Information scrambling at an impurity quantum critical point

The two-channel Kondo impurity model realizes a local non-Fermi-liquid state with finite residual entropy. The competition between the two channels drives the system to an impurity quantum critical point. We show that the out-of-time-ordered (OTO) commutator for the impurity spin reveals markedly distinct behavior depending on the low-energy impurity state. For the one-channel Kondo model with Fermi-liquid ground state, the OTO commutator vanishes for late times, indicating the absence of the butterfly effect. For the two channel case, the impurity OTO commutator is completely temperature independent and saturates quickly to its upper bound 1/4, and the butterfly effect is maximally enhanced. These compare favorably to numerics on spin chain representation of the Kondo model. Our results imply that a large late time value of the OTO commutator does not necessarily diagnose quantum chaos.

Figure 1

Schematic “phase diagram” of the anisotropic two channel Kondo model at zero temperature in the time or energy domain. Only for ${\mathrm{\Gamma }}_b=0$ is the low-energy physics governed by the two channel Kondo effect.

Figure 2

The $s_z$ OTO commutator is shown in the perfect one- and two-channel cases at $T=0$, ${\mathrm{\Gamma }}_a={\mathrm{\Gamma }}_b$ (black), and ${\mathrm{\Gamma }}_b=0$ (blue), respectively, exhibiting also the crossover for ${\mathrm{\Gamma }}_b={\mathrm{\Gamma }}_a/2000$ (red). The dashed line denotes $T\gg {\mathrm{\Gamma }}_a$ for the channel isotropic 1CK, when the imaginary part of the Majorana propagator vanishes. The perfect 2CK is $T$ independent while for the crossover case, all finite $T$ data fall on top of the zero temperature curve.

Figure 3

Comparison of the long-time behavior $t\gtrsim 1/{\mathrm{\Gamma }}_a$ for the $K(t\to \infty )$ and the NRG results obtained by Fourier transforming the spin susceptibility.

Figure 4

Typical comparison between the exact diagonalization (ED), (solid blue lines) and time evolving block decimation (TEBD) (symbols) for the OTO commutator for 1CK (top) and 2CK (bottom). In both cases, the total system size, including the local spin(s) is fixed to N = 21.

Figure 5

The effect of system size on the OTO commutator $C_{2CK}\left(t\right)$ computed using the TEBD algorithm. When the system size is $N=23$, the break down occurs at $t\simeq 12/J_1$ due to entanglement growth, while for larger system sizes much later times are accessible. The parameters are fixed to $J_{R,L}^{\prime }=0.8$.

Figure 6

The ED data for the OTO commutator for the 1CK (red dashed line) and 2CK (blue solid line) is plotted for $J_L^{\prime }=0.7$, $J_R^{\prime }=0$, $N_L=21$ and $J_{R,L}^{\prime }=0.8$, $N_L=N_R=11$, respectively. Finite size effects are already present for later times. The thick green dashed lines denote the expected late time value when the OTO correlator vanishes in, e.g., chaotic systems. The inset shows the scaling of the Kondo temperature for 2CK [43] as $T_K=1.82J_{1}exp(-0.82/J^{\prime })$ (black solid line), deduced from the position of the initial sharp peak (blue circles).