Approximating the long time average of the density operator: Diagonal ensemble

For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e., the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier [M. C. Bañuls, D. A. Huse, and J. I. Cirac, Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a nonintegrable spin chain, for which we observe that local observables converge towards thermal values polynomially with the inverse width of the filter.

Figure 1

Scaling of the variance ${\delta }^2=\langle {\rho }_M|H_C^{†}{H}_C|{\rho }_M\rangle$, as a function of the Chebyshev truncation parameter $M$ for different system sizes $N=20–60$ with bond dimension $D=1000$ and initial state $|X+\rangle$. Except for the smallest values of $M$, we find that our results scale with the expected [31] ${\delta }^2\propto 1/M^2$.

Figure 2

Relation between vector norm of off-diagonal components and inverse off-diagonal width for system sizes $N=20–60$ and bond dimension, $D=1000$, starting with initial state $|X+\rangle$.

Figure 3

Absolute error in local observables ${\sigma }_x$ (upper) and ${\sigma }_z$ (lower figure) between exact diagonal ensemble values and Chebyshev filter results as a function of inverse off-diagonal width for system sizes $N=8,10,12,20$, and 24 with the initial state $|X+\rangle$. The insets indicate the log-log plots of the corresponding figures, where we show the upper bounds with the straight dotted lines. The slope for ${\sigma }_x$ is $-0.52$ and it is $-0.53$ for ${\sigma }_z$.

Figure 4

Absolute error in local observables ${\sigma }_x$ (upper) and ${\sigma }_z$ (lower figure) between thermal values and numerical results based on a Chebyshev filter as a function of inverse off-diagonal width for system sizes $N=30–60$ with the initial state $|X+\rangle$. The insets indicate the log-log plots of the corresponding figures, where we add the upper bounds with the dotted lines and the data points belong to $N=12$ with lighter color as reference values taken from Fig. 3.

Figure 5

Absolute error in local observables ${\sigma }_x$ (upper) and ${\sigma }_z$ (lower figure) between exact diagonal ensemble values and Chebyshev filter results, as a function of inverse off-diagonal width for system sizes $N=8,10,12$, and 20 with the initial state $|Z+\rangle$. The insets indicate the log-log plots of the corresponding figures, where we show the upper bounds with the grey dotted lines. The slope for ${\sigma }_x$ is $-0.62\left(63\right)$ and it is $-0.60\left(47\right)$ for ${\sigma }_z$.

Figure 6

Absolute error in local observables ${\sigma }_x$ (upper) and ${\sigma }_z$ (lower figure) between thermal values and numerical results based on a Chebyshev filter as a function of inverse off-diagonal width for system sizes $N=30–60$ with the initial state $|Z+\rangle$. The insets indicate the log-log plots of the corresponding figures, where we put the upper bounds with the dotted lines and the data points belong to $N=12$ with lighter color as reference values taken from Fig. 5.

Figure 7

Upper figure: Operator space entanglement entropy of the half chain as a function of logarithm of the system size, $N$, with different truncation numbers of Chebyshev filter, $M=f\left(N\right)$, and bond dimension, $D=1000$, for initial state $|X+\rangle$. Our data show that the entropy grows with $logN$ in all cases except that the line for $M=5\sqrt{N}$ stays constant. Lower figure: Behavior of the exponential of the entropy as predicted by Ref. [31] that we have shown in Eq. (8). The dotted line indicates the linear fit where all data points locate on the same line as expected for large system sizes. $D_0$ from fitting the data for all system size is 2.76(40) and the slope of the fit is 1.

Figure 8

Relation between entropy and logarithm of $1/\delta$ based on exact calculation for $N=8,10,12$ with initial state, $|X+\rangle$ (left) and $|Z+\rangle$ (right figure).

Figure 9

Scaling of the bond dimension required to keep a constant precision in the MPS approximation of $T_m\left(H_C\right)|{\rho }_0\rangle$, as a function of the degree $m$ for various values of the truncation error, ${10}^{-2},{10}^{-3},{10}^{-4},{10}^{-5}$, and system sizes $N=20$ (left) and $N=30$ (right) for $D=500$.

Figure 10

Expectation value of local observables ${\sigma }_x$ at the middle of the chain based on Chebyshev filter simulations for initial states $|X+\rangle$ (upper) and $|Z+\rangle$ (lower figure) as a function of inverse off-diagonal width for system sizes $N=10–50$. Red, blue, and green solid lines indicate the long time average values for $N=10,12,20$ respectively. Red, blue, and green dashed lines indicate the thermal values for the same system sizes $N=10,12,20$ respectively, belonging to the same color map in the legend.