Time-polynomial Lieb-Robinson bounds for finite-range spin-network models

The Lieb-Robinson bound sets a theoretical upper limit on the speed at which information can propagate in nonrelativistic quantum spin networks. In its original version, it results in an exponentially exploding function of the evolution time, which is partially mitigated by an exponentially decreasing term that instead depends upon the distance covered by the signal (the ratio between the two exponents effectively defining an upper bound on the propagation speed). In the present paper, by properly accounting for the free parameters of the model, we show how to turn this construction into a stronger inequality where the upper limit only scales polynomially with respect to the evolution time. Our analysis applies to any chosen topology of the network, as long as the range of the associated interaction is explicitly finite. For the special case of linear spin networks we present also an alternative derivation based on a perturbative expansion approach which improves the previous inequality. In the same context we also establish a lower bound to the speed of the information spread which yields a nontrivial result at least in the limit of small propagation times.

Figure 1

Plot of the function $\mathcal{F}\left(x\right)$ entering into the derivation of Eq. (19): for $x=\frac{d(A,B)}{\overline{D}}\ge 1$ it is monotonically increasing, reaching the value $1/e\simeq 0.37$ for $x=1$ and quickly approaching the asymptotic value 1 for large enough $x$.

Figure 2

(a) Pictorial representation of the nested commutator ${\widehat{C}}_{1,2,\cdots ,k}^{\left(k\right)}\left(\widehat{A}\right)$ as a set of boxes, each one fillable with a ${\widehat{h}}_i$. (b) Representation of the only nested commutator which for the case $k=d$ admits a nonzero value for the commutation with $\widehat{B}$. (c) Case $k=d+n$ with $n\ge 1$. Here the boxes indicated with the asterisk can be filled depending on their position; for instance, here the box before ${\widehat{h}}_1$ could contain only ${\widehat{h}}_1$ while the one after ${\widehat{h}}_d$ could contain any.

Figure 3

Simulation of $\parallel [\widehat{A}\left(t\right),\widehat{B}]\parallel$ for different chain lengths $L$ for the Heisenberg $XY$ linear spin chain.

Figure 4

Plot of the value of ${\mathrm{\Gamma }}_d$ defined in Eq. (48) for different values of the chain length $L$, $d$ being fixed equal to $L-1$. Notice that all values are below $\sqrt{2\pi /e^2}$ (dashed line), which is provably the largest value this parameter can achieve.

Figure 5

Simulation and bounds of the function $\parallel [\widehat{A}\left(t\right),\widehat{B}]\parallel$ for a $L=4$ spin chain (upper panel) and for a $L=10$ spin chain (lower panel). The plot shows upper bound (45) (blue curve), lower bound (47) (green curve), simplified lower bound (49) (red), and numerical simulation (black). The colored bars above the plots outline the time domain in which each bound (identified by the same colors) is valid. As expected the simplified bound stands only for sufficiently small times. In all cases the simulation and the simplified lower bound are comparable in magnitude so that their curves are hardly distinguishable. In the case of $L=10$ the complete lower bound (47) (green) is considerably small and hence not visible.