Quantum phase transitions in systems of parallel quantum dots

We study the low-temperature transport properties of systems of parallel quantum dots described by the $N$-impurity Anderson model. We calculate the quasiparticle scattering phase shifts, spectral functions, and correlations as a function of the gate voltage for $N$ up to 5. For any $N$, the conductance at the particle-hole symmetric point is unitary. For $N\ge 2$, a transition from ferromagnetic to antiferromagnetic impurity spin correlations occurs at some gate voltage. For $N\ge 3$, there is an additional transition due to an abrupt change in average impurity occupancy. For odd $N$, the conductance is discontinuous through both quantum phase transitions, while for even $N$ only the magnetic transition affects the conductance. Similar effects should be experimentally observable in systems of quantum dots with ferromagnetic conduction-band-mediated interdot exchange interactions.

Figure 1

(Color online) (a) Zero-temperature conductance through systems of $N$ parallel quantum dots as a function of the gate voltage for a range of magnetic fields. The in-plane field leads only to Zeeman splitting; no magnetic flux pierces the rings formed by pairs of dots. $\Lambda =4$; NRG iterations were performed until the zero-temperature limit was reached. $T_K=3.3\times {10}^{-6}D$ (for $N=2$). The magnetic field is measured in units of $g{\mu }_B$. Only $\delta >0$ is shown due to the symmetry of the problem. (b) Zero-temperature phase diagram delimiting the different regimes as a function of the gate voltage. Filled circles $(●)$ correspond to phase transitions visible in conductance, while the empty circle $(○)$ denotes the phase transition with no associated conductance discontinuity. (c) Conductance and spin conductance in small magnetic field in the transition region.

Figure 2

(Color online) (a) Total occupancy (charge) and spin-spin correlation between pairs of spins as a function of the gate voltage. (b) Schematic representations of the spin configurations in the intermediate AFM ordered phases, their total spins, and degeneracies for $N\ge 3$.

Figure 3

(Color online) Expectation values in the symmetric and asymmetric bases for $N=2$ and $3$ systems. The asymmetric combination of states considered is $d_{\mathrm{asym}}^{†}=1∕\sqrt{2}(d_1^{†}-d_2^{†})$.

Figure 4

(Color online) (a) Normalized spectral function $\pi \Gamma A_{ij}\left(\omega \right)$ and (b) function $g\left(\omega \right)=\pi \Gamma {\sum }_{ij}{A}_{ij}\left(\omega \right)$.