🔬 mesoscale physics

Exact fluctuation relations in voltage- and temperature-biased Laughlin-edge constrictions

This paper establishes that exact non-equilibrium fluctuation-dissipation relations connect experimentally accessible current correlations to tunneling noise and conductance in voltage- and temperature-biased Laughlin edge states, proving their validity for arbitrary tunneling strengths and biases while enabling the reliable reconstruction of local tunneling noise from downstream measurements.

Original authors: Gu Zhang, Gabriele Campagnano, Domenico Giuliano, Igor Gornyi, In`es Safi

Published 2026-09-01

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Gu Zhang, Gabriele Campagnano, Domenico Giuliano, Igor Gornyi, In`es Safi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the strange, ultra-cold world of quantum physics, electrons do not always behave like individual particles. Under extreme conditions, they can organize into a collective state where they move as a single, fluid-like entity. This phenomenon occurs in materials subjected to strong magnetic fields, creating what scientists call fractional quantum Hall states. In these states, the edges of the material act like one-way highways for electric charge. Because the electrons are so tightly coordinated, they can carry fractional charges—fractions of the electron's standard charge—and follow paths that are protected from backscattering, much like a train on a dedicated track that cannot be derailed by small obstacles. Understanding how these exotic states carry electricity and heat is crucial for developing future technologies, including quantum computers that rely on these robust, topological properties.

However, a significant gap exists between the theoretical models physicists use to describe these systems and the actual experiments performed in the lab. Theories often focus on the tiny, local point where two of these edge highways come close enough to allow particles to jump across, a process known as tunneling. This local tunneling generates electrical noise, a fluctuation in the current that holds the secrets to the particles' properties. Yet, in a real laboratory, scientists cannot easily place a sensor right at that tiny tunneling point without disturbing the delicate system. Instead, they must measure the current far downstream, after the particles have traveled away from the tunneling site. The challenge has been to mathematically connect what is measured far away with what actually happens at the tunneling point, especially when the system is pushed far from its calm, equilibrium state by strong voltages or temperature differences.

A team of researchers has now bridged this gap by deriving a set of exact rules that link the local tunneling noise to the measurable currents downstream. They studied a system where two chiral edge states, carrying fractional charges, are brought together at a constriction point. In their setup, these two edges are held at different temperatures and different electrical voltages, creating a complex, non-equilibrium environment. The researchers demonstrated that the noise generated locally at the tunneling point can be precisely reconstructed by measuring the correlations between the currents flowing out of the two edges. These correlations include both the fluctuations within a single edge and the synchronized fluctuations between the two edges. Crucially, their mathematical proof shows that this relationship holds true regardless of how strong the tunneling is, how large the voltage difference is, or how different the temperatures are.

The work goes further by addressing realistic complications that previous theories often ignored. In many real-world experiments, the tunneling region is not a single mathematical point but has a finite size, and the probability of a particle jumping across can depend on its energy or momentum. The researchers proved that their connecting rules remain valid even when the tunneling region is extended and when the tunneling probability varies with momentum. They also examined a scenario where the tunneling probability changes depending on the applied voltage. They found that the original rules still hold perfectly, provided that energy is conserved during the tunneling process. If energy is absorbed or released in a way that breaks conservation at the tunneling site, a specific correction term must be added, but the fundamental structure of the relationship remains intact.

This discovery provides a robust and reliable method for experimentalists to characterize the behavior of these exotic quantum states without needing to probe the most sensitive part of the system directly. By measuring the currents and their fluctuations at the output, researchers can now accurately infer the local noise and conductance at the tunneling point. This is particularly important for the emerging field of "delta-T" noise, which studies how temperature differences drive electrical currents and fluctuations. The ability to handle different temperatures on the two edges makes these new relations directly applicable to these cutting-edge experiments. The findings confirm that the local physics of the tunneling event is faithfully encoded in the downstream measurements, offering a clear and exact path to understanding non-equilibrium transport in chiral edge states.

Technical Summary: Exact Fluctuation Relations in Voltage- and Temperature-Biased Laughlin-Edge Constrictions

Problem Statement
Quantum transport in fractional quantum Hall (FQH) systems, particularly through quantum point contacts (QPCs), serves as a primary tool for probing topological order and quasiparticle statistics. While theoretical models often focus on the local tunneling-current noise generated at the QPC, experimental measurements typically access auto- and cross-correlations of currents collected at downstream drains. Establishing a rigorous, non-perturbative link between these local theoretical quantities and experimentally accessible downstream correlations is challenging, especially in non-equilibrium regimes involving strong drives, temperature gradients, and complex scattering geometries. Previous exact relations were often limited to weak tunneling, identical edge temperatures, energy-independent tunneling amplitudes, or specific filling fractions (e.g., $\nu=1/2$ ).

Methodology
The authors employ a non-perturbative framework based on generalized non-equilibrium linear response theory and bosonization techniques, building upon methods established in Refs. [72, 78, 79, 107]. The system consists of two chiral Laughlin edges (labeled $u$ and $d$ ) with distinct voltages ( $V_u, V_d$ ) and temperatures ( $T_u, T_d$ ) communicating via a QPC.

Key methodological steps include:

Hamiltonian Formulation: The system is described by free chiral Luttinger liquid Hamiltonians ( $H_0$ ), bias terms ( $H_{bias}$ ), and a tunneling Hamiltonian ( $H_T$ ). The authors generalize $H_T$ to accommodate spatially extended QPCs and momentum-dependent transmission amplitudes ( $\xi(x, x')$ ).
Generalized Fluctuation-Dissipation Relations (FDRs): Utilizing a generalized Kubo-type relation for non-equilibrium stationary states, the authors derive an exact identity connecting the frequency-dependent noise matrix $C_{\alpha\alpha'}(\omega)$ and the generalized conductance matrix $G_{\alpha\alpha'}(\omega)$ . This relation, $C_{\alpha\alpha'}(\omega) - C_{\alpha'\alpha}(-\omega) = \hbar\omega[G_{\alpha\alpha'}(\omega) + G_{\alpha'\alpha}(-\omega)]$ , holds for arbitrary stationary states and interaction strengths, provided the bias Hamiltonian depends linearly on the bias.
Derivation of Zero-Frequency Relations: By analyzing the zero-frequency limit ( $\omega \to 0$ ) and utilizing the specific chiral properties of the Laughlin edges, the authors relate the commutator of tunneling current operators to the differential conductance. This allows the elimination of non-observable commutator terms in favor of measurable conductance and temperature parameters.
Handling Generalizations: The derivation explicitly addresses:
- Temperature Gradients: Treating $T_u \neq T_d$ by defining temperatures at the upstream reservoirs, assuming the scattering region is smaller than the equilibration length but larger than the thermal diffusion length.
- Momentum Dependence: Incorporating extended QPCs where the tunneling amplitude depends on momentum.
- Bias Dependence: Investigating cases where the tunneling amplitude $\xi$ depends explicitly on the applied voltage bias, distinguishing between energy-conserving and inelastic scattering scenarios.

Key Contributions and Results
The paper establishes exact, non-perturbative zero-frequency fluctuation relations connecting the local tunneling-current noise ( $S_T$ ) to experimentally accessible downstream current correlations ( $S_{ud}$ for cross-correlation and $S_{\alpha\alpha}$ for auto-correlation). The primary results are summarized in Eq. (19):

Cross-Correlation Relation:
$S_T(0) = -S_{ud}(0) + k_B(T_u + T_d)G$
where $G = \partial I_T / \partial V$ is the tunneling conductance. This relation holds for arbitrary voltage bias, tunneling strength, and temperature differences between edges.
Auto-Correlation Relation:
$S_T(0) = S_{\alpha\alpha}(0) + 2k_B T_\alpha G - 4G_0 k_B T_\alpha$
where $G_0 = \nu e^2/h$ is the quantized conductance of a single edge.

Significance and Claims
The authors claim these results provide a robust, model-independent route to reconstruct the local tunneling noise (which is difficult to measure directly) from downstream correlations. The significance of the work lies in its extension of previous exact relations to experimentally realistic conditions that were previously excluded from exact analytical treatments:

Temperature Gradients: The relations remain exact for $T_u \neq T_d$ , making them directly applicable to the study of $\Delta T$ noise.
Spatial and Momentum Complexity: The validity extends to spatially extended QPCs and momentum-dependent transmission amplitudes, provided the scattering region is well-defined relative to thermal diffusion lengths.
Strong Coupling: The relations are exact for arbitrary tunneling strengths, not limited to the weak-tunneling regime.
Bias Dependence: The authors show that if the transmission amplitude depends on the bias, the relations remain valid provided energy is conserved within the tunneling area. If energy is not conserved (inelastic scattering), a specific correction term $\chi(\xi, V_u, V_d)$ is derived (Eq. 57).

The paper emphasizes that these findings bridge the gap between theoretical calculations of local noise and experimental measurements of downstream correlations, offering a reliable method to characterize non-equilibrium transport in chiral edge states without relying on microscopic details of the strongly correlated system. The work is presented as a complementary approach to non-equilibrium bosonization methods that typically compute noise perturbatively; instead, this work derives exact identities encoding the noise within measurable correlations.

Technical Summary: Exact Fluctuation Relations in Voltage- and Temperature-Biased Laughlin-Edge Constrictions

More like this