Interactions and broken time-reversal symmetry in chaotic quantum dots

When treating interactions in quantum dots within a random-phase-approximation (RPA)-like approach, time-reversal symmetry plays an important role as higher-order terms—the Cooper series—need to be included when this symmetry is present. Here we consider model quantum dots in a magnetic field weak enough to leave the dynamics of the dot chaotic, but strong enough to break time-reversal symmetry. The ground-state spin and addition energy for dots containing 120–200 electrons are found using local spin-density-functional theory, and we compare the corresponding distributions with those derived from an RPA-like treatment of the interactions. The agreement between the two approaches is very good, significantly better than for analogous calculations in the presence of time-reversal-symmetry. This demonstrates that the discrepancies between the two approaches in the time-reversal symmetric case indeed originate from the Cooper channel, indicating that these higher-order terms might not be properly taken into account in the spin-density-functional calculations.

Figure 1

Cumulative density of nearest-neighbor level spacing for the effective Hamiltonian derived from the Thomas-Fermi approximation at $B=0\mathrm{T}$ (dashed gray) and $B=0.15\mathrm{T}$ (dashed black). The solid lines are the corresponding Wigner surmise distributions for the orthogonal (gray) and unitary (dark) Gaussian ensembles.

Figure 2

(Color online) Addition spectra obtained from the LSDA calculations (dark), compared with theoretical predictions derived from RPA/RPW (lighter, blue online). Solid: $N$ even; Dashed: $N$ odd. Top: strength of the RPA interaction obtained by matching the QMC value, $J_s=f_0^a=0.35$ (for $r_s=1.4$). Bottom: better agreement is obtained with a slightly larger value of this parameter, $J_s=0.46$.

Figure 3

(Color online) Spin distribution obtained from the LSDA calculations (left), the RPA/RPW model with $J_s=f_0^a=0.35$ (middle), and $J_s=0.46$ (right). Contrary to the addition spectra, slightly enhancing the RPA interaction does not improve the agreement with the LSDA simulation for the spin distribution; rather, it makes it worse.