf{k}} \left\{ \frac{1}{\hbar}\frac{\partial f_n^{(\text{eq})}}{\partial \epsilon_n} [ v_a^n(\mathbf{k}) \partial_{k_c} G_{bd}^{nn} + v_b^n(\mathbf{k}) \partial_{k_c} G_{ad}^{nn} - v_a^n(\mathbf{k}) \partial_{k_b} G_{cd}^{nn} + (a \leftrightarrow b) ] + \frac{\partial^2 f_n^{(\text{eq})}}{\partial \epsilon_n^2} G_{cd}^{nn} v_a^n v_b^n \right\}$$ where $f_n^{(\text{eq})}(\mathbf{k})$ is the Fermi–Dirac distribution and $v_a^n(\mathbf{k})$ is the band velocity. All terms are manifestly intrinsic and result from Fermi-surface physics, not dissipative relaxation.

3. Role of PT Symmetry: Berry Curvature Suppression

PT symmetry in antiferromagnets enforces the cancellation of Berry curvature ($\Omega^{nn}(\mathbf{k}) = 0$) across the Brillouin zone. Consequently, all Hall-type nonlinear effects and conventional Berry curvature-driven corrections (including nonlinear Hall noise) vanish. The leading geometric response, therefore, arises solely through the quantum metric component of the Berry connection polarizability. This symmetry-forbidden scenario provides an ideal platform for isolating quantum metric-driven effects from other band geometric contributions.

4. Material Example: CuMnAs and Quantum Metric Noise Signature

CuMnAs exemplifies a PT-symmetric antiferromagnet: it features two magnetic sublattices with antiparallel spins and preserves PT, even as both P and T are individually broken. The paper develops a tight-binding model for CuMnAs, calculating the BCP tensor $G_{ab}^{nn}(\mathbf{k})$. Notable observations include:

• BCP tensor peaks near points in k-space where the band gap closes (Dirac points).
• The intrinsic nonlinear noise (from Eq. above) exhibits pronounced peaks as the chemical potential approaches band edges; noise vanishes in the gap regions.
• Intrinsic noise scales with temperature ($k_B T$) and is entirely scattering-time independent.

Thermal noise measurements that reveal these features thus directly probe quantum metric characteristics in the material.

5. Significance for Band Geometry Probes and Quantum Materials

Current fluctuation measurements—specifically nonlinear (second-order in E) thermal noise—serve as an unconventional but powerful window into the quantum geometry of materials. In PT-symmetric antiferromagnets, noise spectroscopy transcends conventional charge transport by unambiguously identifying quantum metric contributions, especially where Berry curvature is symmetry-forbidden. Peaks in the noise spectra as a function of chemical potential reflect underlying band-geometric structure encoded in the quantum metric.

Such measurements:

• Reveal universal, relaxation-time independent quantities tied to the geometry of the Bloch wavefunctions.
• Provide a tool for characterizing topological and quantum geometric phases in PT-symmetric antiferromagnets.
• Enable experiments to extract quantum metric signatures in systems (CuMnAs, etc.) not accessible via conventional Hall or transport probes.

6. Experimental and Theoretical Outlook

Noise spectroscopy focusing on quantum metric-induced nonlinear thermal noise opens pathways for characterizing and exploiting quantum geometry in PT-symmetric antiferromagnetic systems. These results suggest broader exploration in engineered materials where Berry curvature is minimized or suppressed, and quantum metric becomes the sole band-geometric driver of transport phenomena. Future research may leverage these principles for the design of quantum devices where geometric properties, rather than dissipative kinetics, control performance and signal response. In this context, quantum metric-driven thermal noise is an essential experimental observable, connecting abstract geometric principles to concrete measurable quantities in cutting-edge materials (Bhowmick et al., 25 Sep 2025).