Information scrambling and redistribution of quantum correlations through dynamical evolution in spin chains

Abstract

We investigate the propagation of local bipartite quantum correlations, along with the tripartite mutual information to characterize the information scrambling through dynamical evolution of spin chains. Starting from an initial state with the first pair of spins in a Bell state, we study how quantum correlations spread to other parts of the system, using different representative spin Hamiltonians, viz. the Heisenberg Model, a spin-conserving model, the transverse-field XY model, a non-conserving but integrable model, and the kicked Harper model, a spin conserving but nonintegrable model. We show that the local correlations spread consistently in the case of spin-conserving dynamics in both integrable and nonintegrable cases, with a strictly nonnegative tripartite mutual information. In contrast, in the case of non-conserving dynamics, tripartite mutual information is negative and local pair correlations do not propagate.

Appendices

Appendix A: Pairwise correlation functions for the XY model with transverse field

The expectation values of the correlation functions $\langle c^{\dagger }_jc_j\rangle$, $\langle c_jc^{\dagger }_{j+1}\rangle$, $\langle c_jc_{j+1} \rangle$ and $\langle c^{\dagger }_jc_jc^{\dagger }_{j+1}c_{j+1}\rangle$ as a function of time can be calculated analytically as shown as

$$\begin{aligned} \langle c^{\dagger }_jc_j\rangle _t&= \frac{1}{N} \sum _{q_1,q_2} e^{iq_1j-iq_2j} \langle (\chi ^*_{q_1} c^{\dagger }_{q_1} + \xi ^*_{q_1} c_{-q_1}) (\chi _{q_2} c_{q_2} + \xi _{q_2} c_{-q_2}^\dagger )\rangle ,\nonumber \\ \langle c_jc_{j+1}\rangle _t&= \frac{1}{N}\sum _{q_1,q_2} e^{iq_1j-iq_2(j+1)} \langle (\chi _{q_1} c_{q_1} + \xi _{q_1} c_{-q_1}^\dagger ) (\chi _{q_2} c_{q_2} + \xi _{q_2} c_{-q_2}^\dagger )\rangle ,\nonumber \\ \langle c_jc^{\dagger }_{j+1}\rangle _t&= \frac{1}{N}\sum _{q_1,q_2} e^{-iq_1j+iq_2(j+1)}\langle (\chi _{q_1} c_{q_1} + \xi _{q_1} c_{-q_1}^\dagger )(\chi ^*_{q_2} c^{\dagger }_{q_2} + \xi ^*_{q_2} c_{-q_2})\rangle ,\nonumber \\ \langle c^{\dagger }_jc_jc^{\dagger }_{j+1}c_{j+1}\rangle _t&= \frac{1}{N^2}\sum _{q_1,q_2,q_3,q_4}e^{iq_1j-iq_2j+iq_3(j+1)-iq_4(j+1)} \langle (\chi ^*_{q_1} c^{\dagger }_{q_1} + \xi ^*_{q_1} c_{-q_1})\nonumber \\&\quad (\chi _{q_2} c_{q_2} + \xi _{q_2} c_{-q_2}^\dagger ) (\chi ^*_{q_3} c^{\dagger }_{q_3} +\xi ^*_{q_3} c_{-q_3})(\chi _{q_4} c_{q_4} + \xi _{q_4}c_{-q_4}^\dagger )\rangle . \end{aligned}$$

(A1)

Fig. 9

Correlation functions between the sites i and $i+1$ are shown as functions of time t for the transverse field XY model for various Hamiltonian parameters. $\mathrm{Re} \langle \sigma ^+_i \sigma ^-_{i+1}\rangle _t$ is plotted as a function of time for parameters: a $J_x =0.7, J_y = 0.3$ and $h = 0.1$, b $J_x =0.7, J_y = 0.3$ and $h = 1.0$, c $J_x =0.7, J_y = 0.3$ and $h = 10.0$. $\mathrm{Re} \langle \sigma ^+_i \sigma ^+_{i+1}\rangle _t$ is plotted as a function of time for parameters: d $J_x =0.7, J_y = 0.3$ and $h = 0.1$, e $J_x =0.7, J_y = 0.3$ and $h = 1.0$, f $J_x =0.7, J_y = 0.3$ and $h = 10.0$. $\langle \sigma ^z_i \sigma ^z_{i+1}\rangle _t$ for parameters: g $J_x =0.7, J_y = 0.3$ and $h = 0.1$, h $J_x =0.7, J_y = 0.3$ and $h = 1.0$, i $J_x =0.7, J_y = 0.3$ and $h = 10.0$. The results are shown from analytical calculations for the initial state: $\vert \Psi (0)\rangle = (\vert 10\ldots 0\rangle +\vert 010\ldots 0\rangle )/\sqrt{2}$

We see from the above set of equations that the time evolution mixes different operators. Any product of fermion operators in the real space involves N sums over momenta values in real space. The expectation values of products of fermionic operators can be straightforwardly evaluated using Wick’s theorem [44, 45]. However, there will be N sums over the momentum variables. For a large value of N, the above sums are evaluated by converting them to integrals from 0 to $\pi$. The pairwise correlation functions between the nearest neighbors are plotted as functions of site index i and time t in Fig. 9. We have taken three sets of Hamiltonian parameters $(J_x = 0.7, J_y = 0.3, h = 0.1)$; $(J_x = 0.7, J_y = 0.3, h = 1.0)$ and $(J_x = 0.7, J_y = 0.3, h = 10.0)$ to illustrate the results. The total number of down (up) spins in the system is not conserved as $J_x \ne J_y$. However, in the limit $h \rightarrow \infty$, the dynamics is confined to a subspace of the total Hilbert space, as the Hamiltonian in this case commutes with the total number of down (up) spins. The initial state being $\vert \Psi \rangle = \frac{1}{\sqrt{2}}(100...0 + 010...0)$, the initial values of the off-diagonal correlation function $\langle \sigma ^+_i \sigma ^-_{i+1}\rangle$ is 0.5 the diagonal correlation function $\langle \sigma ^z_i \sigma ^z_{i+1}\rangle$ is $-1$ for the first pair and zero for all other pairs.

As time evolves, the correlation function $\langle \sigma ^+_i \sigma ^-_{i+1}\rangle$ becomes nonzero for farther sites for later times implying a finite speed of propagation of correlations) for the case $h = 0.1$ as shown in Fig. 9a. The value of the function $\langle \sigma ^+_i \sigma ^-_{i+1} \rangle$ is nonzero within the ‘light cone’ but zero outside. For the case $h = 1.0$, the correlation function $\langle \sigma ^+_i \sigma ^-_{i+1} \rangle$ decays very quickly and becomes zero beyond the third site as shown in Fig. 9b. For the case $h = 10.0$, the correlations propagate consistently and continuously to further sites with a finite speed as shown in Fig. 9c. The diagonal correlation function $\langle \sigma ^z_i \sigma ^z_{i+1} \rangle$ becomes nonzero for farther sites quickly and propagation does not take place with a finite speed for the cases $h = 0.1$ and $h = 1.0$ as shown in Fig. 9d, e, respectively. For the case $h = 10.0$, the value of the function $\langle \sigma ^z_i \sigma ^z_{i+1} \rangle$ spreads with finite speed and its value is zero outside the light cone as shown in Fig. 9f. The correlation function $\langle \sigma ^+_i \sigma ^+_{i+1} \rangle$ is plotted as a function of time and site index for the same set of Hamiltonian parameters in Fig. 9g–i, respectively. The expectation value of the correlation function $\langle \sigma ^+_i \sigma ^+_{i+1} \rangle$ is initially zero and becomes nonzero as the number of down spins increases in the system. The values of $\langle \sigma ^+_i \sigma ^+_{i+1} \rangle$ for the case $h = 10.0$ in Fig. 9i is much smaller compared to the cases $h = 0.1$ and $h = 1.0$ as shown in Fig. 9g, h, respectively. This indicates that for a high value of h, less number of down spins is generated in the system as time evolves and the dynamics remains confined mainly in the one spin sector.

Appendix B: Pairwise correlation functions for the Ising model

For the Ising model, the time evolved expectation value of any operator O is given as

$$\begin{aligned} \langle O \rangle _t = \langle e^{iJt\sum _i \sigma ^x_{i} \sigma ^x_{i+1}} O e^{-iJt\sum _i \sigma ^x_{i} \sigma ^x_{i+1}} \rangle . \end{aligned}$$

(B1)

The forms of $\langle \sigma ^z_j \rangle _t, \langle \sigma ^\pm _j\rangle _t, \langle \sigma ^z_j \sigma ^z_{j+1}\rangle _t$ and $\langle \sigma ^+_j \sigma ^\mp _{j+1}\rangle _t$ can be given by the following

$$\begin{aligned} \langle \sigma ^z_j\rangle _t&= (C^4 +S^4 - 2C^2 S^2)\langle \sigma ^z_j\rangle + (2CS^3- 2C^3S) (\langle \sigma ^y_j \sigma ^x_{j+1}\rangle + \langle \sigma ^x_{j-1} \sigma ^y_{j}\rangle ) \nonumber \\&\quad -4C^2S^2 \langle \sigma ^x_{j-1} \sigma ^z_{j} \sigma ^x_{j+1}\rangle ,\nonumber \\ \langle \sigma ^\pm _j\rangle _t&= 0,\nonumber \\ \langle \sigma ^z_j \sigma ^z_{j+1}\rangle _t&= (C^6 +S^6 - C^4S^2 -C^2S^4)\langle \sigma ^z_j \sigma ^z_{j+1}\rangle + (CS^5+C^3S^3-2C^5S) (\langle \sigma ^z_j \sigma ^y_{j+1}\sigma ^x_{j+2}\rangle \nonumber \\&\quad + \langle \sigma ^x_{j-1} \sigma ^y_{j}\sigma ^z_{j+1}\rangle )+ (4C^2S^4 +4C^4S^2) \langle \sigma ^x_{j-1} \sigma ^y_{j} \sigma ^y_{j+1} \sigma ^x_{j+2} \rangle ,\nonumber \\ \langle \sigma ^+_j \sigma ^\mp _{j+1}\rangle _t&= (C^6 +S^6)\langle \sigma ^+_j \sigma ^\mp _{j+1}\rangle + (C^4S^2 +C^2S^4) (\langle \sigma ^+_j \sigma ^\pm _{j+1}\rangle + \langle \sigma ^-_j \sigma ^\pm _{j+1}\rangle + \langle \sigma ^-_j \sigma ^\mp _{j+1}\rangle ) \nonumber \\&\quad + (iC^5S - iC^3 S^3) \Big (\frac{1}{2}\langle \sigma ^+_j (1\mp \sigma ^z_{j+1}) \sigma ^x_{j+2} \rangle \nonumber \\&\quad + \frac{1}{2}\langle \sigma ^x_{j-1} (1+\sigma ^z_{j}) \sigma ^\mp _{j+1} \rangle + \frac{1}{4}(1+\sigma ^z_{j})(1\mp \sigma ^z_{j+1})\Big )\nonumber \\&\quad + (iCS^5 - iC^5 S) \Big (\frac{1}{2}\langle \sigma ^+_j (1\pm \sigma ^z_{j+1}) \sigma ^x_{j+2} \rangle \nonumber \\&\quad + \frac{1}{2}\langle \sigma ^x_{j-1} (1-\sigma ^z_{j+1}) \sigma ^\mp _{j+1} \rangle + \frac{1}{4}(1-\sigma ^z_{j})(1\pm \sigma ^z_{j+1})\Big )\nonumber \\&\quad +(C^2S^4+C^4S^2)\Big (\frac{1}{4} \langle \sigma ^x_{j-1} (1+\sigma ^z_{j})(1\pm \sigma ^z_{j+1})\sigma ^x_{j+2}\rangle \nonumber \\&\quad -\frac{1}{4} \langle \sigma ^x_{j-1} (1-\sigma ^z_{j})(1\pm \sigma ^z_{j+1})\sigma ^x_{j+2}\rangle \nonumber \\&\quad -\frac{1}{4} \langle \sigma ^x_{j-1} (1+\sigma ^z_{j})(1\mp \sigma ^z_{j+1})\sigma ^x_{j+2}\rangle +\frac{1}{4} \langle \sigma ^x_{j-1} (1-\sigma ^z_{j})(1\mp \sigma ^z_{j+1})\sigma ^x_{j+2}\rangle \nonumber \\&\quad + \frac{1}{2} \langle \sigma ^x_{j-1}\sigma ^-_{j} (1\mp \sigma ^z_{j+1})\rangle + \frac{1}{2} \langle \sigma ^x_{j-1}\sigma ^-_{j} (1\pm \sigma ^z_{j+1})\rangle - \frac{1}{2} \langle \sigma ^x_{j-1}\sigma ^+_{j} (1\pm \sigma ^z_{j+1})\rangle \nonumber \\&\quad - \frac{1}{2} \langle \sigma ^x_{j-1}\sigma ^+_{j} (1\mp \sigma ^z_{j+1})\rangle \Big ) +\frac{1}{2} \langle (1+\sigma ^z_{j})\sigma ^\pm _{j+1}\sigma ^x_{j+2} \rangle + \frac{1}{2} \langle (1+\sigma ^z_{j})\sigma ^\pm _{j+1}\sigma ^x_{j+2} \rangle \nonumber \\&\quad - \frac{1}{2} \langle (1-\sigma ^z_{j})\sigma ^\pm _{j+1}\sigma ^x_{j+2} \rangle -\frac{1}{2} \langle (1+\sigma ^z_{j})\sigma ^\pm _{j+1}\sigma ^x_{j+2} \rangle ), \end{aligned}$$

(B2)

with $C \equiv \cos (Jt)$ and $S \equiv \sin (Jt)$.