Optimal Lieb–Robinson bounds for long-range systems

Lieb–Robinson bounds give an emergent 'light cone' limiting how fast information and correlations spread under local Hamiltonian dynamics. For short-range interactions the cone is linear with a finite group velocity. For long-range interactions decaying as $1/r^\alpha$, the cone can broaden into a polynomial or logarithmic shape, and the sharp form depends intricately on $\alpha$ and dimension $d$.

What is known: In the standard worst-case operator-norm setting, strictly linear cones are established for $\alpha>2d+1$, not merely $\alpha>2d$. Tran et al. determine the principal optimal exponents above $2d$; the interval $2d<\alpha<2d+1$ has a different, polynomial propagation scale. Matching protocols and commutator estimates must be compared with the same allowed controls and accuracy criterion.

For $\alpha\le d$, whether interactions are Kac-normalized changes the question. Operator norm, state-dependent norms, entanglement generation and state transfer also need not share the same bound. Select a residual exponent/endpoint and specify time-dependent controls, finite-size scaling and error tolerance; do not relabel the known core exponent regime as open.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.