Researchers prove that a standard one-dimensional quantum spin model with randomness spreads information very slowly when we look only at a fixed low-energy part of its spectrum. The model is the random Heisenberg XXZ spin-1/2 chain. The slow spreading they show is a “logarithmic light cone”: to affect sites a distance ℓ away you need times that grow exponentially in ℓ, much slower than the linear-in-time spread allowed in non-random models.

Concretely, they study the time evolution of a local observable and then restrict attention to states with energies in a fixed interval near the bottom of the spectrum. For any local observable T and any chosen scale ℓ, they build an approximating observable Tt that is supported only on sites within distance ℓ of the original support. On average over the randomness, the difference between the true time-evolved observable (projected to that energy window) and the approximation is bounded by a factor that decays exponentially in ℓ, multiplied by a modest power of time. This behavior implies that to keep the approximation accurate as time grows one must increase ℓ only logarithmically in time — the signature of a logarithmic light cone.

To get this result the authors extend earlier finite-size proofs to the infinite chain. Earlier work had shown similar localization and slow dynamics for finite systems or in a particular low-energy “droplet” regime. The new paper removes volume-dependent estimates so the conclusion holds directly in the thermodynamic limit. The allowed parameter range (how strong the randomness is and how strong the spin interactions are) is determined only by the chosen energy interval. That range covers both weak interaction and strong disorder regimes, and requires the anisotropy parameter ∆ to be in the Ising phase (∆>1) and the random field to come from an independent, bounded-density distribution supported inside [0,1].