Phase-space characterization of complexity in quantum many-body dynamics

We propose a phase-space Wigner harmonics entropy measure for many-body quantum dynamical complexity. This measure, which reduces to the well-known measure of complexity in classical systems and which is valid for both pure and mixed states in single-particle and many-body systems, takes into account the combined role of chaos and entanglement in the realm of quantum mechanics. The effectiveness of the measure is illustrated in the example of the Ising chain in a homogeneous tilted magnetic field. We provide numerical evidence that the multipartite entanglement generation leads to a linear increase in entropy until saturation in both integrable and chaotic regimes, so that in both cases the number of harmonics of the Wigner function grows exponentially with time. The entropy growth rate can be used to detect quantum phase transitions. The proposed entropy measure can also distinguish between integrable and chaotic many-body dynamics by means of the size of long-term fluctuations which become smaller when quantum chaos sets in.

Figure 1

(Color online) Time dependence of the entropy measure $S\left(t\right)$ for a chain of ten spins with (a) nonintegrable Hamiltonian as given in Eq. (15) with a longitudinal field $h_z=1.0$ and (b) integrable Hamiltonian as given in Eq. (16). Curves from top to bottom correspond to transverse field $h_x=1.0–0.1$ in decreasing steps of 0.1. Note that during an initial time window $S\left(t\right)$ is clearly seen to grow linearly with time, implying the exponential growth of the number of harmonics in both (a) and (b), for a sufficiently large $h_x$. All the parameters mentioned here, and in the other figures, are dimensionless (we set $\hslash =J=1$).

Figure 2

(Color online) Dependence of entropy $S\left(t\right)$ at time $t=1$ on the strength of the external perturbation $h_x$ for different chain lengths $N$ for (a) a nonintegrable spin chain and (b) an integrable spin chain. For both cases and for sufficiently large $h_x$, it is seen that $S\left(t=1\right)$ scales linearly with $N$.

Figure 3

(Color online) Time dependence of average value of participation number $⟨N_{AB}⟩$ calculated over all balanced bipartitions of the system for (a) nonintegrable and (b) integrable models with parameters discussed in Fig. 1. Curves from top to bottom correspond to transverse field $h_x=1.0–0.1$ in decreasing steps of 0.1. Within a small time window, $⟨N_{AB}⟩\propto e^{At}$ for relatively large $h_x$, as shown by an exponential fit (circles) for $h_x=0.8$ in both panels.

Figure 4

(Color online) Comparison of the dynamics of normalized entropy $S_{\text{norm}}$ and global entanglement $E_{\text{global}}$ for a chain of ten spins with transverse field $h_x=0.8$. (a) corresponds to the nonintegrable model with longitudinal field $h_z=1.0$, and (b) corresponds to the integrable model. Here, $S_{\text{norm}}$ is the entropy $S$ divided by its maximum value $N\text{ln}2$. A close correspondence between the dynamics of two measures is evident.

Figure 5

(Color online) (a) shows $S\left(t=0.5\right)$ normalized by a factor of $N\text{ln}2$ as a function of the transverse field $h_x$ for a transverse Ising chain of ten spins. (b) shows the parallel results for $E_{\text{global}}$ at $t=2.0$. Here, $\lambda \equiv h_x/\left(J+h_x\right)$. Similar to $E_{\text{global}}$, the complexity measure $S$ is seen to peak clearly at the critical point $\lambda =1/2$.

Figure 6

(Color online) Time evolution of entropy $S\left(t\right)$ for a chain of 12 spins with (a) nonintegrable Hamiltonian in Eq. (15) for $h_z=1.0$ and (b) integrable Hamiltonian in Eq. (16). Curves from top to bottom corresponds to transverse field $h_x=1.0$, 0.5, and 0.2. Note that the long-term dynamics of the entropy in the two panels is qualitatively different.

Figure 7

(Color online) Time averaged entropy, denoted $\overline{S}$, vs the strength of the transverse field $h_x$, for a chain of 12 spins. The nonintegrable model corresponds to the Hamiltonian in Eq. (15) with $h_z=1.0$; the integrable model corresponds to the Hamiltonian in Eq. (16). The difference in $\overline{S}$ between integrable and nonintegrable dynamics decreases with increasing $h_x$.

Figure 8

(Color online) Standard deviation $\sigma \left[S\right]$ in the entropy $S\left(t\right)$ vs the strength of the transverse field $h_x$ for a chain of 12 spins. Here, the integrable and nonintegrable models are the same as in Fig. 7. It is observed that as $h_x$ increases, the standard deviation generally decreases in the nonintegrable model, but increases in the integrable model. For large values of $h_x$, $\sigma \left[S\right]$ for a nonintegrable chain is much smaller than that for an integrable chain.

Figure 9

(Color online) Standard deviation of the $x$-polarization expectation value as a function of transverse field $h_x$ for a chain of 12 spins. Here, the integrable and nonintegrable models are the same as in Fig. 7. Similar to the case of entropy $S\left(t\right)$, with increasing perturbation, the standard deviation $\sigma \left[X\right]$ decreases in the nonintegrable model and increases in the integrable model.