Nonequilibrium Transport at a Dissipative Quantum Phase Transition

We investigate the nonequilibrium transport near a quantum phase transition in a generic and relatively simple model, the dissipative resonant level model, that has many applications for nanosystems. We formulate a rigorous mapping and apply a controlled frequency-dependent renormalization group approach to compute the nonequilibrium current in the presence of a finite bias voltage $V$ and a finite temperature $T$. For $V\to 0$, we find that the conductance has its well-known equilibrium form, while it displays a distinct nonequilibrium profile at finite voltage.

Figure 1

Schematic phase diagram as a function of temperature $T$, dissipation strength $\alpha$, and bias voltage $V$.

Figure 2

$g_{\perp ,\mathrm{cr}}(\omega )=-g_{z,\mathrm{cr}}(\omega )$ at $\alpha ={\alpha }_c$; the bare couplings are $g_{\perp }=-g_z=0.1$. We have set $V=0.066D_0$, $0.43D_0$, $0.75D_0$ and $0.99D_0$ where $D_0=1$ for all the figures. The arrows give the values of $g_{\perp ,\mathrm{cr}}(\omega =0)$ at these bias voltages.

Figure 3

Nonequilibrium conductance at the KT transition. $G_0$ is the equilibrium conductance at the transition for $T=D_0$: $G_0=G_{\mathrm{eq}}({\alpha }_c,T=D_0)=0.005\pi$ with the bare couplings $g_{\perp }=-g_z=0.1$. Again, we set the charge $e=\hslash =1$.

Figure 4

Conductance in the localized phase (in units of $\pi /\hslash$). (a) $G(V)$ at low bias follows the equilibrium scaling (dashed lines). (b) The conductance $G(V)/G_c$ is a function of $V/T^{*}$ where we have defined $G_c=G({\alpha }_c,V)$ and $T^{*}=D_0{e}^{-\pi /\sqrt{g_z^2-g_{\perp }^2}}$. (c) At large bias voltages $V$, the nonequilibrium conductance $G(V)$ (solid lines) is distinct from the equilibrium form (dashed lines). The dot-dashed lines stem from an analytical approximation via Eq. (8).

Figure 5

Scaling of the conductance at the KT transition (same unit as in Fig. 3). (a) For $V>T$, the conductance follows the nonequilibrium scaling $G({\alpha }_c,V)$. (b) For $V<T$, now the conductance follows the equilibrium scaling $G({\alpha }_c,T)$.