Tunable quantum phase transitions in a resonant level coupled to two dissipative baths

We study tunneling through a resonant level connected to two dissipative bosonic baths: one is the resistive environment of the source and drain leads, while the second comes from coupling to potential fluctuations on a resistive gate. We show that several quantum phase transitions (QPT) occur in such a model, transitions which emulate those found in interacting systems such as Luttinger liquids or Kondo systems. We first use bosonization to map this dissipative resonant level model to a resonant level in a Luttinger liquid, one with, curiously, two interaction parameters. Drawing on methods for analyzing Luttinger liquids at both weak and strong coupling, we obtain the phase diagram. For strong dissipation, a Berezinsky-Kosterlitz-Thouless QPT separates strong-coupling and weak-coupling (charge localized) phases. In the source-drain symmetric case, all relevant backscattering processes disappear at strong coupling, leading to perfect transmission at zero temperature. In fact, a QPT occurs as a function of the coupling asymmetry or energy of the resonant level: the two phases are (i) the system is cut into two disconnected pieces (zero transmission), or (ii) the system is a single connected piece with perfect transmission, except for a disconnected fractional degree of freedom. The latter arises from the competition between the two fermionic leads (source and drain), as in the two-channel Kondo effect.

Figure 1

Schematic of a spinless quantum dot coupled to two conducting leads and a gate. The source and drain junctions are characterized by tunneling amplitudes $V_S$ and $V_D$, as well as capacitances $C_S$ and $C_D$. The dot-leads system is symmetrically biased by a voltage $V$ through the lead resistances $R_S$ and $R_D$. The gate is capacitively coupled to the dot (capacitance $C_G$) through a resistance $R_G$. We consider the simplified situation in which $C_S=C_D\equiv C$ and $R_S=R_D\equiv R/2$.

Figure 2

Schematic representation of the RG flow for the symmetric case ($V_S=V_D\equiv V$) on resonance. For $r_{\mathrm{eff}}<2$, one has $K_1^{\mathrm{bare}}<4/(1+r)-1$, and the system flows to the strong-coupling fixed point at which there is a uniform system and perfect transmission. For $r_{\mathrm{eff}}>2$, as the bare coupling $V$ decreases, for instance along the red dashed line, there is a BKT type quantum phase transition at $V=V^{*}$. For smaller $V$, resonant tunneling is destroyed, and the flow is toward the decoupled, zero transmission state (blue line of fixed points on the horizontal axis). For $r>3$ (not shown), the flow is always toward the decoupled state, indicating that resonant tunneling is not possible in this regime.

Figure 3

Schematic representation of the RG flow of the two tunneling amplitudes, $V_S$ and $V_D$, in regime (i): the level is on resonance and the dissipation is not too strong, $r_{\mathrm{eff}}<2$. The diagonal is the symmetric barrier case: it flows into the strong-coupling quantum critical point at $(1,1)$ which corresponds to a uniform system and so perfect transmission. At point $(1,0)$, the level is fully incorporated into the D lead ($V_D=1$) while completely disconnected from the S lead ($V_S=0$); the roles of source and drain are reversed at $(0,1)$. Single barrier scaling is expected along the straight lines from $(1,1)$ to either $(1,0)$ or $(0,1)$.

Figure 4

Schematic representation of the RG flow of the two tunneling amplitudes, $V_S$ and $V_D$, when the level is on resonance and the dissipation is strong. (a) Regime (ii), $r_{\mathrm{eff}}>2$ but $r<3$. (b) Regime (iii), $r>3$. The dotted line marks the BKT transition between a localized state in which the level is disconnected from the leads, $(0,0)$, and an extended state in which the level joins seamlessly with either one [(0,1) or (1,0)] or both leads [(1,1)].

Figure 5

Effective impedance seen by an electron tunneling across the S barrier in the sequential tunneling regime. The real part of this impedance is $r_s^{\prime }{R}_Q$, which controls the low temperature scaling of the tunneling rate.