Operator entanglement entropy of the time evolution operator in chaotic systems

We study the growth of the operator entanglement entropy (EE) of the time evolution operator in chaotic, many-body localized (MBL) and Floquet systems. In the random-field Heisenberg model we find a universal power-law growth of the operator EE at weak disorder, a logarithmic growth at strong disorder, and extensive saturation values in both cases. In a Floquet spin model, the saturation value after an initial linear growth is identical to the value of a random unitary operator (the Page value). We understand these properties by mapping the operator EE to a global quench problem evolved with a similar parent Hamiltonian in an enlarged Hilbert space with the same chaotic, MBL, and Floquet properties as the original Hamiltonian. The scaling and saturation properties reflect the spreading of the state EE of the corresponding time evolution. We conclude that the EE of the evolution operator should characterize the propagation of information in these systems.

Figure 1

Matrix product state (MPS) and matrix product operator (MPO). Upper panel shows a diagrammatic representation of the matrix product state. Each tip of the vertical bond is a physical index; horizontal bonds contract auxiliary matrix indices. After performing a left and right canonicalization up to one bond, the chosen bond will contain all the Schmidt eigenvalues that determine the entanglement entropy. Bottom panel shows that the matrix product operator can be viewed as a matrix product state with two copies of physical indices on each site; then its entanglement entropy can be similarly defined and calculated by a matrix product algorithm.

Figure 2

Operator entanglement entropy $S\left[\widehat{U}\left(t\right)\right]$ of the time evolution operator $\widehat{U}\left(t\right)$ for the models introduced in Sec. 4 as a function of time $t$. Top panel shows chaotic models. Bottom panel shows integrable (MBL) models. For all models, the opEE grows fast for short times and, for the considered nonintegrable models, the opEE reaches a large value close to that of a random operator ($Lln2-\frac{1}{2}$). For the clean integrable cases, the opEE fluctuates at large times due to commensurate periods of integrals of motion. The MBL model exhibits the slowest growth of the opEE with time. For the Floquet model, the opEE grows nearly linearly and saturates at the limit of a random unitary operator. The results for disordered models are averaged over 100 to 1000 realizations.

Figure 3

Comparison of saturation value of the opEE of the time evolution operator $S\left[\widehat{U}\left(\infty \right)\right]$ (crosses and error bars illustrate the size of fluctuations around the saturation value) of the Floquet model with the result for a random unitary operator (Page value, given by full lines) for all possible smaller subsystem sizes $\ell$. The opEE for the Floquet system perfectly matches the Page result for all system sizes and partitions.

Figure 4

Saturation value of the disorder-averaged opEE for the equal bipartition for the random-field Heisenberg model as a function of system size for different disorder strengths. For strong disorder $W\gtrsim 3.7$, the system is in the MBL phase and we observe a suppressed but extensive saturation value. At weak disorder, the saturation value scales as $Lln2$ but has a constant deficit.

Figure 5

Block-diagonal form of the reduced density matrix.

Figure 6

Probability distribution of the entropy $S_i$ of different blocks of the operator reduced density matrix $\rho$. The blocks are labeled by the number of up spins on subsystem $A$ in the first and second index of $\rho$ and the Page value for each block is indicated by a dashed line, clearly showing that it is an upper bound for the block entropy.

Figure 7

Average weight $p_i$ of different blocks of the operator reduced density matrix $\rho$ vs the value of the block entropy $S_i$.

Figure 8

Growth of the operator entanglement entropy in the Floquet model (31). The thick black lines correspond to a power-law fit. Left panel shows Floquet model with time-reversal symmetry ($h_y^i=0$), a small deviation from a linear growth is visible in the exponent. Right panel shows Floquet model without time-reversal symmetry ($h_y^i=0.3457$), the linear growth is almost perfect.

Figure 9

(left) Growth of the disorder-averaged operator entanglement entropy of the evolution operator in the random-field Heisenberg model ($S_z=0$) at short times for various disorder strengths. (right) Growth of the opEE in units of the saturation value $S_{\infty }$ (dashed horizontal lines) as a function of time in units of the half saturation time $t_{0.5}$. We observe a power-law growth at weak disorder with an exponent $\alpha <1$, which is valid for intermediate times, when $t\approx t_{0.5}$. At stronger disorder, the exponent decreases and finite-size effects are stronger, seemingly leading to a long domain of sup-power-law growth of the opEE.

Figure 10

Growth of operator entanglement entropy in the MBL phase of the random-field Heisenberg model ($S_z=0$).

Figure 11

Channel-state duality point of view of opEE. The vertical lines correspond to the two copies of the original system and the bipartition of the system into $A$ and $B$ has to be performed equally in both copies.

Figure 12

Diagrammatic representation of $\mathrm{Tr}\left({\rho }^2\left[U\right]\right)$.

Figure 13

Eight indices in $\mathrm{Tr}\left({\rho }^2\left[U\right]\right)$, where contractions are performed for the four pairs.

Figure 14

$\delta$ functions for each permutation element $\sigma$ will close these diagrams. Each loop will contribute a ${\left(2^{{\ell }_A}\right)}^2$ or $\left(2^{{\ell }_B}\right)$, where the square is for two copies of indices.