Disordered Lieb-Robinson Bounds in One Dimension

By tightening the conventional Lieb-Robinson bounds to better handle systems that lack translation invariance, we determine the extent to which “weak links” suppress operator growth in disordered one-dimensional spin chains. In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths $\mu (J)$ has a sufficiently heavy tail at small $J$ and we identify the correct dynamical exponent to use instead. Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics—we must distinguish between bounds that hold for all sites of the chain and bounds that hold for a subsequence of sites and we show by explicit example that these two can have dramatically different behaviors. All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent. We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.

Popular Summary

The dynamics of quantum many-body systems far from equilibrium is a notoriously difficult topic of study, especially when disorder is involved. Significant progress has been made in recent years concerning how disorder impedes quantum dynamics, yet there continues to be very few sharp mathematical results. Here we use Lieb-Robinson bounds to identify rigorous constraints on how information can propagate and correlations can spread in one dimension due to disorder. This has implications not only for quantum dynamics but also for the robustness of quantum information protocols.

Our main result is a proof that when a one-dimensional system has a sufficient number of “weak links,” in the sense of anomalously weak interactions, it is impossible for information to propagate ballistically. We determine the correct shape of the optimal front to use instead and prove that it is tight by explicit example. Nonetheless, we find that the standard formulation of Lieb-Robinson bounds must be generalized in order to better describe the complexity of the quantum dynamics in disordered systems by distinguishing between information propagation that reaches all sites and information propagation that reaches only a subsequence of sites. We use Lieb-Robinson techniques to prove that these two can have very different behaviors.

These results advance our understanding of the extent to which disorder impedes information propagation in one dimension. Going forward, it will be important to investigate the degree to which these conclusions apply more broadly, both in higher dimensions and in the presence of more moderate disorder.

Figure 1

The geometry of the systems studied in the present work, i.e., 1D chains with nearest-neighbor interactions. Sites are indicated by black squares, labeled by $i$. Terms of the Hamiltonian are indicated by solid lines, with $H_l(t)$ acting on the two sites connected by that link. Given a set $\{J_l\}$, we require that $\Vert H_l(t)\Vert \le J_l$.

Figure 2

The possibility of a Hamiltonian causing operator spreading at dynamical exponent $z$, i.e., in a time growing asymptotically as $t(r)\propto r^z$ to reach a large distance $r$. The horizontal axis is the exponent $\alpha$ characterizing the number of weak links in the couplings $\{J_l\}$—the fraction of links between sites 0 and $r$ having $J_l\le J$ is assumed to go as $J^{\alpha }$ at small $J$ and large $r$. The vertical axis is $z$. The blue region is where no Hamiltonian with such $\alpha$ can reach any large-distance site at such $z$. The red region is where a Hamiltonian exists with such $\alpha$ that reaches every large-distance site at (or faster than) such $z$. The boundary between the two is given by $z_c(\alpha )=max[1/\alpha ,1]$. In the special case of independent random couplings, the $\alpha >1$ portion of the boundary is included in the red region, while the $\alpha <1$ portion (shown in purple) is where a Hamiltonian exists that reaches a subsequence of sites faster than $z$ but where no Hamiltonian can reach every site at such $z$. Lastly, at point $(\alpha ,z)=(1,1)$, no Hamiltonian can reach every site ballistically but it is unknown whether any Hamiltonian can reach some sites ballistically.

Figure 3

The circuit diagram of a simple protocol that transfers the state on site 0 to site $r$ (here, the sites are arranged vertically and the solid lines indicate the absence of any unitary). The definition of the swap gate acting on sites $i$ and $i+1$ is given at the bottom of the figure—$a$ and $b$ label basis states of the local Hilbert space (each taking values in $\{1,\dots ,d\}$).

Figure 4

The visual interpretation of Eq. (18). $\omega$ consists of the sites in red and $\mathrm{\Omega }/\omega$ the sites in blue. $\lambda$ consists of the dashed links and $\mathrm{\Lambda }/\lambda$ the solid links. Evolution under only $H_{\mathrm{\Lambda }/\lambda }$ cannot transform a string having the identity on $\omega$ into one with a nonidentity element, nor vice versa.

Figure 5

A sketch of a possible spatial profile $\Vert [A_0^t,B_r]\Vert$ as a function of $r$, indicated by dots (the black line merely connects the points). The red dashed line shows an a.a. bound consistent with the profile—the red and blue dots indicate points contained within the bound. There are implied to be only a finite number of black dots. The blue dashed line shows an i.o. bound—the blue dots indicate the subsequence $\{r_k\}$ to which the bound applies.

Figure 6

The transformation from the original picture (top line) to the interaction picture with respect to “large” terms (bottom line). The dashed links (labeled $l_1$ through $l_4$) are those on which $\Vert H_l(t)\Vert \le ϵ$ and the solid links those on which $\Vert H_l(t)\Vert >ϵ$. The squares on the top line represent individual sites but the rectangles on the bottom line represent the collection of all corresponding sites connected by solid links—the transformed interaction ${\tilde{H}}_{l_i}^t$ is supported on the entire neighboring rectangles [but no further by virtue of Eq. (18)].

Figure 7

An example of a “ladder,” with $M=3$ sites per rung. To apply our analysis, simply treat all $M$ sites connected vertically as a single “site,” indicated by the red shading.

Figure 8

The plot of Eq. (88), using $\alpha =2/3$ for concreteness. The dashed red line corresponds to $n=2$ with ${\beta }_i\in \{0,3/4,3/2\}$ (each of the individual curves being minimized among in Eq. (88) is shown as a dotted black line). The solid red line corresponds to $n=30$ and ${\beta }_i$ equally spaced by $1/20$, which closely approximates the curve at large $n$. The blue line is the conventional LR bound for comparison. To reproduce the exact numbers in the plot, set $r=25$ and $C=C^{\prime }=1$.

Figure 9

The implementation of a swap gate between sites $i$ and $j$. The ${\tilde{X}}_{ij}$ gate is given in Eq. (C1), the ${\mathrm{ft}}_j$ gate is given in Eq. (C3), and the ${\mathrm{cz}}_{ij}$ gate is given in Eq. (C4).