The relaxation of uniform quantum systems with finite-range interactions after a quench is generically driven by the ballistic propagation of long-lived quasiparticle excitations triggered by a sufficiently small quench. Here we investigate the case of long-range (1/r^{α}) interactions for a d-dimensional lattice spin model with uniaxial symmetry, and show that, in the regime d<α<d+2, the entanglement and correlation buildup is radically altered by the existence of a nonlinear dispersion relation of quasiparticles, ω∼k^{z<1}, at small wave vectors, leading to a divergence of the quasiparticle group velocity and superballistic propagation. This translates in a superlinear growth of correlation fronts with time and sublinear growth of relaxation times of subsystem observables with size when focusing on k=0 fluctuations. Yet the large dispersion in group velocities leads to an extreme wavelength dependence of relaxation times of finite-k fluctuations, with entanglement being susceptible to all of them. Our predictions are directly relevant to current experiments probing the nonequilibrium dynamics of trapped ions, or ultracold magnetic and Rydberg atoms in optical lattices.
The relaxation of uniform quantum systems with finite-range interactions after a quench is generically driven by the ballistic propagation of long-lived quasiparticle excitations triggered by a sufficiently small quench. Here we investigate the case of long-range (1/r^{α}) interactions for a d-dimensional lattice spin model with uniaxial symmetry, and show that, in the regime d<α<d+2, the entanglement and correlation buildup is radically altered by the existence of a nonlinear dispersion relation of quasiparticles, ω∼k^{z<1}, at small wave vectors, leading to a divergence of the quasiparticle group velocity and superballistic propagation. This translates in a superlinear growth of correlation fronts with time and sublinear growth of relaxation times of subsystem observables with size when focusing on k=0 fluctuations. Yet the large dispersion in group velocities leads to an extreme wavelength dependence of relaxation times of finite-k fluctuations, with entanglement being susceptible to all of them. Our predictions are directly relevant to current experiments probing the nonequilibrium dynamics of trapped ions, or ultracold magnetic and Rydberg atoms in optical lattices.
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.
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∣α, the correlation spreading is substantially different due to a possible divergence of group velocity when tuning the power-law exponent α. 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.
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, 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 excitations; and fully non-local for α<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.
We study the out-of-equilibrium dynamics induced by quantum quenches in quadratic Hamiltonians featuring both short- and long-range interactions. The spreading of correlations in the presence of algebraic decaying interactions, 1/R α , is studied for lattice Bose models in arbitrary dimension D. These models are exactly solvable and provide useful insight in the universal description of more complex systems as well as comparisons to the known universal upper bounds for the spreading of correlations. Using analytical calculations of the dominant terms and full numerical integration of all quasi-particle contributions, we identify three distinct dynamical regimes. For strong decay of interactions, α>D+1, we find a causal regime, qualitatively similar to what previously found for short-range interactions. This regime is characterized by ballistic (linear cone) spreading of the correlations with a cone velocity equal to twice the maximum group velocity of the quasi-particles. For weak decay of interactions, α < D, we find instantaneous activation of correlations at arbitrary distance. This signals the breaking of causality, which can be associated with the divergence of the quasi-particle energy spectrum. Finite-size scaling of the activation time precisely confirms this interpretation. For intermediate decay of interactions, D<α<D+1, we find a sub-ballistic, algebraic (bent cone) spreading and determine the corresponding exponent as a function of α. These outcomes generalize existing results for one-dimensional systems to arbitrary dimension. We precisely relate the three regimes to the first- and second-order divergences of the quasi-particle energy spectrum for any dimension. The long-range transverse Ising model in dimensions D = 1, 2, and 3 in the (quadratic) spin-wave approximation is more specifically studied and we also discuss the shape of the correlation front in dimension higher than one. Our results apply to several condensed-matter systems as well as atomic, molecular, and optical systems, and pave the way to the observation of causality and its breaking in diverse experimental realization.
We study the out-of-equilibrium dynamics induced by quantum quenches in quadratic Hamiltonians featuring both short- and long-range interactions. The spreading of correlations in the presence of algebraic decaying interactions, 1/ R α , is studied for lattice Bose models in arbitrary dimension D . These models are exactly solvable and provide useful insight in the universal description of more complex systems as well as comparisons to the known universal upper bounds for the spreading of correlations. Using analytical calculations of the dominant terms and full numerical integration of all quasi-particle contributions, we identify three distinct dynamical regimes. For strong decay of interactions, , we find a causal regime, qualitatively similar to what previously found for short-range interactions. This regime is characterized by ballistic (linear cone) spreading of the correlations with a cone velocity equal to twice the maximum group velocity of the quasi-particles. For weak decay of interactions, α < D , we find instantaneous activation of correlations at arbitrary distance. This signals the breaking of causality, which can be associated with the divergence of the quasi-particle energy spectrum. Finite-size scaling of the activation time precisely confirms this interpretation. For intermediate decay of interactions, , we find a sub-ballistic, algebraic (bent cone) spreading and determine the corresponding exponent as a function of α . These outcomes generalize existing results for one-dimensional systems to arbitrary dimension. We precisely relate the three regimes to the first- and second-order divergences of the quasi-particle energy spectrum for any dimension. The long-range transverse Ising model in dimensions D = 1, 2, and 3 in the (quadratic) spin-wave approximation is more specifically studied and we also discuss the shape of the correlation front in dimension higher than one. Our results apply to several condensed-matter systems as well as atomic, molecular, and optical systems, and pave the way to the observation of causality and its breaking in diverse experimental realization.
We investigate the dynamics following a global parameter quench for two one-dimensional models with variable-range power-law interactions: a long-range transverse Ising model, which has recently been realized in chains of trapped ions, and a long-range lattice model for spinless fermions with long-range tunneling. For the transverse Ising model, the spreading of correlations and growth of entanglement are computed using numerical matrix product state techniques, and are compared with exact solutions for the fermionic tunneling model. We identify transitions between regimes with and without an apparent linear light cone for correlations, which correspond closely between the two models. For long-range interactions (in terms of separation distance r, decaying slower than 1/r), we find that despite the lack of a light cone, correlations grow slowly as a power law at short times, and that---depending on the structure of the initial state---the growth of entanglement can also be sublinear. These results are understood through analytical calculations, and should be measurable in experiments with trapped ions.