Causality and quantum criticality in long-range lattice models

TL;DR

This paper investigates how long-range interactions in quantum lattice models affect their critical behavior and causal structure, revealing conditions under which relativistic dynamics emerge.

Contribution

It provides a field-theoretic analysis of long-range quantum models, deriving critical exponents and identifying the transition to relativistic behavior based on the interaction decay.

Findings

Dynamic critical exponent calculated up to two-loop order.

Emergence of relativistic dynamics beyond a critical long-range exponent.

Characterization of correlation decay and causal cone deviations.

Abstract

Long-range quantum lattice systems 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 with long-range interactions, and a fermionic model with long-range hopping and pairing terms, 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 the long-range couplings, 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.