Quantum quench of the Sachdev-Ye-Kitaev model

We describe the nonequilibrium quench dynamics of the Sachdev-Ye-Kitaev models of fermions with random all-to-all interactions. These provide tractable models of the dynamics of quantum systems without quasiparticle excitations. The Kadanoff-Baym equations show that, at long times, the fermion two-point function has a thermal form at a final temperature determined by energy conservation, and the numerical analysis is consistent with a thermalization rate proportional to this temperature. We also obtain an exact analytic solution of the quench dynamics in the large $q$ limit of a model with $q$ fermion interactions: in this limit, the thermalization of the two-point function is instantaneous.

Figure 1

The ratio of the Keldysh and spectral function is shown before and after a quench from a $J_2$+$J_4$ model to a purely $J_4$ model. Fits to this data allow determination of the effective temperature as a function of time after the quench ${\beta }_{\text{eff}}\left(\mathcal{T}\right)$ using Eq. (3.8).

Figure 2

Fits to the values of the effective temperature ${\beta }_{\text{eff}}\left(\mathcal{T}\right)=1/T_{\text{eff}}\left(\mathcal{T}\right)$ from results as in Fig. 1 to Eq. (1.8) to allow determination of the thermalization rate $\mathrm{\Gamma }$ for each quench.

Figure 3

The values of the thermalization rate $\mathrm{\Gamma }$ obtained from Fig. 2 are shown as a function of the final temperature $T_f$ for each quench. One of our main numerical results is the proportionality of $\mathrm{\Gamma }$ to $T_f$ at small $T_f\ll J$. At low temperatures the relaxation towards equilibrium is expected to be controlled by quantum and classical fluctuations, and hence we expect an equilibration rate proportional to $T_f$. At higher temperatures $\mathrm{\Gamma }$ is of order $J_4$, and since at high temperatures the relaxation is essentially dominated by thermal fluctuations, the relaxation rate is dominated by the microscopic interaction energy scale of the fermions.

Figure 4

The regions in the $t_1-t_2$ plane.

Figure 5

The spectral function of the random-hopping model A long before ($\mathcal{T}=-14.7$) and after ($\mathcal{T}=14.7$) a parameter quench from $J_{2,i}=1$ to $J_{2,f}=0.5$ and 2.0. The width depends only on the value of $J_f$, becoming wider if $J_f>J_i$ and narrower if $J_f<J_i$.

Figure 6

The Keldysh component of the Green's function of the random-hopping model A long before ($\mathcal{T}=-14.7$) and after ($\mathcal{T}=14.7$) a parameter quench from $J_{2,i}=1$ to $J_{2,f}=0.5$ and 2.0.

Figure 7

The spectral function for Majorana fermions is shown long after a quench starting with free Majorana fermions and switching on a $q=4$ SYK interaction term. Here $J_{4,f}=1$.

Figure 8

The propagation of the conjugacy property in the $t_1-t_2$ plane.