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.
Long-range interacting quantum systems exhibit interesting phenomena beyond the nearest-neighbours regime. Such interactions can reach further separated particles and change the behaviour of many-body correlations. Here, we analyse their structure in the antiferromagnetic long-range interaction transverse field Ising model with power-law interactions decaying as
1/rα. We examine the spatial behaviour of basis-independent total multipartite correlations, called sector lengths
Sk, of its ground state in the vicinity of critical points for different interaction exponents
α and fixed system size. These correlation functions were shown to indicate entanglement and the existence of local realistic models, allowing us to make claims about the Bell nonlocality and entanglement in the system. We further investigate them in the context of non-stabiliserness, often referred to as quantum magic, which holds information on their classical simulability and usefulness for quantum computation. We evaluate R{é}nyi stabiliser entropy near and outside the criticality and show that cumulative magic, up to the critical point, increases with
α for all of the examined system sizes. Our results provide new insights into the interplay between the long-range interactions, observable non-classical phenomena and their classical simulability.
Until now multiscale quantum problems have appeared to be out of reach at the many-body level relevant to strongly correlated materials and current quantum information devices. In fact, they can be modeled with
q-th order fractional derivatives, as we demonstrate in this work, treating classical and quantum phase transitions in a fractional Ising model for
0<q≤2 (
q=2 is the usual Ising model). We show that fractional derivatives not only enable continuous tuning of critical exponents such as
ν,
δ, and
η, but also define the Hausdorff dimension
HD of the system tied geometrically to the anomalous dimension
η. We discover that for classical systems,
HD is precisely equal to the fractional order
q. In contrast, for quantum systems,
HD deviates from this direct equivalence, scaling more gradually, driven by additional degrees of freedom introduced by quantum fluctuations. These results reveal how fractional derivatives fundamentally modify the fractal geometry of many-body interactions, directly influencing the universal symmetries of the system and overcoming traditional dimensional restrictions on phase transitions. Specifically, we find that for
q<1 in the classical regime and
q<2 in the quantum regime, fractional interactions allow phase transitions in one dimension. This work establishes fractional derivatives as a powerful tool for engineering critical behavior, offering new insights into the geometry of multiscale systems and opening avenues for exploring tunable quantum materials on NISQ devices.
The nonequilibrium response of a quantum many-body system defines its fundamental transport properties and how initially localized quantum information spreads. However, for long-range-interacting quantum systems little is known. We address this issue by analyzing a local quantum quench in the long-range Ising model in a transverse field, where interactions decay as a variable power law with distance ∝r(-α), α>0. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for α>2, weakly long range for 1<α<2 without a clear light cone but with a finite propagation speed of almost all excitations, and fully nonlocal for α<1 with instantaneous transmission of correlations. This last regime breaks generalized Lieb-Robinson bounds and thus locality. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasiparticles remains valid, allowing an intuitive interpretation of our findings via divergences of quasiparticle velocities. Our results may be tested in state-of-the-art trapped-ion experiments.
Locality imposes stringent constraints on the spreading of information in nonrelativistic quantum systems, which is reminiscent of a ``light cone,'' a causal structure arising in their relativistic counterparts. Long-range interactions can potentially soften such constraints, allowing almost instantaneous long jumps of particles, into the ``spacelike'' region, thus defying causality. Since interactions decaying as a power law with distance ${r}^{\ensuremath{-}\ensuremath{\alpha}}$ are ubiquitous in nature, it is pertinent to understand what is the fate of causality and information spreading in such systems. Using a numerically exact technique we address these questions by studying the out-of-time-order correlation function of a representative generic system in one dimension. We show that while the interactions are long range, their effect on information spreading is asymptotically negligible as long as $\ensuremath{\alpha}>1$. In this range we find a complex compound behavior, where, after a short transient, a fully local behavior emerges, yielding asymptotic light cones virtually indistinguishable from light cones in corresponding local models. The long-range nature of the interaction is only expressed in the power-law leaking of information from the light cone, with the same exponent as the exponent of the interaction $\ensuremath{\alpha}$. Our results directly apply also to corresponding response functions and suggest that previously obtained rigorous bounds on information spreading in long-range interacting systems are not tight, and could be potentially improved.
We use an out-of-time-order commutator (OTOC) to diagnose the propagation of chaos in one-dimensional long-range power law interaction system. We map the evolution of OTOC to a classical stochastic dynamics problem and use a Brownian quantum circuit to exactly derive the master equation. We vary two parameters: The number of qubits
N on each site (the on-site Hilbert space dimension) and the power law exponent $\ensuremath{\alpha}$. Three light cone structures of OTOC appear at
N=1: (1) logarithmic when $0.5<\ensuremath{\alpha}\ensuremath{\lesssim}0.8$, (2) sublinear power law when $0.8\ensuremath{\lesssim}\ensuremath{\alpha}\ensuremath{\lesssim}1.5$, and (3) linear when $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. The OTOC scales as $exp(\ensuremath{\lambda}t)/{x}^{2\ensuremath{\alpha}}$ and ${t}^{2\ensuremath{\alpha}/\ensuremath{\zeta}}/{x}^{2\ensuremath{\alpha}}$, respectively, beyond the light cones in the first two cases. When $\ensuremath{\alpha}\ensuremath{\ge}2$, the OTOC has essentially the same diffusive broadening as systems with short-range interactions, suggesting a complete recovery of locality. In the large
N limit, it is always a logarithmic light cone asymptotically, although a linear light cone can appear before the transition time for $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$. This implies the locality is never fully recovered for finite $\ensuremath{\alpha}$. Our result provides a unified physical picture for the chaos dynamics in a long-range power law interaction system.