Prethermalization and transient dynamics of multichannel Kondo systems under generic quantum quenches: Insights from large-$N$ Schwinger-Keldysh approach

Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-$N$ Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an $O\left(1\right)$ timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the $1/N$ correction, we demonstrate that the system attains complete thermalization on an $O\left(N\right)$ timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-$N$ quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.

Figure 1

(a) Phase diagram of prequench and postquench states. $T, \mathrm{\Lambda }, N_f$, and $J$ are temperature, cutoff of conduction electron, conduction electron's density of states near the Fermi energy, and Kondo coupling constant, respectively. The circle ($\circ$) and star ($★$) denote the $f$ fermion and $\varphi$ boson, respectively. The blue-colored ones correspond to the prequench states with $J=J_0$ while the red-colored ones denote the postquench states with $J=J_1^{<}(<J_0)$ and $J_1^{>}(>J_0)$. The green dotted line shows the Kondo temperature as a function of the Kondo coupling constant $J$. Here the results of the two quench protocols (i) $J_0(=10)\to J_1^{<}(=6)$ and (ii) $J_0(=10)\to J_1^{>}(=15)$ of the initial states with $J_0=10$ and $T_0=0.4T_{K,0},1.6T_{K,0}$, and $5.0T_{K,0}$ (denoted by the horizontal dashed lines) are depicted. $T_{K,0}$ is the Kondo temperature with $J=J_0$. The inset is a zoomed-in picture of the temperature changes. (b) Different timescales for the SU(N) MCKI model under a sudden quench.

Figure 2

Keldysh contour $\mathcal{C}={\mathcal{C}}_1\cup {\mathcal{C}}_2\cup {\mathcal{C}}_3$. The initial state with the temperature $1/{\beta }_0$ is prepared at $t=t_0$.

Figure 3

Feynman diagram of the spin-spin correlation functions with the order $O\left(N^0\right)$.

Figure 4

(a) Spectral functions of $f$ fermion and $\varphi$ boson, (b) Self-energies of the fermion $f$ and (c) Self energies of the boson $\varphi$ with the temperatures $5T_K, 1.6T_K$, and $0.4T_K$.

Figure 5

(a) $G_f^{>}\left(t\right)$ with temperatures $T_0=10T_K, 5T_K, 2T_K$. (b) $G_f^{>}\left(t\right)$ with temperatures $T_0=0.8T_K, 0.6T_K, 0.4T_K$, and $0.22T_K$. Red dashed dotted line denotes the $G_{f,\mathrm{conf}}^{>}\left(t\right)$ solution (35) with $b_f=0.2$. (c) Log-log plot of the real part of $G_f^{>}\left(t\right)$ with low temperatures with the power-law Green’s function $G_f>\left(t\right)\sim \frac{1}{{\left|t\right|}^{2{\mathrm{\Delta }}_f}}$ (Red dashed line).

Figure 6

(a) ${\chi }_{\mathrm{imp}}\left(t\right)$ in a scaled form and its log-log plot with temperatures $T_0=10T_K, 5T_K, 2T_K$. (b) ${\chi }_{\mathrm{imp}}\left(t\right)$ in a scaled form and its log-log plot with temperatures $T_0=0.8T_K, 0.6T_K, 0.4T_K$ and $0.22T_K$.

Figure 7

(a) $C_{\mathrm{imp}}\left(t\right)$ in a scaled form and its log-log plot with temperatures $T_0=10T_K, 5T_K, 2T_K$. (b) $C_{\mathrm{imp}}\left(t\right)$ in a scaled form and its log-log plot with temperatures $T_0=0.8T_K, 0.6T_K, 0.4T_K$ and $0.22T_K$.

Figure 8

Evolutions of the $\langle {\vec{S}}_{\mathrm{imp}}·{\vec{s}}_c\left(t\right)\rangle /N$ for (a) $T_0=0.4T_K$ and (b) $T_0=5T_K$ cases under a quench from $J_0=10$ to different values of $J_1$. Here the value of $\gamma$ is 1.

Figure 9

Evolution of the Kondo order parameter under the quench ($J_1=15$ and 6) with different $\gamma$ at $T_0=0.4T_K$. (a) Quench from $J_0=10$ to $J_1=15$ case. (b) Quench from $J_0=10$ to $J_1=6$ case.

Figure 10

Logarithm plots of (a) $C_{\text{imp}}(\mathcal{T},t_r)$ and (b) ${\chi }_{\text{imp}}(\mathcal{T},t_r)$ for $T_0=0.4T_K$ case. The color of the graph denotes the values of $\mathcal{T}$. Dashed lines with blue, green, and brown colors for both $J_1=6$ and 15 cases are the linear fitting lines of the graphs at $\mathcal{T}=-5.6, -0.4$, and 7.4, respectively. Slopes of dashed lines in time order are given by $(-0.54,-0.40,-0.26)$ for $C_{\text{imp}}$ with $J_1=6, (-0.54,-0.54,-0.56)$ for $C_{\text{imp}}$ with $J_1=15, (-0.54,-0.40,-0.26)$ for ${\chi }_{\text{imp}}$ with $J_1=6$, and $(-0.54,-0.54,-0.52)$ for ${\chi }_{\text{imp}}$ with $J_1=15$.

Figure 11

Logarithm plots of (a) $C_{\text{imp}}(\mathcal{T},t_r)$ and ${\chi }_{\text{imp}}(\mathcal{T},t_r)$ for $T_0=5T_K$ case. The color of the graph denotes the values of $\mathcal{T}$. Dashed lines with blue, green, and brown colors are the linear fitting lines of the graphs at $\mathcal{T}=-1.6$, 0.0, and 2.4, respectively. Slopes of dashed lines in time order are given by $(-3.8,-2.4,-1.1)$ for $C_{\text{imp}}$ with $J_1=6, (-3.8,-5.3,-6.2)$ for $C_{\text{imp}}$ with $J_1=15, (-3.7,-2.4,-1.0)$ for ${\chi }_{\text{imp}}$ with $J_1=6$, and $(-3.7,-6.2,-6.5)$ for ${\chi }_{\text{imp}}$ with $J_1=15$.

Figure 12

(a) Plot of ratio of exponential decaying rate as a function of $ln(J_1/J_0)$ for $T_0=0.4T_K$ case. (b) Logarithm plot of ratio of exponential decaying rate as a function of $ln(J_1^2/J_0^2)$ for $T_0=5T_K$ case.

Figure 13

Evolution of the spectral functions of $f$ fermion and $\varphi$ boson for $J_1=6$ and 15 cases with the initial temperature $T_0=0.4T_K$.

Figure 14

Final temperatures of (a) $f$-fermion and (b) $\varphi$-boson after quenching as a function of $J_1-J_0$ for the cases with the different initial temperatures $0.4T_K, 1.6T_K$, and $5T_K$. (c) Difference between final temperatures of $f$ fermion and $\varphi$ boson.

Figure 15

Logarithmic ratio between final temperatures and the Kondo temperatures ($T_{K,1}=\mathrm{\Lambda }{e}^{-\frac{1}{J_1{N}_F}}$) of (a) $f$-fermion and (b) $\varphi$-boson after quench as a function of $J_1-J_0$ for the cases with the different initial temperatures $0.4T_K, 1.6T_K$, and $5T_K$. The dashed red line denotes the zero values of the logarithmic ratio and $J_1-J_0$ for clarity.

Figure 16

Thermalization time of (a) $f$ fermion and (b) $\varphi$ boson as a function of $J_1-J_0$ for thermalized cases with the initial temperatures $0.4T_K, 0.6T_K, 0.8T_K$, and $1.0T_K$.

Figure 17

Thermalization time of (a) $f$ fermion and (b) $\varphi$ boson as a function of $J_1-J_0$ for thermalized cases with the initial temperatures $1.6T_K, 5.0T_K$, and $10T_K$.

Figure 18

One-loop self-energy correction of the order $O(1/N)$ to the conduction electron.

Figure 19

Temperature difference of the conduction electron with the self-energy correction of $O\left(N^{-1}\right)$. Here $T_0=5T_K$ case is considered.

Figure 20

$T_{1,\varphi }$ as a function of $T_{1,f}$ for different values of $\mathrm{\Gamma }$ with $\alpha =0$. (b) $T_{1,\varphi }$ as a function of $T_{1,f}$ for different values of $\alpha$ with $\mathrm{\Gamma }/T_0=0.4$.

Figure 21

Flow chart of the Kadanoff-Baym equation solver.

Figure 22

Evolution of $C_{\text{imp}}(\mathcal{T},t_r)$ and ${\chi }_{\text{imp}}(\mathcal{T},t_r)$ for $T_0=0.4T_K$ case with $J_1$ are given by 6 and 15.

Figure 23

Evolution of $C_{\text{imp}}(\mathcal{T},t_r)$ and ${\chi }_{\text{imp}}(\mathcal{T},t_r)$ for $T_0=5T_K$ case with $J_1$ are given by 6 and 15.

Figure 24

The evolution of the Kondo order parameter $\langle {\mathbf{S}}_{\mathrm{i}}·{\mathbf{s}}_{\mathrm{c}}\rangle$ for (a) the initial state being $\psi =\frac{1}{\sqrt{2}}\left(\right|\uparrow \downarrow \rangle -|\downarrow \uparrow \rangle )$, and (b) the initial state is a thermal state with $\beta =0.0001$. The parameters are $(J,\mathrm{\Gamma })=(6,0.01)$.