Level dynamics and avoided level crossings in driven disordered quantum dots

The statistical properties of the dynamics of energy levels are investigated in the case of two two-dimensional disordered quantum dot models with nearest-neighbor hopping subjected to external time-dependent perturbations. While in the first model the external drivings are realized by a continuous variation of the onsite energies, in the second one it is generated by deformations of a parabolic potential. We concentrate on the effects of the potential on the localization properties and investigate the statistics of the energy-level velocities and curvatures regarding their typical magnitudes and domain of agreement with the predictions of random matrix theory for the Gaussian orthogonal, unitary and symplectic ensembles. Moreover, the statistical properties of the avoided level crossings are investigated in terms of the corresponding Landau-Zener parameters. We find that the strength of the Landau-Zener transitions exhibit universal behavior which also imply universal single-particle dynamics for slow perturbations independent of the disorder and potential strength, the system size and the symmetry class. These results can be verified experimentally by measurements of single-particle energy spectra in quantum dots.

Figure 1

Localization length as a function of the potential strength for various system sizes and disorder strengths. For weak potentials they agree with those in the corresponding Anderson models and for the GSE ensembles even for $W=3<W_c$ finite localization length is observed for $V_0>300J$. Inset shows velocity variance scaling as $\sim W^2/L^2$ with the same prefactor for the three ensembles.

Figure 2

Velocity distributions normalized such that their maximum values are 1 with variances scaled to unity. In the onsite model the statistics transforms from Gaussian to uniform distribution as disorder increases (diamonds), while for large potential strength completely featureless statistics is observed (crosses) in the potential model. Inset shows curvature unit dependence in the onsite model for the three ensembles, exhibiting ${\beta }^2{W}^4$ power-law behavior.

Figure 3

Comparison of the distribution of the gap and the asymptotic slope at the avoided level crossings obtained in the two-dimensional models and the RMT analytical results for the GOE ensemble (symbols). (a) Typical shape of an avoided crossing with width $\delta \lambda$, slope $\tilde{\gamma }$ and gap ${\tilde{\mathrm{\Delta }}}_{\min }$, with the dashed line fitting the Landau-Zener approximation of the level distance. (b) Gap distribution for the onsite and potential model. Red squares show potential model for system size, variance and potential strength, $L=28, W=J,$ and $V_0=100J$, respectively. Blue diamonds show onsite model results for $L=24$ and $W=J$. Good agreement is observed between the distributions with their means scaled to unity and the analytical results (dashed line). Inset shows disorder dependence of the gap in the potential model for $V_0=50J$ showing insensitivity, up to numerical precision, for all the three ensembles. (c) Distributions of the asymptotic slope for the same parameters. Numerical data collected from around zero energy and scaled to have unit mean values are in good agreement with the RMT analytical results (dashed line). Inset shows disorder dependence of the number of the avoided level crossings in the potential model for $V_0=50J$, exhibiting again constant behavior up to numerical precision.

Figure 4

Statistics of the Landau-Zener parameters at the avoided crossings for the GUE and GSE ensembles comparing the RMT numerical results with the onsite and potential model (symbols). (a) Gap distribution for the GUE ensemble for system sizes, disorder strengths, and potential strengths, $L=60, W=2J, J$, and $V_0=75J$ for the potential model and onsite model, respectively. Inset shows average number of the avoided level crossings in the potential model with $V_0=50J$ and $W=J$ as a function of the system size growing as $\sim L^2$ (dashed line) for all the three ensembles (b) Gap distribution for the GSE ensemble for parameters $L=48, W=J, 2J$, and $V_0=75J$, respectively, for the potential and the onsite model. In both cases remarkable agreement is observed with the RMT analytical results (dashed line). Inset shows average number of the anticrossings as function of the system size at $W=J$ for the onsite model growing linearly (dashed line). (c), (d) Slope distributions for the same parameters, with the two disordered models following the same curves up to high precision (dashed line). Inset of panel (c) shows scaling of the gap as a function of the disorder strength in the onsite model showing constant behavior up to numerical precision. Inset of panel (d) shows average number of the avoided crossings scaled with the size of the system for the onsite model growing linearly with the disorder strength (dashed line) and with approximately the same coefficient for all the three ensembles.

Figure 5

(a) Verification of the universal single-particle dynamics based on the universal distribution of the avoided level crossing parameters. Occupation profiles are plotted for all the three symmetry classes for velocities and quench times such that the occupation broadenings are the same. When scaling velocity and time as indicated in the main text these profiles become independent of the symmetry class, system size, and disorder and potential strength and fall on top of each other on a universal Gaussian curve. Inset: Time evolution of the variance of the occupation profile for $L=30,W=2J$, and $V_0=300J$ for the GUE ensemble in the potential model and for $L=50,W=J$ for the GOE ensemble in the on-site model exhibiting linear dependence. (b) Velocity dependence of the diffusion coefficient, exhibiting a universal power-law behavior with an anomalous dependence, $\sim {\tilde{v}}^{\beta /2+1}$ in the slow process limit with the slightly smaller values for the potential model. Numerical simulations were performed for various system sizes, disorder, and potential strengths, for the three symmetry classes being identical on the two panels.