Quantum Metric-Induced Nonlinear Thermal Noise

• Quantum metric-induced nonlinear thermal noise arises from the intrinsic geometric tensor of quantum states, fundamentally altering noise properties in quantum systems.
• It exhibits a nonlinear field dependency with resonant peaks at ω = 2|μ| and diverges at low temperatures, distinguishing it from conventional Johnson–Nyquist noise.
• Experimental investigations in Dirac and Weyl materials validate the theory, enabling noise spectroscopy to directly extract quantum geometric features.

Quantum metric-induced nonlinear thermal noise refers to a class of current or metric fluctuations in quantum systems where the amplitude, functional form, or temperature dependence is intrinsically governed by the quantum geometric tensor—specifically, the quantum metric—of the underlying Bloch bands or quantum state manifold. This effect emerges in regimes where field-induced modifications to quantum states and selection rules intertwine with Fermi surface properties, resulting in nonlinear and sometimes anomalous noise characteristics that are inaccessible via conventional (linear, relaxation-limited) noise mechanisms.

1. Theoretical Foundations and Formalism

The quantum metric is defined as the symmetric part of the quantum geometric tensor for Bloch bands:

$$g_{ab}(k) = \mathrm{Re} \langle \partial_a u_n(k) | [1 - |u_n(k)\rangle \langle u_n(k)|] | \partial_b u_n(k) \rangle,$$

where $|u_n(k)\rangle$ are cell-periodic parts of Bloch states, and derivatives are with respect to crystal momentum $k_a$.

Quantum metric-induced nonlinear thermal noise is distinguished from Johnson–Nyquist noise, which exhibits a temperature scaling $S_{JN} = 2 k_B T G$ and vanishes as $T \to 0$. In contrast, under suitable electron-photon or coherent field perturbations, the DC thermal noise (DTN) includes a contribution $S_T^{ab}$ such that

$$S_T^{ab} = \frac{1}{2} \sum_{mn} \int_{\mathrm{BZ}} \!\! dk\, J^a_{mn} J^b_{nm} [ (f_n - f_m)^2 + f_n (1-f_n) + f_m (1-f_m) ],$$

where $J^a = -e \partial H/\partial k_a$, $f_n$ is the Fermi function, and the first term gives the shot noise (DSN), while the $f_n(1-f_n)$ terms yield the nonlinear, quantum-metric–induced DTN (Xiang et al., 17 Dec 2024).

Electron–photon (or field) interactions, particularly under monochromatic or static fields, select for resonant interband transitions via $\delta(\omega - \omega_{mn})$. At low temperatures, the expressions

$$S_T = -T \int_{\mathrm{BZ}} dk\, [f_n'' - f_m''] N^{(2)}_{nm} \delta(\omega - \omega_{mn})$$

demonstrate that $S_T$ can persist or even diverge at $T \to 0$, in strong contrast to classical noise, whenever the quantum metric $g_{ab}(k)$ appears in the prefactor $N^{(2)}_{nm}$ and selection rules pin $k$ to shells on the Fermi surface.

2. Origin of Anomalous Nonlinear Noise and Low-Temperature Behavior

In two-band Dirac models, the DC thermal noise decomposes into “injection” and “shift” terms. The principal result is that, as $T \to 0$,

• The injection term $S_T^{(1)}(\omega)$ tends to a $\delta$-function at the resonant frequency $\omega = 2|\mu|$,
• The shift term $S_T^{(2)}(\omega)$ exhibits a $1/T$ divergence, producing two antisymmetric peaks centered at $\omega = 2|\mu|$.

Explicitly,

$$\mathcal{D}(\omega) = \pm \left[\frac{\hbar \omega}{4 k_B T}\right] \mathrm{sech}^2 \left(\frac{\hbar \omega \mp 2\mu}{4 k_B T}\right) \tanh\left(\frac{\hbar \omega \mp 2\mu}{4 k_B T}\right)$$

and for $T \rightarrow 0$ this yields a non-vanishing, even divergent, noise amplitude at resonance.

The appearance of $\delta(\omega - \omega_{mn})$ in these expressions locks the noise generation to interband transitions that match the photon (or field) energy, imposing a geometric constraint determined by the Fermi surface and resulting in robust, temperature-anomalous noise.

3. Quantum Metric as the Governing Geometric Quantity

The quantum metric enters the noise expressions through interband matrix elements in both two-band models and in general multiband systems. For bands $n$ and $m$,

$$g^{bc}_{nm}(k) = r^b_{nm} r^c_{mn} + r^c_{nm} r^b_{mn},\qquad r^b_{nm} = i\langle u_n | \partial_{k_b} u_m \rangle$$

appear in all $N^{(i)}$ prefactors for the nonlinear terms.

In \textit{PT}-symmetric systems, the Berry curvature vanishes ($\Omega(\mathbf{k}) = 0$), but the quantum metric, being even under both parity and time reversal, survives. Second-order corrections to current and noise, which are forbidden to originate from Berry curvature in such systems, are governed exclusively by the quantum metric (Bhowmick et al., 25 Sep 2025). The central formula for the quantum-metric–induced second-order thermal noise is

$$S^{(2)}_{ij}(T) = e^2 k_B T\, E_c E_d \! \int\! \frac{d^d k}{(2\pi)^d}\, \partial_{k_i} \partial_{k_j} [ g_{cd}(\mathbf{k}) f^0(\varepsilon) ]$$

where $f^0$ is the equilibrium Fermi distribution.

Thus, quantum-metric–induced noise serves as an intrinsic, disorder-independent signature of the quantum geometry of wavefunctions.

4. Nonlinearity and Experimental Characteristics

Quantum metric-induced noise is fundamentally nonlinear. It emerges at second order in the applied field (either static electric or optical), with no linear contribution in noise in \textit{PT}-symmetric structures due to symmetry exclusion of lower-order geometric terms. Key characteristics are:

• Nonlinearity in field amplitude: $S^{(2)} \sim E^2$ for static fields, or $\sim |E|^2$ for monochromatic light.
• Resonant structure: In systems with time-reversal symmetry, the noise universally peaks at photon energy $\hbar\omega = 2|\mu|$, corresponding to the chemical potential. The noise is locked to interband transitions by $\delta(\omega-2d(k))$.
• Persistence and anomaly at low $T$: DTN remains finite or even divergent as $T \to 0$, unlike classical thermal or Johnson–Nyquist noise.

The table below summarizes the main distinctions:

Feature	Johnson–Nyquist Noise	Quantum Metric-Induced DTN
Field dependence	Linear ($\sim E$)	Nonlinear ($\sim E^2$)
Temperature scaling	$\propto T$	Persists, can diverge as $1/T$
Origin	Scattering-limited, local	Interband, geometric
Resonant structure	None (white)	Peaks at $\omega=2|\mu|$

5. Material Implementations and Case Studies

Quantum metric-induced nonlinear thermal noise manifests in a range of quantum materials. Notable systems include:

• Dirac Materials: Two-dimensional graphene ($H = v_F(k_x \sigma_x + k_y \sigma_y)$, $g^{bc}(k) = (k^2 \delta_{bc} - k_b k_c)/(2 k^4)$) and surfaces of 3D topological insulators both host Dirac cones whose quantum metric structure directly enters the nonlinear noise amplitude (Xiang et al., 17 Dec 2024).
• Weyl Semimetals: Three-dimensional Weyl fermion systems ($H = v_F \vec{k} \cdot \vec{\sigma}$) feature quantum-metric–driven noise with tensors $\eta_{L,2}^{aaa}(\omega)$ and $\eta_{L,2}^{abb}(\omega)$ determined by the Brillouin-zone metric integral at resonance.
• PT-symmetric Antiferromagnets: In systems such as two-dimensional CuMnAs, noise spectroscopy isolates $E^2$-dependent contributions. Using the four-band tight-binding models, the theory predicts a sharp noise peak as chemical potential passes through the conduction band edge, with magnitude in the $10^{-27}$--$10^{-25}~$A$^2~$Hz$^{-1}$ range for accessible fields. The quantum metric can be extracted via four-probe noise measurements, providing a direct characterization independent of extrinsic scattering (Bhowmick et al., 25 Sep 2025).

Measurement protocols typically require low-frequency current noise detection under d.c. or optical drive, with temperature and chemical potential (gate voltage) sweeps to map the characteristic scaling and quantum-metric resonances.

6. Total Noise Spectrum, Resonances, and Physical Interpretation

The total noise spectrum in the presence of both DSN and anomalous DTN contributions is governed by the quantum metric at resonance:

$$S_{\mathrm{tot}}(\omega) = [\mathcal{D}(\omega) + \mathcal{D}_1(\omega)]\, \mathcal{N}^{(2)},$$

where $\mathcal{D}(\omega)$ is the anomalous $1/T$-divergent weight from DTN and $\mathcal{D}_1(\omega)$ captures the injected shot noise; $\mathcal{N}^{(2)}$ is a geometric factor set by the Brillouin zone integral over quantum metric and velocity prefactors.

Both theory and explicit calculations find a universal peak at $\omega_p = 2|\mu|$, reflecting resonant electron–hole creation at the chemical-potential surface. This peak is sharpened by the $1/T$ divergence as $T \rightarrow 0$, signaling interband quantum-metric fluctuations locked to the optical selection rule (Xiang et al., 17 Dec 2024). The magnitude and antisymmetric shape of the noise spectrum around this resonance serve as unambiguous, geometry-driven experimental signatures.

A plausible implication is that quantum-metric–induced nonlinear noise provides a means to access otherwise elusive quantum geometric quantities in solids and insulators, without reliance on Hall or Berry curvature effects, especially in symmetry-protected phases such as PT-symmetric antiferromagnets.

7. Extensions, Cosmological Contexts, and Broader Significance

While the above mechanisms arise in crystalline solids, analogous metric-induced nonlinear noise phenomena appear in quantum field theory in curved backgrounds, where thermal fluctuations of quantum fields (e.g., thermal Maxwell) induce metric fluctuations related to the geometric structure of quantum states. In stochastic gravity, the variance of induced metric perturbations is determined by the thermal noise kernel, itself governed (in the conformal case) by the quantum stress–energy geometry (Fröb et al., 29 Jun 2025). At high temperatures or curvatures, these effects can become nonlinear and set fundamental noise floors for measurement of gravitational waves or cosmological tensors.

This suggests that quantum metric-induced nonlinear noise is a pervasive and unifying concept across both condensed matter and high-energy/quantum gravity contexts, providing a geometric bridge between quantum information, transport, and foundational fluctuation phenomena.