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arXiv:1010.4576 · Physical Review A

Information propagation for interacting-particle systems

Norbert Schuch, Sarah K. Harrison, Tobias J. Osborne, Jens Eisert

Published Sep 8, 2011• 62 citations
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Abstract

We study the speed at which information propagates through systems of interacting quantum particles moving on a regular lattice and show that for a certain class of initial conditions there exists a maximum speed of sound at which information can propagate. Our argument applies equally to quantum spins, bosons such as in the Bose-Hubbard model, fermions, anyons, and general mixtures thereof, on arbitrary lattices of any dimension. It also pertains to dissipative dynamics on the lattice, and generalizes to the continuum for quantum fields. Our result can be seen as an analog of the Lieb-Robinson bound for strongly correlated models.

DOI
10.1103/physreva.84.032309

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We study the speed at which information propagates through systems of interacting quantum particles moving on a regular lattice and show that for a certain class of initial conditions there exists a maximum speed of sound at which information can propagate. Our argument applies equally to quantum spins, bosons such as in the Bose-Hubbard model, fermions, anyons, and general mixtures thereof, on arbitrary lattices of any dimension. It also pertains to dissipative dynamics on the lattice, and generalizes to the continuum for quantum fields. Our result can be seen as an analog of the Lieb-Robinson bound for strongly correlated models.
On physical grounds, one expects locally interacting quantum many-body systems to feature a finite group velocity. This intuition is rigorously underpinned by Lieb-Robinson bounds that state that locally interacting Hamiltonians with finite-dimensional constituents on suitably regular lattices always exhibit such a finite group velocity. This also implies that causality is always respected by the dynamics of quantum lattice models. It had been a long-standing open question whether interacting bosonic systems also feature finite speeds of sound in information and particle propagation, which was only recently resolved. This work proves a strikingly simple such bound for particle propagation - shown in literally a few elementary, yet not straightforward, lines - for generalized Bose-Hubbard models defined on general lattices, proving that appropriately locally perturbed stationary states feature a finite speed of sound in particle numbers.
The Lieb-Robinson bound asserts the existence of a maximal propagation speed for the quantum dynamics of lattice spin systems. Such general bounds are not available for most bosonic lattice gases due to their unbounded local interactions. Here we establish for the first time a general ballistic upper bound on macroscopic particle transport in the paradigmatic Bose-Hubbard model. The bound is the first to cover a broad class of initial states with positive density including Mott states, which resolves a longstanding open problem. It applies to Bose-Hubbard-type models on any lattice with not too long-ranged hopping. The proof is rigorous and rests on controlling the time evolution of a new kind of adiabatic spacetime localization observable via iterative differential inequalities.
We propose Lieb-Robinson bounds (bounds on the speed of propagation of information in quantum systems) with an explicit dependence on the interaction strengths of the Hamiltonian. For systems with more than two interactions it is found that the Lieb-Robinson speed is not always algebraic in the interaction strengths. We consider Hamiltonians with any finite number of bounded operators and also a certain class of unbounded operators. We obtain bounds and propagation speeds for quantum systems on lattices and also general graphs possessing a kind of homogeneity and isotropy. One area for which this formalism could be useful is the study of quantum phase transitions which occur when interactions strengths are varied.
When locally exciting a quantum lattice model, the excitation will propagate through the lattice. This effect is responsible for a wealth of nonequilibrium phenomena, and has been exploited to transmit quantum information. It is a commonly expressed belief that for local Hamiltonians, any such propagation happens at a finite "speed of sound". Indeed, the Lieb-Robinson theorem states that in spin models, all effects caused by a perturbation are essentially limited to a causal cone. We show that for meaningful translationally invariant bosonic models with nearest-neighbor interactions (addressing the challenging aspect of an experimental realization) this belief is incorrect: We prove that one can encounter accelerating excitations under the natural dynamics that allow for reliable transmission of information faster than any finite speed of sound. It also implies that the simulation of dynamics of strongly correlated bosonic models may be much harder than that of spin chains even in the low-energy sector.
The Lieb-Robinson bound shows the existence of a maximum speed of signal propagation in discrete quantum mechanical systems with local interactions. This generalizes the concept of relativistic causality beyond field theory, and provides a powerful tool in theoretical condensed matter physics and quantum information science. Here, we extend the scope of this seminal result by considering general markovian quantum evolution, where we prove that an equivalent bound holds. In addition, we use the generalized bound to demonstrate that correlations in the stationary state of a Markov process decay on a length scale set by the Lieb-Robinson velocity and the system's relaxation time.
We prove that quantum information propagates with a finite velocity in any model of interacting bosons whose (possibly time-dependent) Hamiltonian contains spatially local single-boson hopping terms along with arbitrary local density-dependent interactions.More precisely, with the density matrix ρ ∝ exp½-μN (with N the total boson number), ensemble-averaged correlators of the form h½A 0 ; B r ðtÞi, along with outof-time-ordered correlators, must vanish as the distance r between two local operators grows, unless t ≥ r=v for some finite speed v.In one-dimensional models, we give a useful extension of this result that demonstrates the smallness of all matrix elements of the commutator ½A 0 ; B r ðtÞ between finite-density states if t=r is sufficiently small.Our bounds are relevant for physically realistic initial conditions in experimentally realized models of interacting bosons.In particular, we prove that v can scale no faster than linear in number density in the Bose-Hubbard model: This scaling matches previous results in the highdensity limit.The quantum-walk formalism underlying our proof provides an alternative method for bounding quantum dynamics in models with unbounded operators and infinite-dimensional Hilbert spaces, where Lieb-Robinson bounds have been notoriously challenging to prove.