Dissipation-induced quantum phase transition in a quantum box

In a recent work, Le Hur has shown, using perturbative arguments, that dissipative coupling to gate electrodes may play an important role in a quantum box near its degeneracy point [K. Le Hur, Phys. Rev. Lett. 92, 196804 (2004)]: While quantum fluctuations of the charge of the dot tend to round Coulomb blockade charging steps of the box, strong enough dissipation suppresses these fluctuations and leads to the reappearance of sharp charging steps. In the present paper, we study this quantum phase transition in detail using bosonization and the numerical renormalization group in the limit of vanishing level spacing and map out the phase diagram using these nonperturbative methods. We also discuss the properties of the renormalized lead-dot conductance in the vicinity of the phase transition and determine the scaling properties of the dynamically generated crossover scale analytically.

Figure 1

(a) Sketch of the single electron box. (b) Top view of the regime where electrons move in the two-dimensional electron gas. Black areas indicate various electrodes necessary to shape the electron gas.

Figure 2

The number of electrons on a single electron box as a function of the dimensionless gate voltage. (a) The sudden jumps of an isolated box become smeared out due to quantum fluctuation as soon as we couple the box to a lead. (b) Large enough dissipation restores the discrete jumps.

Figure 3

(Color online) Relation between the phase shift ${\delta }_g$ extracted from the finite size spectrum and the fermionic coupling $g$. The continuous line shows the fit with Eq. (43).

Figure 4

(Color online) Finite size spectra obtained by NRG for the two distinct regimes. The upper part shows the spectrum for $\alpha =0.653,j_z=0.2,j_{\perp }=0.7$, while the lower part stands for the same $j\mathrm{s}$ but for $\alpha =0.667$. It is transparent that while on the left figure the levels cross over to the well-known Kondo spectrum, on the right panel no such crossover occurs.

Figure 5

(Color online) The critical surface in the parameter space separating the Kondo regime from the localized regime. The heavy continuous line indicates the phase transition line for the case relevant for the single electron box $j_z=0$. The heavy dashed line indicates the critical line, $j_{\perp }=0$. The arrows show the scaling trajectories for $\alpha ={\alpha }_c$. All these critical scaling trajectories end up at the critical (dashed) line.

Figure 6

(Color online) Top: The physically relevant $j_z=0$ cut of the critical surface. Bottom: Phase boundary as a function of ${\delta }_z∕\pi$ for two values of $j_{\perp }$. These phase boundaries compare very well with the analytical expression Eq. (31).

Figure 7

(Color online) Kondo temperature as a function of dissipation for $j_{\perp }=0.4$ and $j_z=0.1$

Figure 8

(Color online) The expectation value of $\widehat{N}-1∕2$ for $j_z=0.2,j_{\perp }=0.8$, and different values of $\alpha$. For moderate values of the bosonic coupling, the step is smeared out by quantum fluctuations of the charge of the dot. The step gets sharper and sharper as $\alpha$ approaches ${\alpha }_c$, and for $\alpha >{\alpha }_c$, a jump appears in $⟨\widehat{N}⟩$.

Figure 9

(Color online) The imaginary part of the fermions’ $T$ matrix for $j_z=0.15,j_{\perp }=0.15$, and for different values of the bosonic coupling. For decoupled bosonic heat bath, the usual Kondo resonance shows up. As $\alpha$ is increased, the width of the Kondo resonance is decreasing and at the critical bosonic coupling $({\alpha }_c=0.304)$ it disappears and the electrons become decoupled from the spin.