Theory of ac spin current noise and spin conductance through a quantum dot in the Kondo regime: Equilibrium case

We analyze the equilibrium frequency-dependent spin current noise and spin conductance through a quantum dot in the local moment regime. Spin current correlations behave markedly differently from charge correlations. Equilibrium spin correlations are characterized by two universal scaling functions in the absence of an external field: one of them is related to charge correlations, while the other one describes cross-spin correlations. We characterize these functions using a combination of perturbative and nonperturbative methods. We find that at low-temperatures spin cross correlations are suppressed at frequencies below the Kondo scale, $T_K$, and a dynamical spin accumulation resonance is found at the Kondo energy, $\omega \sim T_K$. At higher temperatures, $T>T_K$, surprising low-frequency anomalies related to overall spin conservation appear in the spin noise and spin conductance, and the Korringa rate is shown to play a distinguished role. The transient spin current response also displays universal and singular properties.

Figure 1

Sketch of the setups to measure the spin-dependent left-right conductance, $G_{LR}^{\mathbf{n}{\mathbf{n}}^{\prime }}\left(\omega \right)$, and the noise, $S_{LR}^{\mathbf{n}{\mathbf{n}}^{\prime }}\left(\omega \right)$. The arrows indicate the direction of the spin polarization in each lead, denoted by $\mathbf{n}$ and ${\mathbf{n}}^{\prime }$ for the left and right leads, respectively. In the upper setup, the spin filter on the right side detects the spin-resolved current at time $t^{\prime }$, $J_R^{{\mathbf{n}}^{\prime }}\left(t^{\prime }\right)$, induced by applying a spin-dependent voltage to the left lead at time $t$, $V_L^{\mathbf{n}}\left(t\right)$. The lower setup allows for injecting spins and measuring the spin-resolved currents in both electrodes.

Figure 2

Sketch of the properties of the real part of the universal function, $-g(\omega ,T)\sim G_{LR}^{\uparrow \downarrow }\left(\omega \right)$, characterizing the cross-spin conductance through the dot.

Figure 3

Real part of the conductance $G_{LR}^{\uparrow \uparrow }\left(\omega \right)$ through the dot for $T\gg T_K$. The anomaly below the Korringa relaxation rate, $E_K$, is a consequence of correlations between consecutive spin-flip processes and is related to the dip in $\left|\Re eg\right(\omega \left)\right|$ (see Fig. 2).

Figure 4

Real time response of the current $\langle J_R^{\downarrow }\rangle \left(t\right)$ as generated by a sudden change of amplitude, $\delta V_L^{\uparrow }$, with $\Delta \propto \max \{T,T_K\}$.

Figure 5

Structure of the spin-noise components $S_{LR}^{\uparrow \uparrow }\left(\omega \right)$ and $S_{LR}^{\uparrow \downarrow }\left(\omega \right)$ for $T\gg T_K$.

Figure 6

The zero-temperature universal functions $g$, $\tilde{g}$ for the ac conductance (upper panel) and for the spin noise $s$ and $\tilde{s}$ (lower panel) computed by NRG.

Figure 7

Real parts of the universal functions $g_z$ and $\tilde{g}$, and the corresponding noise scaling functions, $s_z$ and $\tilde{s}$, as obtained by performing zero-temperature NRG calculations with a magnetic field $B_z$ along the $z$th direction.

Figure 8

Zero-temperature NRG results for the real part of the universal function $g_{\perp }$ (upper part), and the corresponding noise scaling function $s_{\perp }$ (lower part) for several values of magnetic field $B_x$ pointing along the $x$ direction.

Figure 9

Upper panel: frequency-dependent spin current cross correlations ($S_{LR}^{\uparrow \downarrow }$) and auto-correlations ($S_{LR}^{\uparrow \uparrow }$) in the absence of spin relaxation, as calculated using the master equation approach (ME, solid line) and second-order perturbation theory (PT, dashed line). In the calculations, we assumed $j=0.2$. Lower panel: effect of a finite (rather large) external spin relaxation, with a rate $1/{\tau }_s\equiv E_{K,0}/2$.

Figure 10

Sketch of the universal scaling functions (upper panel) for the spin noise $s$ (continuous line) and $\tilde{s}$ (dashed line), and (lower panel) the real parts of $g$ (continuous line) and $\tilde{g}(\omega /T_K,T/T_K)$ (dashed line) for $T\gg T_K$. The charge conductance and noise functions, $\tilde{g}$ and $\tilde{s}$ show no particular feature, while the spin scaling functions exhibit an anomaly below the Korringa rate, $E_K$.

Figure 11

Sketch of the universal scaling functions (upper panel) for the spin noise $s$ (continuous line) and $\tilde{s}$ (dashed line), and (lower panel) the real parts of $g$ (continuous line) and $\tilde{g}$ (dashed line) for $T\ll T_K$.

Figure 12

Pole structure of the integrand in Eq. (62), for the spin-up/spin-down (upper panel) and spin-up/spin-up (lower panel) channels. In the spin-$\uparrow \downarrow$ channel, the pole at $\omega =0$ is canceled by the ${\omega }^2$ dependence of $g(\omega ,T)$, while in the case of spin-$\uparrow \uparrow$ channel, this pole survives and gives a finite response as $t\to \infty$.

Figure 13

Transient current response in the spin-down (upper panel) and spin-up (lower panel) channels of the right lead, upon a constant bias applied at $t=0$ in the spin-up channel to the left lead.

Figure 14

Sketch of the dependence of spin cross correlations $S_{LR}^{\uparrow \downarrow }(\omega =0,T)$ on the spin relaxation rate $1/{\tau }_s$ in the Fermi-liquid regime ($T\ll T_K$) (upper panel) and in the perturbative regime ($T\gg T_k$) (lower panel).

Figure 15

Correspondence between a single quantum dot device and a spinless double-dot device.