Statistics of Deep Energy States in Coulomb Glasses

We study the statistics of local energy minima in the configuration space and the energy relaxation due to activated hopping in a system of interacting electrons in a random environment. The distribution of the local minima is exponential, which is in agreement with extreme value statistics considerations. The relaxation of the system energy shows logarithmic time dependence reflecting the ultrametric structure of the system.

Figure 1

Illustration of a 3-tree, an element of the hierarchical tree representing the configurational space with ultrametric topology.

Figure 2

Energy distribution of about 12 000 local minimum states in half-log representation with linear fit. Linear scale with exponential fit (triangular inset).

Figure 3

Distribution of the difference of the local minima ${\Delta }_{\alpha \beta }$ (0 for identical states, 1 for complementary ones). The distribution is fitted by a Gaussian curve, with mean at 0.054 and width 0.007. The calculation includes ${10}^7$ overlap integrals. The inset shows the dependence of the mean (crosses) and width (triangles) of the distribution on system size.

Figure 4

Energy relaxation of the system energy for two different temperatures. In the two upper plots logarithmic fits are shown. At lower temperatures ($T=0.01$) the initial “flat” region is due to a very long equilibration phase of the system to reach the steady state from the local minimum. Even after averaging over several time series the fluctuations at higher temperatures are still large. At even higher temperatures the fluctuations dominate and the logarithmic behavior cannot be recognized since due to computational limits no more averaging can be done in reasonable time. The lower graph shows the energy relaxation for the case when the Coulomb interaction range is artificially limited to 3 lattice sites at $T=0.05$. One sees (1) that the behavior deviates from the log behavior and (2) the amplitude of the fluctuations are much smaller as for the case with full Coulomb interaction.