Landau level mixing in the perturbative limit

We study the effects of Landau level mixing in the limit of weak electron interaction. We use a numerical method to obtain the two- and three-body corrections to quantum Hall pseudopotentials, which are exact to lowest order in the Landau level mixing parameter. Our results are in general agreement with certain analytic results (some derived here; some derived by other authors) in the thermodynamic limit. We find that the convergence to this thermodynamic limit can be slow. This suggests that errors could occur if one tries to use pseudopotentials derived in a thermodynamic limit for numerical work on finite systems.

Figure 1

Sample double extrapolations. The multiple curves in the main plots correspond to different $N_{\varphi }$. The intercepts (infinite $L{L}_{\text{max}}$) are then plotted in the inset and extrapolated to infinite $N_{\varphi }$. Top: Extrapolation of the two-body LL-mixing correction terms $\delta V_3-\delta V_1$ for electrons in the LLL. Here ${\mathrm{LL}}_{\text{max}}$ ranges from 2 to 6. $N_{\varphi }$ ranges from 12 to 24. The double extrapolation yields a value 0.0273 compared to the analytically exact result $0.0272267$ (see Table , Refs. 10 and 11, and the Appendix below in this paper). Bottom: Extrapolation of the two-body LL-mixing correction terms $\delta V_3-\delta V_1$ for electrons in the first excited LL. Here ${\mathrm{LL}}_{\text{max}}$ ranges from 3 to 6. $N_{\varphi }$ ranges from 12 to 20. Due to the curvature in these plots, one must assume a bit more error on the extrapolation. The double extrapolation yields a value 0.1089. The analytical result of Ref. 10 yields 0.1104. This seems well within the range of possible extrapolation error.

Figure 2

Sample double extrapolations of the three-body pseudopotential $V_5-V_3$ (top) and $V_6-V_3$ (bottom) in LL1. The extrapolated values are 0.109 and 0.0027, respectively. Note that there is quite a bit of slope with respect to $N_{\varphi }$, which makes accurate extrapolation difficult. Secondly, note that the extrapolated result is close to zero on the scale of the values measured at finite $N_{\varphi }$ which also makes the extrapolation difficult.

Figure 3

Sample double extrapolations of the three-body pseudopotential $V_5-V_3$ in the LLL. The extrapolated value is 0.0214. Note that there the value of the pseudopotential taken at finite $N_{\varphi }$ can be quite different from the thermodynamic limit.