Lyapunov growth in quantum spin chains

The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analog of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.

Figure 1

The level spacing distribution for the Ising spin chain $\left(j=1/2\right)$ for an $L=14$ chain at integrable (leftmost) and chaotic (rightmost) points in parameter space follow the Poisson distribution (2.3) and the Wigner surmise (2.2), respectively. In between these two parameter points, there is a continuous transition from Poisson to Wigner-Dyson statistics. The middle figure, with magnetic fields very near the integrable line, displays this crossover.

Figure 2

The commutator squared as a function of time depicted for various positions of the second operator. No exponential regime develops as the distance between the two operators grows. The figure on the left uses a semi-log scale where exponential growth would appear linear, whereas the figure on the right uses a log-log scale to exhibit the early time power-law growth. The BCH prediction for this early time power-law growth is superimposed (dashed line).

Figure 3

The commutator squared for sites 2, 4, 6, and 8 of an eight-site chain. A fit to the ansatz (2.9) with $p=1$ for the near saturation behavior (red dashed line) and the early time BCH behavior (blue dashed line) are superimposed.

Figure 4

The level spacing statistics for a two-site chain for $j=45$ (left) and a six-site chain for $j=2$ (right) with a purely transverse magnetic field. For both chain lengths we see no sign of the integrability at $h_z=0$ present in the $j=1/2$ case.

Figure 5

The level spacing statistics for a two-site chain for $j=45$ as we vary the transverse magnetic field $h_x$. The center figure is at the conventional strongly chaotic point $(h_x^{*},h_z^{*})=(-1.05,0.5)$ where we observe Wigner-Dyson statistics. As we decrease or increase the magnetic field $h_x$ (in the figures to the left and right, respectively) we see that the Wigner-Dyson statistics start to break down.

Figure 6

The level spacing statistics for a six-site chain for $j=2$ as we vary the transverse magnetic field $h_x$. The center figure is at the conventional strongly chaotic point $(h_x^{*},h_z^{*})=(-1.05,0.5)$ where we observe Wigner-Dyson statistics. As we decrease or increase the magnetic field $h_x$ we see that the Wigner-Dyson statistics start to break down, although the spread in the magnetic fields for which one finds Wigner-Dyson statistics is larger than for the two-site chain.

Figure 7

A semi-log plot of the commutator squared is shown as a function of time for various spins, with separation $x=0$ on the left and $x=1$ on the right. The lowermost (light blue) curve corresponds to the spin 1/2 case. The different curves represent increasing spin representations as one moves towards the darkest curve, for which $j=61$. An exponential regime emerges at intermediate values of the commutator squared and grows as we increase the spin.

Figure 8

A semi-log plot of the commutator squared displaying its late time saturation for various spins. The horizontal black lines depicts the naive saturation of the commutator squared at the value of 2.

Figure 9

The various time intervals used for the fits of the Lyapunov exponents at the strongly chaotic point. The lower (upper) bounds of the time interval $t_i\left(t_f\right)$ are taken in between the two vertical red (blue) lines. The left figure is for $x=0$ and the right for $x=1$.

Figure 10

The left figure demonstrates the spread of the linear fits (gray) to the region of exponential growth for the commutator squared at the highest spin $\left(j=61\right)$ we studied for various choices of the time interval to which we fit. The commutator squared is shown for $j=1,7,16,20,25,33,45$, and 61, to illustrate its approach to these linear fits. The right figure depicts commutator squared and its corresponding fit for $j=20$ in blue and $j=61$ in red. The dashed vertical lines illustrate the time at which the data starts to deviates from the fit. The region where the data fits well to a line corresponds to an exponential growth of roughly three and four e-foldings for $j=20$ and 61, respectively.

Figure 11

The Lyapunov exponents extracted from the fit to the exponential regime of the commutator squared as a function of the spin $j$ at infinite temperature and at the strongly chaotic point $(h_x^{*},h_z^{*})=(-1.05,0.5)$ for the largest time range $[t_i,t_f]$. A power law (red, dashed), an exponential (blue, solid), and a logarithmic (green, dash-dotted) function have been fitted to the data for separations of $x=0$ (left) and $x=1$ (right). We see that both the power law and exponentials provide a good fit, whereas the logarithm (which grows without bound at large spin) does not.

Figure 12

A comparison between the Lyapunov exponent at the highest computed spin (yellow diamonds) and the ones obtained from a power law (blue circles) or an exponential (red circles) extrapolation to infinite spin. The transverse field $h_x$ is varied around the strongly chaotic value of $h_x^{*}=-1.05$ and the longitudinal field is fixed at $h_z^{*}=0.5$. The error bars denote one standard deviation of the distribution of Lyapunov exponents found using the procedure outlined in Sec. 4b2.

Figure 13

The commutator squared for the highest spin at the strongly chaotic point together with the slope (black) found by averaging the Lyapunov exponents over both extrapolation methods and over the two sites. The blue (orange) curve is for $x=0\left(x=1\right)$.

Figure 14

A semi-log plot of the time dependence of the commutator squared at the strongly chaotic point for a three-site chain. The small fluctuations on top of the exponential regime that are present when the perturbation is close to the probe site seem to become already almost imperceptible for a distance of 2 between the perturbation and the probe site.

Figure 15

The left figure shows the classical version of the commutator squared (blue solid line), which is the average over initial conditions of a Poisson bracket squared, between $S_z^{\left(1\right)}\left(t\right)$ and $S_z^{\left(1\right)}\left(0\right)$. This is plotted for a two-site chain at the strongly chaotic point. The classical Lyapunov exponent is extracted from a fit (purple dashed line) to the exponential regime. On top of the Poisson bracket, the commutator squared [times the usual factor of $j(j+1)]$ is shown for increasing spins in red. In the right figure, the classical (blue solid) and highest spin quantum (red dotted line) commutator squared are displayed at early times for both sites, with their respective fits. The slope of the fit to the classical data (purple dashed line) is obtained by averaging the Lyapunov exponents at the two sites, while an average of the power-law extrapolated Lyapunov exponent at both sites is used as slope to show the fit to the quantum data (purple dashed-dotted line).

Figure 16

A comparison between the classical Lyapunov exponents (yellow diamonds) and those obtained from the classical limit of the quantum model using the power-law extrapolation (blue circles) and the exponential extrapolation (red squares) for a range of transverse field strengths. The error bars depict an estimate of the uncertainty involved in extracting these Lyapunov exponents as described in Secs. 4b2 and 5a. The longitudinal magnetic field is fixed at $h_z^{*}=0.5$. We find a good match between the exponents extracted from the power-law approach and the classical exponents in the strongly chaotic domain.

Figure 17

The classical version of the commutator squared (blue solid line) together with the the quantum commutator squared [times the usual factor of $j(j+1)]$ at $j=61$ (red dotted line), for a two-site chain with both sites superimposed and for various strengths of the transverse magnetic field in our analysis. The purple dashed and dashed-dotted straight lines represent the fits to the classical and quantum Lyapunov regime, where respectively the average of the classical and the average of the power-law extrapolated exponents at the two sites were used as a slope. The left figure shows the time evolution until the saturation of the Poisson bracket while the right figure emphasizes the early time part.