Macroscopic quantum violation of the fluctuation-dissipation theorem in equilibrium

We examine the Hall conductivity of a macroscopic two-dimensional quantum system and show that the observed quantities can sometimes violate the fluctuation dissipation theorem (FDT), even in the linear response (LR) regime infinitesimally close to equilibrium. At low temperature and in strong magnetic field, which are experimentally accessible, the violation can be by an order of magnitude larger than the Hall conductivity itself. We further generalize the results and obtain a necessary condition for such large-scale violation to happen. This violation is a genuine quantum phenomenon that appears on a macroscopic scale when the time-reversal symmetry is broken (by, e.g., a magnetic field). Our results are not only bound to the development of the fundamental issues of nonequilibrium physics, but the idea is also meaningful for practical applications, since the FDT is widely used for the estimation of noises from the LRs. We give an alternative formula evaluating precisely the fluctuations from the LRs even in the quantum systems with large FDT violation.

Figure 1

(a) $\mathrm{Im}{\sigma }_{xy}\left(\omega \right)$ as a function of frequency $\omega$ for ${\omega }_{\mathrm{c}}\tau =4$, 8, and 12 at temperature, $T=\hslash /20\tau$, and electron density, $n=10m/h\tau$. Cyclotron frequency and the scattering time at $B=0$ are denoted as ${\omega }_{\mathrm{c}}$ and $\tau$, respectively. (b) $|{\sigma }_{xy}\left(0\right)|$ and ${\sigma }_{xx}\left(0\right)$ as a function of filling factor, $\nu$, at $T=\hslash /20\tau$ (low $T$) and $T=\hslash /5\tau$ (high $T$) for fixed electron density, $n=10m/h\tau$. (c) ${\sigma }_{xy}\left(0\right)$ and $\beta S_{xy}\left(0\right)$ as a function of $\nu$ for the same values of parameters as in (b).