Emergence of quantum chaos and thermalization in finite Fermi systems

The Random Matrix Theory (RMT) was developed to explain the general properties of complex energy spectra in many-body interacting systems such as heavy nuclei, many electron atoms and molecules. Later, it has found many other successful applications in different physical systems. Among the most recent of them we can quote models of quantum chaos where RMT appears due to the classically chaotic but deterministic underlying dynamics. One of the most direct indications of the emergence of quantum chaos is the transition of the level spacing statistics $P(s)$ from Poisson to Wigner-Dyson (WD) distribution. This property has been widely used to detect the transition from integrability to chaos not only in systems with few degrees of freedom (O.Bohigas et al.) but also in solid-state models with many interacting electrons. It was also applied to determine the Anderson delocalization threshold in noninteracting disordered systems .

While the conditions for the appearance of the WD distribution in noninteracting systems is qualitatively well understood the situation is more intricate in presence of interaction. Indeed, in this case the size of the total Hamiltonian matrix grows exponentially with the number of particles and it becomes very sparse as a result of the two-body nature of the interaction. According the common lore in nuclear physics the level density grows exponentially with the number of particles and therefore an exponentially small interaction is sufficient to mix nearby levels. However recent estimates on few-particle models ($n=2,3,4$) showed that in spite of the high many-body density of states, only an interaction strength comparable to the two-particle level spacings can give a level mixing and WD distribution for $P(s)$ [79]. This result was confirmed later by Weinmann, Pichard and Imry. The generalization to the case of large number of particles was done in [94]. There, for a model with a random two-body interaction it was shown that there is a smooth crossover from Poisson to WD distribution for interaction $U > U_c \approx 1/\rho_c \sim 1/(\rho_2 n^2)$ where $\rho_2$ is the density of two-particle states and $n$ is the number of effectively interacting particles. For fixed interaction being small comparing to one-particle level spacing $\Delta \gg U$ the number of effectively interacting particles depends on excitation energy $\delta E$ above the Fermi level: $\delta n \sim T/\Delta \sim \sqrt{\delta E/\Delta}$, where $T$ is the temperature of this finite closed Fermi system. Due to that as it is found in [94] the crossover to WD distribution takes place only for $\delta E > \delta E_{ch} \approx \Delta (\Delta/U)^{2/3}$. Since without random matrix properties termalization cannot set in the result of [94] implies that the system is thermalized only for $T>T_{ch} \approx \Delta (\Delta/U)^{1/3}$ (or $\delta E > \delta E_{ch}$).

The obtained estimates for the quantum chaos border can be applied to different finite interacting Fermi systems such as complex nuclei with residual interaction, atoms and molecules, clusters and quantum dots. Here we briefly discuss the case of metallic quantum dots studied in the experiments of Sivan et al.. In this case the interparticle interaction is relatively weak so that $U/\Delta \sim 1/g$ with $g = E_c/\Delta \gg 1$ being the conductance of the dot and $E_c$ the Thouless energy. According to above estimates the thermalization will take place above the excitation energy $\delta E_{ch} \sim \Delta g^{2/3}$. This is in a satisfactory agreement with the experimental results where a dense spectrum of excitations in dots with $g \sim 100$ appears at excitation energies $\delta E_{ch} \sim 10 \Delta$. The above border for thermalization and chaos $\delta E_{ch}$ is higher than the border for quasiparticle disintegration on many modes $\delta E_D \sim \Delta g^{1/2}$ proposed by Altshuler, Gefen, Kamenev and Levitov. The parametrically different dependence on $g$ suggested there appears because the effect of energy redistribution between many excited modes was neglected while the results of [94,96] show that it plays an important role.

The investigation of the eigenstate properties was done in [96]. It was shown that for $U>U_c$ the number of noninteracting states contributing to one eigenstate, which is proportional to the inverse participation ratio, is $\xi \approx \Gamma \rho_n$. Here $\Gamma \sim U^2 \rho_c$ is the Briet-Wigner width of the local density of states and $\rho_n$ is the multi-particle density of states. As the result for $U \sim U_c$ interaction mix exponentially many states ( $\xi_c \sim \rho_n/\rho_c$). Near the Fermi level $\Gamma \sim ({\delta E}/{\Delta})^{3/2} \; (U^2/\Delta)$. The last expression for $\Gamma$ has a simple meaning. Indeed, $\Gamma$ is the total Breit-Wigner width for $\delta n$ effectively interacting particles. Therefore, the partial width $\Gamma_D \sim \Gamma/\delta n$ is the usual quasi-particle decay rate which in agreement with the theory of Landau Fermi liquid is proportional to $T^2$. At the quantum chaos border $\delta E = \delta E_{ch}$, when the crossover to the Wigner-Dyson statistics takes place, the IPR becomes exponentially large $\xi_c \sim (T_{ch}/\Delta) \exp(2.6 T_{ch}/\Delta)$. We note that in the Landau Fermi liquid theory quasiparticles are well defined if $\Gamma_D < T (T< T_L = \Delta (\Delta/U)^2)$. In this regime the level statistics $P(s)$ can be as in chaotic ( $P(s)=P_{WD}(s)$ for $T_{ch} < T < T_L$) or integrable ($P(s)=P_{P}(s)$ for $T < T_{ch} < T_L$) systems.

As a result, the Landau Fermi liquid in finite systems can be integrable or chaotic depending on temperature (or excitation energy) and interaction strength.