Quantum Violation of the Fluctuation–Dissipation Theorem in Macroscopic Two-dimensional Electronic Systems

The fluctuation–dissipation theorem (FDT) was experimentally presented by Johnson and formulated by Nyquist [1], and is now known as an essential and universal result of the nonequilibrium statistical physics in the linear response regime. The FDT was originally discussed in classical systems as a relationship between the diagonal conductivity and the diagonal current fluctuation. It has also been investigated in quantum systems and for off-diagonal components. As the FDT links the temporal current fluctuation in equilibrium to the conductivity, which is a non-equilibrium quantity, it is regarded as an important theorem in fundamental physics.

While the FDT has been utilized in various systems, there have been several discussions on its violation. For example, the FDT is violated in non-equilibrium systems beyond the linear response regime and complex systems, such as colloidal and glassy systems. In the last three decades, the FDT has been recognized as a fully established textbook formula in the linear response regime, and other relations, such as the fluctuation theorem and thermodynamic uncertainty relation, have attracted attention in non-equilibrium statistical mechanics. However, it was actually pointed out in the 1950s, although it is often overlooked, that the FDT may be violated even without considering general nonequilibrium situations, and it can be said that the FDT still has points to be discussed.

In Refs. 2 and 3, Fujikura and Shimizu showed that the quantum violation of the FDT occurs because of the disturbance caused by quantum measurements, even if we consider the quasi-classical measurement, which reduces the measurement error and the disturbance in an appropriate way, by using the quantum measurement theory and sophisticated theorems for macroscopic systems, such as the quantum central limit theorem [4] and the Lieb–Robinson bound [5]. In other words, the FDT can be violated owing to the quantum effect. While the conductivity is given by the canonical correlation according to the Kubo formula [6], the current fluctuation is given by another correlation (symmetrized correlation) according to the quantum measurement theory. This difference between these correlations appears as a quantum violation of the FDT. Notably, the response is more insensitive to the disturbance of the measurement, and thus, the Kubo formula may be valid even under the disturbance by quantum measurement.

Recently, Kubo, Asano, and Shimizu [7] have theoretically and numerically shown the quantum violation of the FDT in two-dimensional electronic systems under a magnetic field. They focused on σ_xy and βS_xy, where σ_xy is the off-diagonal conductivity, β is the inverse temperature and S_xy is the current correlation, because the quantum violation of the off-diagonal component is easier to measure. Reference 7 is similar to their previous paper, which approximately calculated the violation [8], but differs significantly because of the rigorous analytical and numerical methods adopted. In Ref. 7, the authors first considered a pure system without impurity potentials and analytically demonstrated that the FDT was violated. In the pure system, the form of the violation is simple and the FDT recovers in the classical limit βℏ|ω| ≪ 1. However, the quantum violation of the FDT is more pronounced in the disordered systems because localized states affect σ_xy and βS_xy in different ways. Using numerically exact diagonalization, they showed that the filling factor dependences of σ_xy and βS_xy were quite different (see Fig. 1). To discuss the physical origin of this violation, they divided σ_xy and βS_xy into intra- and inter-Landau level contributions. They then evaluated the dependence of σ_xy and βS_xy on impurities, which clarified why βS_xy is more insensitive to impurities than the conductivity σ_xy.

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Fig. 1. Filling factor dependence of σ_xy and βS_xy in the disordered systems. This figure was obtained from Fig. 1 of Ref. 7.

The authors also further discussed the possibility of experiments. Quasiclassical measurements of macroscopic systems are required. For example, experimental techniques related to heterodyne and quantum nondemolition measurements, which do not increase the uncertainty, may be available.

References

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