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arXiv:1508.00906 · Physical review. B./Physical review. B

Causality and quantum criticality in long-range lattice models

Mohammad F. Maghrebi, Zhexuan Gong, Michael Foss‐Feig, Alexey Vyacheslavovich Gorshkov

Published Mar 17, 2016• 72 citations
arXivHTMLPDF

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.

DOI
10.1103/physrevb.93.125128

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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.
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We study causality and criticality in a one-dimensional fractional multiscale transverse-field Ising model, where fractional derivatives generate long range interactions beyond the scope of standard power laws. Such fractional responses are common in classical systems including the anomalous stress-strain behaviour of viscoelastic polymers, Lévy-like contaminant transport in heterogeneous porous media, and the non-Debye dielectric relaxation of glassy dielectrics. Furthermore, these unique interactions can be implemented in current quantum information architectures, with intriguing consequences for the many-body dynamics. Using a truncated Jordan-Wigner approach, we show that in the long wavelength limit of the mean field, the dynamical critical exponent is set by the fractional order q as z=q/2z=q/2z=q/2. To probe genuine many-body dynamics, we apply matrix-product-state simulations with the time-dependent variational principle adapted to nonlocal couplings. Tracking the entanglement-entropy light cone and performing finite-size scaling of the many-body gap for $0
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In this thesis, we have investigated the spreading of quantum correlations in isolated lattice models with short- or long-range interactions driven far from equilibrium via sudden global quenches. A general theoretical approach relying on a quasiparticle theory is presented. The latter has permitted to unveil a generic expression for the equal-time connected correlation functions valid both for short-range and long-range interacting particle and spin lattice models on a hypercubic lattice. Relying on stationary phase arguments, we have shown that its causality cone displays a universal twofold structure consisting of a correlation edge and a series of local extrema defining the outer and inner structure of the space-time correlations. For short-range interactions, the motion of each structure is ballistic and the associated spreading velocities are related to the group and phase velocities of the quasiparticle dispersion relation of the post-quench Hamiltonian. For long-range interactions of the form 1/∣R∣α1/|R|^α1/∣R∣α, the correlation spreading is substantially different due to a possible divergence of group velocity when tuning the power-law exponent ααα. For a divergent group velocity, i.e. the quasi-local regime, we have presented evidence of a universal algebraic structure for the causality cone. While, the correlation edge motion has been found to be always slower than ballistic, the local extrema propagate faster than ballistically and ballistically for gapless and gapped quantum systems respectively. For the local regime implying a well-defined group velocity, we have recovered similar scaling laws and spreading velocities than the short-range case for the causality cone of correlations.
Whether long-range interactions allow for a form of causality in nonrelativistic quantum models remains an open question with far-reaching implications for the propagation of information and thermalization processes. Here, we study the out-of-equilibrium dynamics of the one-dimensional transverse Ising model with algebraic long-range exchange coupling. Using a state of the art tensor-network approach, complemented by analytic calculations and considering various observables, we show that a weak form of causality emerges, characterized by nonuniversal dynamical exponents. While the local spin and spin correlation causal edges are sub-ballistic, the causal region has a rich internal structure, which, depending on the observable, displays ballistic or superballistic features. In contrast, the causal region of entanglement entropy is featureless and its edge is always ballistic, irrespective of the interaction range. Our results shed light on the propagation of information in long-range interacting lattice models and pave the way to future experiments, which are discussed.