Steady-State Dynamics and Effective Temperature for a Model of Quantum Criticality in an Open System

We study the thermal and nonthermal steady-state scaling functions and the steady-state dynamics of a model of local quantum criticality. The model we consider, i.e., the pseudogap Kondo model, allows us to study the concept of effective temperatures near fully interacting as well as weak-coupling fixed points. In the vicinity of each fixed point we establish the existence of an effective temperature—different at each fixed point—such that the equilibrium fluctuation-dissipation theorem is recovered. Most notably, steady-state scaling functions in terms of the effective temperatures coincide with the equilibrium scaling functions. This result extends to higher correlation functions as is explicitly demonstrated for the Kondo singlet strength. The nonlinear charge transport is also studied and analyzed in terms of the effective temperature.

Figure 1

(a) Sketch of the model: a spin interacts with two fermionic leads which are characterized by their respective density of states ${\rho }_{c,L/R}^{-}(\omega )$ and chemical potentials ${\mu }_{L/R}$. (b) Phase diagram of the multichannel PKM with gap exponent $r<r_{\max }$: A QCP (C) separates the multichannel Kondo (MCK) fixed point from the (weak-coupling) local moment (LM) fixed point.

Figure 2

(a) ${\chi }^{\prime }{(0)}^{-1}$ vs $J$ for different $T$. (b) ${\varphi }_s$ vs $J$ for different $T$. The $T=0$ curve is approached from below in the MCK and from above in the LM phases. (c) Scaling $T/T_{\mathrm{eff}}$ vs $V/T$ at fixed points LM, C, and MCK: $T_{\mathrm{eff}}\sim V$ for $V\gg T$. (d) ${\mathrm{FDR}}_{\chi }^{-1}(\omega )$ vs $\omega /T_{\mathrm{eff}}$ near fixed point C, shown for $V/D=10^{-2},10^{-3},10^{-4},10^{-5},10^{-6}$. The grey line is ${\mathrm{FDR}}^{-1}(\omega )=\tanh (\beta \omega /2)$.

Figure 3

Scaling of observables with $T_{\mathrm{eff}}$ at different fixed points for the values of $V$ as in Fig. 2: (a) ${\chi }^{\prime }{(0)}^{-1}$ vs $T_{\mathrm{eff}}$; (b) ${\partial }_{\ln \omega }\ln {\chi }^{\prime \prime }(\omega )$ vs $\omega /T_{\mathrm{eff}}$; (c) ${\varphi }_s$ vs $T_{\mathrm{eff}}$. For each fixed point, the equilibrium scaling form (grey curves) is compared with the same quantity under nonequilibrium conditions and $T$ substituted by $T_{\mathrm{eff}}$.

Figure 4

Conductance $G$ normalized to the MCK fixed point conductance $2\pi G_0=0.415$. (a) $G(T)$ vs $T$ computed for the lowest nonzero value of $V$ at different values of $J$ (see color coding). (b) $G$ vs $V$ for fixed $T$. (c) $G={\mathcal{J}}_P/V$ vs $T_{\mathrm{eff}}$ at different fixed points. The equilibrium form is given by the grey curves.