Lieb-Robinson Bounds and Quasi-Locality for the Dynamics of Many-Body Quantum Systems

Sims, Robert

doi:10.1142/9789814350365_0007

Cited by 8 publications

(14 citation statements)

References 40 publications

(96 reference statements)

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“…clearly satisfies Assumption (57) with a convolution constant that is no bigger than the one of F. In fact, as observed in [Si,Section 3.1], the multiplication of such a function F with a non-increasing weight f :…”

Section: Banach Spaces Of Short-range Interactionsmentioning

confidence: 84%

Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

Bru

Pedra

2017

SpringerBriefs in Mathematical Physics

124

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We generalize to multi-commutators the usual Lieb-Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expansions) in order to estimate time-dependent multi-commutators for interacting fermions. Lieb-Robinson bounds for multi-commutators are effective mathematical tools to handle analytic aspects of the dynamics of quantum particles with interactions which are non-vanishing in the whole space and possibly time-dependent. To illustrate this, we prove that the bounds for multi-commutators of order three yield existence of fundamental solutions for the corresponding non-autonomous initial value problems for observables of interacting fermions on lattices. We further show how bounds for multi-commutators of an order higher than two can be used to study linear and non-linear responses of interacting fermions to external perturbations. All results also apply to quantum spin systems, with obvious modifications. However, we only explain the fermionic case in detail, in view of applications to microscopic quantum theory of electrical conduction discussed here and because this case is technically more involved.

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Section: Banach Spaces Of Short-range Interactionsmentioning

confidence: 84%

Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

Bru

Pedra

2017

SpringerBriefs in Mathematical Physics

124

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“…clearly satisfies Condition (11) with a convolution constant that is no bigger than the one of F. In fact, as observed in [Si,Section 3.1], the multiplication of such a function F with a non-increasing weight f : In all the paper, (10)-(11) are assumed to be satisfied.…”

Section: Banach Space Of Short-range Interactionsmentioning

confidence: 92%

“…which has convolution constant D ≤ 2 d+1+ǫ F 1,L for ǫ ∈ R + . See [NOS, Eq. (1.6)] or [Si,Example 3.1]. Note that the exponential function F (r) = e −ςr , ς ∈ R + , satisfies (10) but not (11).…”

Section: Banach Space Of Short-range Interactionsmentioning

confidence: 99%

Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles

Bru

Pedra

2015

Lett Math Phys

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We apply Lieb-Robinson bounds for multi-commutators we recently derived [BP3] to study the (possibly non-linear) response of interacting fermions at thermal equilibrium to perturbations of the external electromagnetic field. This analysis leads to an extension of the results for quasi-free fermions of [BPH1, BPH2] to fermion systems on the lattice with shortrange interactions. More precisely, we investigate entropy production and charge transport properties of non-autonomous C * -dynamical systems associated with interacting lattice fermions within bounded static potentials and in presence of an electric field that is time-and space-dependent. We verify the 1st law of thermodynamics for the heat production of the system under consideration. In linear response theory, the latter is related with Ohm and Joule's laws. These laws are proven here to hold at the microscopic scale, uniformly with respect to the size of the (microscopic) region where the electric field is applied. An important outcome is the extension of the notion of conductivity measures to interacting fermions.

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“…Consider a statistical model on the one-dimensional lattice 1 Z: a C * -algebra of "local" observables U, a state (positive linear functional) ω giving their statistical averages, and space and time-translation * -isomorphisms under which ω is invariant; a(x, t) is the translate of a ∈ U by distance x ∈ Z and time t ∈ R (see e.g. [29][30][31][32][33]). Under what conditions and for what (say countable) set of observables {q i } ⊂ U -representing ballistic waves -do the Fourier transforms of two-point correlation functions C i j (k, t) = x∈Z e ikx ω q i (x, t)q j (0, 0) − ω(q i )ω(q j ) satisfy the linear differential equation…”

Section: Linearised Euler Equation and Boltzmann-gibbs Principlementioning

confidence: 99%

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

Doyon

2022

Commun. Math. Phys.

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One of the most profound questions of mathematical physics is that of establishing from first principles the hydrodynamic equations in large, isolated, strongly interacting many-body systems. This involves understanding relaxation at long times under reversible dynamics, determining the space of emergent collective degrees of freedom (the ballistic waves), showing that projection occurs onto them, and establishing their dynamics (the hydrodynamic equations). We make progress in these directions, focussing for simplicity on one-dimensional systems. Under a model-independent definition of the complete space of extensive conserved charges, we show that hydrodynamic projection occurs in Euler-scale two-point correlation functions. A fundamental ingredient is a property of relaxation: we establish ergodicity of correlation functions along almost every direction in space and time. We further show that to every extensive conserved charge with a local density is associated a local current and a continuity equation; and that Euler-scale two-point correlation functions of local conserved densities satisfy a hydrodynamic equation. The results are established rigorously within a general framework based on Hilbert spaces of observables. These spaces occur naturally in the $$C^*$$ C ∗ algebra description of statistical mechanics by the Gelfand–Naimark–Segal construction. Using Araki’s exponential clustering and the Lieb–Robinson bound, we show that the results hold, for instance, in every nonzero-temperature Gibbs state of short-range quantum spin chains. Many techniques we introduce are generalisable to higher dimensions. This provides a precise and universal theory for the emergence of ballistic waves at the Euler scale and how they propagate within homogeneous, stationary states.

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Lieb-Robinson Bounds and Quasi-Locality for the Dynamics of Many-Body Quantum Systems

Abstract: We review a recently proven Lieb-Robinson bound for general, many-body quantum systems with bounded interactions. Several basic examples are discussed as well as the connection between commutator estimates and quasi-locality.

Cited by 8 publications

References 40 publications

Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

Lieb-Robinson Bounds for Multi-Commutators and Applications to Response Theory

Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles

Hydrodynamic Projections and the Emergence of Linearised Euler Equations in One-Dimensional Isolated Systems

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