Propagation of Correlations in Quantum Lattice Systems

Nachtergaele, Bruno; Ogata, Yoshiko; Sims, Robert

doi:10.1007/s10955-006-9143-6

Cited by 193 publications

(277 citation statements)

References 13 publications

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“…The proof of this fact is a straightforward consequence of the existence of the infinite volume dynamics (see [19,49,50]) and of the existence of the β, L → ∞ limit of the Gibbs state. In appendix C, we reproduce this proof; that is, we prove that the limit lim β→∞ lim L→∞ J…”

Section: Reconstruction Of the Real-time Kubo Formulamentioning

confidence: 94%

“…The proof is a simple adaptation of [19,49], the only difference being the choice of boundary conditions (periodic, rather than free). We consider two bounded operators A, B on the fermionic Fock space, even in the fermionic operators, with supports X and Y , respectively, independent of L. We shall think the torus Λ L as a subset of Λ 'centered' at the barycenter of X and Y , to be denoted z 0 .…”

Section: Appendix C: Infinite Volume Dynamicsmentioning

confidence: 99%

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Universality of the Hall Conductivity in Interacting Electron Systems

Giuliani

Mastropietro

Porta

2016

Commun. Math. Phys.

101

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Section: Reconstruction Of the Real-time Kubo Formulamentioning

confidence: 94%

Section: Appendix C: Infinite Volume Dynamicsmentioning

confidence: 99%

Universality of the Hall Conductivity in Interacting Electron Systems

Giuliani

Mastropietro

Porta

2016

Commun. Math. Phys.

101

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“…The measurement statistics of the two observables can show correlations only after the dynamics of the system had enough time to correlate the two regions X and Y (see Ref. [44] for a similar discussion in the context of Hamiltonian dynamics).…”

Section: Theorem 2 (Quasi-locality Of Local Liouvillian Dynamics [6])mentioning

confidence: 99%

“…Outside the space time cone defined by this speed, any signal is typically exponentially suppressed in the distance. The results of Lieb and Robinson, originally derived in the setting of translation invariant 1D spin systems with short range, or exponentially decaying interactions [38] have since been tightened [27,43] and extended to more general graphs [31,47] and to interactions decaying only polynomially with the distance, both, for spin systems [44] and fermionic systems [31] (see also Ref. [45] for a review).…”

Section: Introductionmentioning

confidence: 99%

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Kliesch

Gogolin

Eisert

2014

Many-Electron Approaches in Physics, Chemistry and Mathematics

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This is an introductory text reviewing Lieb-Robinson bounds for open and closed quantum many-body systems. We introduce the Heisenberg picture for time-dependent local Liouvillians and state a Lieb-Robinson bound that gives rise to a maximum speed of propagation of correlations in many body systems of locally interacting spins and fermions. Finally, we discuss a number of important consequences concerning the simulation of time evolution and properties of ground states and stationary states.

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Non-local propagation of correlations in quantum systems with long-range interactions

Richerme

Gong

Lee

et al. 2014

Nature

760

922

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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 correlated [1][2][3][4] and how difficult the system will be to describe numerically [5]. 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, since the best long-range bound [7] is too loose to give the correct light-cone shape for any known spin model and since analytic solutions rarely exist. In this work, we experimentally determine the spatial and time-dependent correlations of a far-from-equilibrium quantum many-body system evolving under a long-range Ising-or XY-model Hamiltonian. For several different interaction ranges, we extract the shape of the light cone and measure the velocity with which correlations propagate through the system. In many cases we find increasing propagation velocities, which violate the Lieb-Robinson prediction, and in one instance cannot be explained by any existing theory. Our results demonstrate that even modestly-sized quantum simulators are well-poised for studying complicated many-body systems that are intractable to classical computation.Lieb-Robinson bounds [6] have strongly influenced our understanding of locally-interacting quantum many-body systems. These bounds restrict the many-body dynamics to a well-defined causal region outside of which correlations are exponentially suppressed [8], analogous to causal light cones that arise in relativistic theories. Their existence has enabled proofs linking the decay of correlations in ground states to the presence of a spectral gap [7,9], as well as the area law for entanglement entropy [5,10,11], which can indicate the computational complexity of classically simulating a quantum system. Furthermore, Lieb-Robinson bounds constrain the timescales on which quantum systems might thermalize [12][13][14] and the maximum speed with which information can be sent through a quantum channel [15]. Recent experimental work has observed an effective Lieb-Robinson (i.e. linear) light cone in a 1D quantum gas [16].When interactions in a quantum system are longrange, the speed with which correlations build up between distant particles is no longer guaranteed to obey the Lieb-Robinson prediction. Indeed, for sufficiently long-ranged interactions, the notion of locality is expected to break down completely [17]. Violation of the Lieb-Robinson bound means that comparatively little can be predicted about the growth and propagation of correlations in long-range interacting systems, though there have been several recent theoretical and numerical advances [2,3,7,[17][18][19].Here we report an experiment that directly measures the shape of the causal region and the speed at which correlations propagate within Ising and XY spin chains. To induce the spread of correlations, we perform a global q...

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Propagation of Correlations in Quantum Lattice Systems

Cited by 193 publications

References 13 publications

Universality of the Hall Conductivity in Interacting Electron Systems

Universality of the Hall Conductivity in Interacting Electron Systems

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

Non-local propagation of correlations in quantum systems with long-range interactions

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