Quantum lattice systems with long-range interactions often exhibit
drastically different behavior than their short-range counterparts. In
particular, because they do not satisfy the conditions for the Lieb-Robinson
theorem, they need not have an emergent relativistic structure in the form of a
light cone. Adopting a field-theoretic approach, we study the one-dimensional
transverse-field Ising model and a fermionic model with long-range
interactions, explore their critical and near-critical behavior, and
characterize their response to local perturbations. We deduce the dynamic
critical exponent, up to the two-loop order within the renormalization group
theory, which we then use to characterize the emergent causal behavior. We show
that beyond a critical value of the power-law exponent of long-range
interactions, the dynamics effectively becomes relativistic. Various other
critical exponents describing correlations in the ground state, as well as
deviations from a linear causal cone, are deduced for a wide range of the
power-law exponent.
