The maximum speed with which information can propagate in a quantum many-body system directly affects how quickly disparate parts of the system can become correlated1,2,3,4 and how difficult the system will be to describe numerically5. For systems with only short-range interactions, Lieb and Robinson derived a constant-velocity bound that limits correlations to within a linear effective ‘light cone’6. However, little is known about the propagation speed in systems with long-range interactions, because analytic solutions rarely exist and because the best long-range bound7 is too loose to accurately describe the relevant dynamical timescales for any known spin model. Here we apply a variable-range Ising spin chain Hamiltonian and a variable-range XY spin chain Hamiltonian to a far-from-equilibrium quantum many-body system and observe its time evolution. For several different interaction ranges, we determine the spatial and time-dependent correlations, extract the shape of the light cone and measure the velocity with which correlations propagate through the system. This work opens the possibility for studying a wide range of many-body dynamics in quantum systems that are otherwise intractable.
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We thank J. Preskill, A. M. Rey, K. Hazzard, A. Daley, J. Schachenmayer, M. Kastner, S. Manmana and L.-M. Duan for discussions. This work is supported by the US Army Research Office (ARO) Award W911NF0710576 with funds from the DARPA Optical Lattice Emulator Program, ARO award W911NF0410234 with funds from the IARPA MQCO Program, and the US NSF Physics Frontier Center at JQI. M.F.-F. thanks the NRC for support. S.M. acknowledges funding provided by the Institute for Quantum Information and Matter, an NSF Physics Frontier Center with the support of the Gordon and Betty Moore Foundation (through grant GBMF1250).
The authors declare no competing financial interests.
Extended data figures and tables
Extended Data Figure 1 A long-time partial revival in the long-range Ising model.
a, Spatial correlations measured at long times after a global quench of an Ising model with α = 0.63. b, A small partial revival in correlation between sites 1 and 2 is evident, showing quantum coherence at long times. The black line shows the exact solution predicted from equation (4). Error bars, 1 s.d.
Extended Data Figure 2 Numeric calculation of XY model correlations.
Calculated spatial and time-dependent correlations for an N = 22-spin XY model with spin–spin couplings Jij ≈ J0/|i − j|1.19, found by numerically evolving the Schrödinger equation.
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Richerme, P., Gong, ZX., Lee, A. et al. Non-local propagation of correlations in quantum systems with long-range interactions. Nature 511, 198–201 (2014). https://doi.org/10.1038/nature13450
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