Magnetization dynamics in clean and disordered spin-1 XXZ chains

We study spin transport in the one-dimensional anisotropic $S=1$ Heisenberg model. Particular emphasis is given to dynamics at infinite temperature, where current autocorrelations and spatiotemporal correlation functions are obtained by means of an efficient pure-state approach based on the concept of typicality. Our comprehensive numerical analysis unveils that high-temperature spin transport is diffusive in the easy-axis regime for strong exchange anisotropies. This finding is based on the combination of numerous signatures, such as (i) Gaussian spreading of correlations, (ii) a time-independent diffusion coefficient, (iii) power-law decay of equal-site correlations, (iv) exponentially decaying long-wavelength modes, and (v) Lorentzian line shapes of the dynamical structure factor. Moreover, we provide evidence that some of these signatures are not exclusively restricted to the infinite-temperature limit but can persist at lower temperatures as well, where we complement our results by additional quantum Monte Carlo simulations of large systems. In contrast to the easy-axis regime, we show that in the case of an isotropic chain, the signatures (i)–(v) are much less pronounced or even entirely absent, suggesting the existence of anomalous spin transport despite the nonintegrability of the model. Eventually, upon introducing a random on-site magnetic field, we observe a breakdown of diffusion and distinctly slower dynamics. In particular, our results exhibit qualitative similarities to disordered spin-$1/2$ chains and might be consistent with the onset of many-body localization in the $S=1$ model for sufficiently strong disorder.

Figure 1

(a) Current autocorrelation $\langle j\left(t\right)j\rangle /L$ obtained for different system sizes $L=12,14,16,18$ by ED and DQT. As a comparison, we also show data for $L=16,18$ calculated in the $S^z=0$ subsector only. (b) Corresponding diffusion coefficient $D\left(t\right)$, cf. Eq. (5). Rescaled data from time-dependent density matrix renormalization group (tDMRG) calculations for $T=10$ are depicted [83, 116]. (c) Conductivity $\sigma \left(\omega \right)$ calculated according to Eq. (6) for frequency resolutions $\delta \omega =\pi /10, \pi /20$, and $\pi /100$ ($S^z=0$). As a comparison, we depict data obtained by the microcanonical Lanczos method (MCLM) from Ref. [60]. The other parameters are $\mathrm{\Delta }=1$ and $\beta =0$.

Figure 2

Analogous data as in Fig. 1, but now for a larger anisotropy $\mathrm{\Delta }=1.5$. Note that we plot the absolute value $|\langle j(t)j\rangle |/L$ in (a) for better visibility. All calculations are performed in the full Hilbert space.

Figure 3

Broadening of density profiles $C_{l,L/2}\left(t\right)$ at infinite temperature $\beta =0$ for (a) $\mathrm{\Delta }=1$ and (b) $\mathrm{\Delta }=1.5$. We have $L=18$ in both cases.

Figure 4

(a) Equal-site correlation $C_{L/2,L/2}\left(t\right)$ for $\mathrm{\Delta }=1$ and system sizes $L=10$ (ED), $L=16,18$ (DQT), in a logarithmic plot. The dashed line indicates power-law decay $\propto 1/\sqrt{t}$. The constant long-time value scales as $\frac{2}{3}/L$ [117, 118]. (b) Same data as in (a) but now for the larger anisotropy $\mathrm{\Delta }=1.5$. We have $\beta =0$ in both cases.

Figure 5

(a) Density profile $C_{l,L/2}\left(t\right)$ at fixed times $t=0,1,2,4$. Dashed curves are Gaussian fits to the data. (b) The width $\mathrm{\Sigma }\left(t\right)$, as obtained from these density profiles [symbols, Eq. (12)], is compared to the width $\mathrm{\Sigma }\left(t\right)$, as obtained from current autocorrelations [curve, Eq. (13)]. The derivative $D\left(t\right)$ in Eq. (13) as well as the quantity ${\mathrm{\Sigma }}^2/\left(2t\right)$ are shown as well. The other parameters are $\mathrm{\Delta }=1, L=18$, and $\beta =0$.

Figure 6

Analogous data as in Fig. 5, but now for the larger anisotropy $\mathrm{\Delta }=1.5$. The density profiles in (a) are well described by Gaussians over 3 orders of magnitude.

Figure 7

Structure factors for the smallest nonzero momentum $q=\pi /9$ in a chain with $L=18$. (a) $C_q\left(t\right)$ for $\mathrm{\Delta }=1,1.5$ in a semilogarithmic plot. (b) $C_q\left(\omega \right)$ for $\mathrm{\Delta }=1$. (c) $C_q\left(\omega \right)$ for $\mathrm{\Delta }=1.5$. The dashed lines indicate an exponential decay, as well as Gaussian or Lorentzian line shapes, respectively. We have $\beta =0$ in all cases, and $\delta \omega =\pi /50$ in (b) and (c).

Figure 8

Equal-site correlation $C_{L/2,L/2}\left(t\right)$ for $\mathrm{\Delta }=1.5$ and $L=12,14,16$ at the finite temperature $\beta =1$. Data are averaged over $N_S=50$ random initial states, and the shaded area indicates the standard deviation of the mean. The dashed line shows the prediction $\propto 1/\sqrt{t}$ from diffusion phenomenology.

Figure 9

(a), (b) Dynamical structure factor $C_q\left(\omega \right)$ at infinite temperature $\beta =0$ for $\mathrm{\Delta }=1$ and $\mathrm{\Delta }=1.5$, obtained by DQT for system size $L=18$. (c, d) $C_q\left(\omega \right)$ at $\beta =1$. Data obtained by DQT ($N_S=50$) for $L=16$ and $q<\pi$ are compared to QMC simulations for $L=64$ at $q>\pi$.

Figure 10

Dynamical structure factor $C_q\left(\omega \right)$ at finite temperature $\beta =1$ and various momenta $q$ both for $\mathrm{\Delta }=1$ (left) and $\mathrm{\Delta }=1.5$ (right). The data are obtained by DQT (symbols) and QMC (curves) for chains with $L=16$. Note that the data in panels (c)–(h) have been multiplied by a factor for better visibility.

Figure 11

Conductivity $\sigma \left(\omega \right)$ for disorder $W=1,2,4$ and anisotropies (a) $\mathrm{\Delta }=1$, (b) $\mathrm{\Delta }=1.5$. In the low-$\omega$ regime, the conductivity is well described by power laws, $\text{Re}\sigma \left(\omega \right)\approx {\sigma }_{\text{dc}}+a{\left|\omega \right|}^{\alpha }$, with $\alpha =1$, cf. Refs. [123, 124]. We have $\beta =0, L=16$, and $N=100$ in all cases.

Figure 12

Broadening of density profiles $C_{l,L/2}\left(t\right)$ for different values of disorder (a) $W=1$, (b) $W=4$. We have $\mathrm{\Delta }=1.5, \beta =0, L=16$, and $N=100$ in both cases.

Figure 13

$C_{l,L/2}\left(t\right)$ at fixed times $t=1,5,10$ for $W=1$ [(a)–(c)] and $W=4$ [(d)–(f)] in a semilogarithmic plot. Curves indicate Gaussian or exponential fits. We have $\mathrm{\Delta }=1.5, \beta =0, L=16$, and $N=100$ in all cases.

Figure 14

$C_{L/2,L/2}\left(t\right)$ for disorder $W=0,1,2,4$ in a logarithmic plot. We have $\mathrm{\Delta }=1.5, \beta =0, L=16$, and $N=100$ in all cases.

Figure 15

(a) $C_q\left(t\right)$ at momentum $q=\pi /8$ for disorder $W=0,1,2,4$ in a semilogarithmic plot. (b, c) Contour plots of $C_q\left(\omega \right)$ for disorder $W=1$ and $W=4$, respectively. (d) $C_q\left(\omega \right)$ at momentum $q=\pi$ for disorder $W=0,1,2,4$. For $W=0$, we additionally compare to ED ($L=10$). The curves for $W>0$ are shifted by constant offsets in order to improve the visibility. We have $\mathrm{\Delta }=1.5, \beta =0, L=16$, and $N=100$ in all cases.