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arXiv:1805.06895 · Physical Review A

Emergent locality in systems with power-law interactions

David J. Luitz, Yevgeny Bar Lev

Published Jan 30, 2019• 52 citations
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Abstract

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.

DOI
10.1103/physreva.99.010105

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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−\ensuremathα{r}^{\ensuremath{-}\ensuremath{\alpha}}r\ensuremath−\ensuremathα 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 \ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>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 \ensuremathα\ensuremath{\alpha}\ensuremathα. 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.
In nonrelativistic quantum theories with short-range Hamiltonians, a velocity vvv can be chosen such that the influence of any local perturbation is approximately confined to within a distance rrr until a time $t\ensuremath{\sim}r/v$, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law ($1/{r}^{\ensuremath{\alpha}}$) interactions, when $\ensuremath{\alpha}$ exceeds the dimension DDD, an analogous bound confines influences to within a distance rrr only until a time $t\ensuremath{\sim}(\ensuremath{\alpha}/v)\mathrm{log}r$, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are bounded by a polynomial for $\ensuremath{\alpha}>2D$ and become linear as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems.
Understanding the dynamics of many-body systems is crucial for understanding, e.g., thermalization or transmission of information. Nevertheless, little is known in the case of quantum systems with long-range interactions. Here, we analyze the long-range Ising model in a transverse field, where interactions decay as a power-law with distance ∝r−α\propto r^{-\alpha}∝r−α, α>0\alpha>0α>0. Using complementary numerical and analytical techniques, we identify three dynamical regimes: short-range-like with an emerging light cone for α>2\alpha>2α>2; weakly long-range for 1<α<21<\alpha<21<α<2 without a clear light cone but with a finite propagation speed of excitations; and fully non-local for α<1\alpha<1α<1 with instantaneous transmission of correlations. This last regime breaks generalized Lieb--Robinson bounds. Numerical calculation of the entanglement spectrum demonstrates that the usual picture of propagating quasi-particles remains valid for long-range interactions. This allows an intuitive interpretation in terms of qualitative changes to the spin-wave dispersion, leading to diverging quasi-particle velocities in the long-range regime. Our results may be tested in state-of-the-art trapped-ion experiments.
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.
Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators C(r, t ) = [A i (t ), B i+r ] of local operators.Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law 1/r α with distance r.Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for α < 1 and to linear light cones for α > 1.The asymptotic form of the tails of C(r, t ) ∝ t/r α is described by short-time perturbation theory, which is valid at short times and long distances.
We study the nonequilibrium dynamics of correlations in quantum lattice models in the presence of long-range interactions decaying asymptotically as a power law. For exponents larger than the lattice dimensionality, a Lieb-Robinson-type bound effectively restricts the spreading of correlations to a causal region, but allows supersonic propagation. We show that this decay is not only sufficient but also necessary. Using tools of quantum metrology, for any exponents smaller than the lattice dimension, we construct Hamiltonians giving rise to quantum channels with capacities not restricted to a causal region. An analytical analysis of long-range Ising models illustrates the disappearance of the causal region and the creation of correlations becoming distance independent. Numerical results obtained using matrix product state methods for the XXZ spin chain reveal the presence of a sound cone for large exponents and supersonic propagation for small ones. In all models we analyzed, the fast spreading of correlations follows a power law, but not the exponential increase of the long-range Lieb-Robinson bound.
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.