Quench dynamics in randomly generated extended quantum models

We analyze the thermalization properties and the validity of the eigenstate thermalization hypothesis in a generic class of quantum Hamiltonians where the quench parameter explicitly breaks a $Z_2$ symmetry. Natural realizations of such systems are given by random matrices expressed in a block form where the terms responsible for the quench dynamics are the off-diagonal blocks. Our analysis examines both dense and sparse random matrix realizations of the Hamiltonians and the observables. Sparse random matrices may be associated with local quantum Hamiltonians and they show a different spread of the observables on the energy eigenstates with respect to the dense ones. In particular, the numerical data seem to support the existence of rare states, i.e., states where the observables take expectation values that are different compared to the typical ones sampled by the microcanonical distribution. In the case of sparse random matrices, we also extract the finite-size behavior of two different time scales associated with the thermalization process.

Figure 1

Dense random matrices (with $N=4000$). Overlaps for the quench process. Left: $|c_a{|}^2$ for the ground state. Right: $|c_a{|}^2$ for the 2000$\mathrm{th}$ state, in the middle of the energy band.

Figure 2

Dense random matrices (with $N=4000$). Typical IPR of the initial states.

Figure 3

Dense random matrices. IPR vs matrix size. The red lines are linear fits $y=ax$. Left panel: average IPR on all initial states ($a=0.3336$). Right panel: maximum IPR ($a=0.3764$).

Figure 4

Dense random matrices. EEV $\langle E_a\left|\mathcal{O}\right|E_a\rangle$ vs $E_a$. Left: even observable. Right: odd observable.

Figure 5

Dense random matrices. EEV variance (averaged over the entire spectrum) vs matrix size. The continuous line is the fit $a/x^b$ with $b=0.9\pm 0.1$. The data points for the even and odd observables exactly overlap in this plot.

Figure 6

Dense random matrices. Maximum-minimum EEV vs matrix size for the range of even (left) and odd (right) observables (again the behavior of the two is indistinguishable). The continuous line is the fit $a\sqrt{\frac{\ln N}{N}}$.

Figure 7

Dense random matrices. $\sigma =\sqrt{{({\mathcal{O}}_{\mathrm{micro}}-{\mathcal{O}}_{\mathrm{diag}})}^2}$ vs matrix size for initial states laying in the central part of the spectrum $\overline{e}\approx 0$. The continuous line is the fit $a/x^b$. Left: even observable $b=1.9\pm 0.1$, and right: odd observable $b=2.1\pm 0.1$.

Figure 8

Sparse random matrices (with $N=4000$). Left: density of states, right: level spacing statistics, the continuous line is the Wigner surmise for GOE matrices.

Figure 9

Sparse random matrices. Behavior of the overlaps $|c_a{|}^2$ for a state midspectrum (left) and for one in the upper portion of the spectrum (right).

Figure 10

Sparse random matrices (with $N=4000$). The IPR for a specific realization. The ordering of the initial states in this plot is according to their energy relative to the postquench Hamiltonian, $\langle {\psi }_0\left|H^{>}\right|{\psi }_0\rangle$.

Figure 11

Sparse random matrices. IPR vs matrix size. The continuous line is the $y=ax$ fit. Left: average IPR on the whole spectrum ($a=0.0738$). Right: maximum IPR ($a=0.2096$).

Figure 12

Sparse random matrices. IPR with respect to local basis vs matrix size. The continuous line is the $y=ax$ fit. Left: average IPR on the whole spectrum ($a=0.0321$). Right: maximum IPR ($a=0.1632$).

Figure 13

Sparse random matrices (with $N=4000$). The IPR of the postquench eigenstates with respect to the local basis for a specific realization. The ordering of the initial states in this plot is according to their energy eigenvalue.

Figure 14

Sparse random matrices. EEV $\langle E_a\left|\mathcal{O}\right|E_a\rangle$ vs $e_a$. Left: even observable. Right: odd observable.

Figure 15

Sparse random matrices. EEV variance vs matrix size for the full spectrum. The continuous lines are the fit $a/x^b$. Left: even observable $b=0.50\pm 0.01$ Right: odd observable $b=0.75\pm 0.01$.

Figure 16

Sparse random matrices. Maximum-minimum EEV vs matrix size for the full spectrum. Left: even observable. Right: odd observable.

Figure 17

Sparse random matrices. EEV variance vs matrix size for a small energy window around $e=0$. The continuous line is the fit $a/x^b$. Left: even observable $b=1.29\pm 0.01$. Right: odd observable $b=1.2\pm 0.1$.

Figure 18

Sparse random matrices. Maximum-minimum EEV vs matrix size for a small energy window around $e=0$ in logarithmic scale. The continuous line is the fit $a/x^b+c$. Left: even observable $e=0$, $b=0.65\pm 0.01$, and $c=0.067\pm 0.001$. Right: odd observable $e=0$, $b=0.50\pm 0.01$, and $c=0.01\pm 0.01$.

Figure 19

Sparse random matrices. Maximum-minimum EEV vs matrix size for a small energy window around $e=1$. The continuous line is the fit $a/x^b+c$. Left: even observable $e=1$, $b=0.35\pm 0.01$, and $c=1.7\pm 0.1$. Right: odd observable $e=1$, $b=0.33\pm 0.01$, and $c=0.14\pm 0.01$.

Figure 20

Sparse random matrices. $\sigma =\sqrt{{({\mathcal{O}}_{\mathrm{micro}}-{\mathcal{O}}_{\mathrm{diag}})}^2}$ vs matrix size for initial states laying in the central part of the spectrum $\overline{e}\approx 0$. The continuous lines are the fits $a/x^b$. Left: even observable $b=1.15\pm 0.01$. Right: odd observable $b=1.30\pm 0.01$.

Figure 21

Sparse random matrices. $\sigma =\sqrt{{({\mathcal{O}}_{\mathrm{micro}}-{\mathcal{O}}_{\mathrm{diag}})}^2}$ vs matrix size for initial states laying in a small window around $e=1$. The continuous lines are the fits $a/x^b$. Left: even observable $b=0.31\pm 0.01$. Right: odd observable $b=0.44\pm 0.01$.

Figure 22

Sparse random matrices. Kullback-Leibler entropy vs matrix size for the center of the spectrum for the uniform (squares) and the Gaussian distribution (circles).

Figure 23

Left: time evolution of the observable ${\mathcal{O}}_t$ vs time $t$. Right: time evolution of the fluctuations ${\Delta }_t^{\mathcal{O}}/{\Delta }_0^{\mathcal{O}}$ vs time $t$, logarithmic scale on $y$ axis. The straight line is the exponential fit whose slope defines ${\tau }_S$.

Figure 24

Scaling with the matrix size of the two time scale ${\tau }_S$ (left) with a linear fit ($-3.94+0.066N$) and ${\tau }_F$ (right) with a logarithmic fit ($-28.61+8.60\ln N$).