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The fluctuation-dissipation theorem (FDT) claims that the current fluctuation in a macroscopic equilibrium system is equal to the product of the temperature and electrical conductivity. This “theorem” was proved for classical systems for all components of fluctuations including off-diagonal fluctuations, namely cross-time correlations between currents flowing in different directions.

However, the validity of the FDT in quantum systems was questioned, because disturbances by quantum measurement often play a crucial role, which was ignored in the proof of the FDT for classical systems. Recently, this long-standing question was formally solved, and the FDT was shown to be violated even when the fluctuation is measured in a way that simulates the classical ideal measurement as closely as possible. However, this formal solution neither gave concrete systems that exhibit the FDT violation nor estimated the magnitude of violation.

We propose a two-dimensional electron system in a magnetic field as a real physical system in which the FDT is violated. We clarify the conditions for large violations and show that the magnitude of violation is macroscopically large. In fact, the FDT for the off-diagonal component is significantly violated in a strong magnetic field at low temperatures, whereas the FDT for the diagonal component holds for any values of the parameters. In the standard setup used in the quantum Hall effect experiments, the off-diagonal current fluctuation is several tens of times larger than the product of temperature and Hall conductivity (off-diagonal conductivity).

Such a large violation implies novel properties of off-diagonal current fluctuations that are yet to be studied. Localized states of electrons contribute to the off-diagonal current fluctuation to the same extent as extended states, and hence, the off-diagonal fluctuation is insensitive to system imperfections. This is in sharp contrast to the Hall conductivity that is very sensitive to the imperfections because only extended states contribute. Moreover, as an application of this finding, we propose a new method for estimating the electron number density by measuring the off-diagonal fluctuation. Because fluctuations are a cause of error and noise, our results are expected to provide fundamental design guidelines for applications.

(Written by K. Kubo, K. Asano, and A. Shimizu)