Elliott H. Lieb, Anna Vershynina
This paper has been withdrawn by the authors because of an error in the proof. We can, however, prove a weaker spatial fall-off that is still superlinear, namely exp[-x log x].
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We prove Lieb-Robinson bounds for the dynamics of systems with an infinite dimensional Hilbert space and generated by unbounded Hamiltonians. In particular, we consider quantum harmonic and certain anharmonic lattice systems.
I review the role of Lieb-Robinson bounds in characterizing and utilizing the locality properties of the Heisenberg dynamics of quantum lattice systems. In particular, I discuss two definitions of gapped ground state phases and show that they are essentially equivalent.
We introduce a class of UV-regularized two-body interactions for fermions in $\mathbb{R}^d$ and prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a step toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrödinger operators. We apply the propagation bound to prove the existence of infinite-volume dynamics as a strongly continuous group of automorphisms on the CAR algebra.
We consider the isotropic XY quantum spin chain in a random external field in the $z$ direction, with single site distributions given by i.i.d. random variables times the critical decaying envelope $j^{-1/2}$. Our motivation is the study of many-body localization. We investigate transport properties in terms of polynomial Lieb-Robinson (PLR) bounds. We prove a zero-velocity PLR bound for large disorder strength $λ$ and for small $λ$ we show a partial converse, which suggests the existence of non-trivial transport in the model.
We give a Lieb-Robinson bound for the group velocity of a large class of discrete quantum systems which can be used to prove that a non-vanishing spectral gap implies exponential clustering in the ground state of such systems.
We prove Lieb-Robinson bounds and the existence of the thermodynamic limit for a general class of irreversible dynamics for quantum lattice systems with time-dependent generators that satisfy a suitable decay condition in space.
We generalize the Lieb-Robinson theorem to systems whose Hamiltonian is the sum of local operators whose commutators are bounded.
This is a short review on the applications of Lieb-Robinson bounds for a general readership of mathematical physicists.
We consider the possibility of developing a Lieb-Robinson bound for the double bracket flow. This is a differential equation $$\partial_B H(B)=[[V,H(B)],H(B)]$$ which may be used to diagonalize Hamiltonians. Here, $V$ is fixed and $H(0)=H$. We argue (but do not prove) that $H(B)$ need not converge to a limit for nonzero real $B$ in the infinite volume limit, even assuming several conditions on $H(0)$. However, we prove Lieb-Robinson bounds for all $B$ for the double-bracket flow for free fermion systems, but the range increases \emph{exponentially} with the control parameter $B$.
Lieb-Robinson bounds, automorphic equivalence and LPPL for long-range interacting fermions
本文证明了长程相互作用费米子系统的Lieb-Robinson界，揭示了自守等价与LPPL原理的成立条件，并指出某些新界不适用于费米子系统。
We consider the dynamics of the Bose-Hubbard model on general lattices and prove a Lieb-Robinson bound for observables whose supports are separated by an initially almost particle-free region. We further obtain a maximal velocity bound for particle transport through an initially empty region which also applies to long-range hopping. Our techniques originate in the proofs of maximal velocity bounds for Schrödinger operators and scattering theory in non-relativistic QED.
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.
Discrete lattice models are a cornerstone of quantum many-body physics. They arise as effective descriptions of condensed matter systems and lattice-regularized quantum field theories. Lieb-Robinson bounds imply that if the degrees of freedom at each lattice site only interact locally with each other, correlations can only propagate with a finite group velocity through the lattice, similarly to a light cone in relativistic systems. Here we show that Lieb-Robinson bounds are equivalent to the locality of the interactions: a system with k-body interactions fulfills Lieb-Robinson bounds in expone…
We give an overview of recent results on Lieb-Robinson bounds and some of their applications in the study of quantum many-body models in condensed matter physics.
For a mixture of interacting Bose gases initially prepared in a regime of condensation (uncorrelation), it is proved that in the course of the the time evolution observables of disjoint sets of particles of each species have correlation functions that remain asymptotically small in the total number of particles and display a controlled growth in time. This is obtained by means of ad hoc estimates of Lieb-Robinson type on the propagation of the interaction, established here for the multi-component Bose mixture.
We present a recent result on the existence of the dynamics in the thermodynamic limit of a class of anharmonic quantum oscillator lattices, which was obtained using Lieb--Robinson bounds.
We consider translation invariant gapped quantum spin systems satisfying the Lieb-Robinson bound and containing single-particle states in a ground state representation. Following the Haag-Ruelle approach from relativistic quantum field theory, we construct states describing collisions of several particles, and define the corresponding $S$-matrix. We also obtain some general restrictions on the shape of the energy-momentum spectrum. For the purpose of our analysis we adapt the concepts of almost local observables and energy-momentum transfer (or Arveson spectrum) from relativistic QFT to the la…
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.
For a class of infinite lattices of interacting anharmonic oscillators, we study the existence of the dynamics, together with Lieb-Robinson bounds, in a suitable algebra of observables
We improve the Lieb-Robinson bound for a wide class of quantum many-body systems with long-range interactions decaying by power law. As an application, we show that the group velocity of information propagation grows by power law in time for such systems, whereas systems with short-range interactions exhibit a finite group velocity as shown by Lieb and Robinson.
We establish locality estimates, known as Lieb-Robinson bounds, for the Toda lattice. In contrast to harmonic models, the Lieb-Robinson velocity for these systems do depend on the initial condition. Our results also apply to the entire Toda as well as the Kac-van Moerbeke hierarchy. Under suitable assumptions, our methods also yield a finite velocity for certain perturbations of these systems.
Lieb-Robinson bounds in the continuum via localized frames
本文通过引入格点局域化框架，证明了连续介质中相互作用费米子的一般Lieb-Robinson界，并应用于准自由朗道哈密顿量，为分数量子霍尔效应提供动力学基础。
Lieb-Robinson bounds for Bose-Hubbard Hamiltonians: A review with a simplified proof
本文综述了玻色- Hubbard哈密顿量状态依赖的李-罗宾逊界最新进展，提出简化证明方法，给出多项式速度界。
We prove locality estimates, in the form of Lieb-Robinson bounds, for classical oscillator systems defined on a lattice. Our results hold for the harmonic system and a variety of anharmonic perturbations with long range interactions. The anharmonic estimates are applicable to a special class of observables, the Weyl functions, and the bounds which follow are not only independent of the volume but also the initial condition.
We derive a Lieb-Robinson bound for the propagation of spin correlations in a model of spins interacting through a bosonic lattice field, which satisfies itself a Lieb-Robinson bound in the absence of spin-boson couplings. We apply these bounds to a system of trapped ions, and find that the propagation of spin correlations, as mediated by the phonons of the ion crystal, can be faster than the regimes currently explored in experiments. We propose a scheme to test the bounds by measuring retarded correlation functions via the crystal fluorescence.
Lieb-Robinson bounds for irreversible dynamics via adaptive Dyson expansion
本文针对量子自旋系统的不可逆动力学，提出自适应戴森展开方法，导出与生成元哈密顿量及耗散在位项无关的利布-罗宾逊界，拓展了利布-罗宾逊界的适用范围。
On Lieb-Robinson bounds for a class of continuum fermions
本文研究了连续介质多费米子系统的量子动力学，在放宽势场假设下给出了 Lieb-Robinson 界并将其改进为近弹道形式，同时发展了条件期望概念以逼近时间演化的费米子可观测量。
We study the velocity of the propagation of information for a class of local dissipative quantum dynamics. This finite velocity is expressed by the so-called Lieb-Robinson bound. Besides the properties of the already studied dynamics, we consider an additional relation that expresses the propagation of certain subspaces. The previously derived bounds did not reflect the dissipative character of the dynamics and yielded the same result as for the reversible case. In this article, we show that for this class the velocity of propagation of information is time dependent and decays in time towards …
Unitary dynamics with a strict causal cone (or "light cone") have been studied extensively, under the name of quantum cellular automata (QCAs). In particular, QCAs in one dimension have been completely classified by an index theory. Physical systems often exhibit only approximate causal cones; Hamiltonian evolutions on the lattice satisfy Lieb-Robinson bounds rather than strict locality. This motivates us to study approximately locality preserving unitaries (ALPUs). We show that the index theory is robust and completely extends to one-dimensional ALPUs. As a consequence, we achieve a converse …
The Lieb-Robinson condition and the Fréchet topology
本文研究了量子自旋系统中局部性概念与弗雷歇拓扑的关系，揭示了李-罗布里逊界与连续性之间的非等价性。
We consider a quantum spin chain with nearest neighbor interactions and sparsely distributed on-site impurities. We prove commutator bounds for its Heisenberg dynamics which incorporate the coupling strengths of the impurities. The impurities are assumed to satisfy a minimum spacing, and each impurity has a non-degenerate spectrum. Our results are proven in a broadly applicable setting, both in finite volume and in thermodynamic limit. We apply our results to improve Lieb-Robinson bounds for the Heisenberg spin chain with a random, sparse transverse field drawn from a heavy-tailed distribution…
In this work, we investigate how quickly local perturbations propagate in interacting boson systems with Bose-Hubbard-type Hamiltonians. In general, these systems have unbounded local energies, and arbitrarily fast information propagation may occur. We focus on a specific but experimentally natural situation in which the number of bosons at any one site in the unperturbed initial state is approximately limited. We rigorously prove the existence of an almost-linear information-propagation light-cone, thus establishing a Lieb--Robinson bound: the wave-front grows at most as $t\log^2 (t)$. We pro…
We prove Lieb-Robinson bounds for a general class of lattice fermion systems. By making use of a suitable conditional expectation onto subalgebras of the CAR algebra, we can apply the Lieb-Robinson bounds much in the same way as for quantum spin systems. We preview how to obtain the spectral flow automorphisms and to prove stability of the spectral gap for frustration-free gapped systems satisfying a Local Topological Quantum Order condition.
We propose new 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 whi…
Lieb-Robinson bounds in classical oscillating lattice systems
本文证明了与任意多邻居相互作用的经典谐振子系统在格子和一般结构上存在 Lieb-Robinson 界，并在交换预解代数上证明了全局动力系统的存在性。
We rigorously prove a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for the isotropic XY chain with Fibonacci external magnetic field at arbitrary coupling. It is anomalous in that the usual exponential decay in $x-vt$ is replaced by exponential decay in $x-vt^α$ with $0<α<1$. In fact, we can characterize the values of $α$ for which such a bound holds as those exceeding $α_u^+$, the upper transport exponent of the one-body Fibonacci Hamiltonian. Following the approach of \cite{HSS11}, we relate Lieb-Robinson bounds to dynamical bounds for the one-body Hamiltonian corresponding t…
We announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson bound for an isotropic XY chain in a quasi-periodic transversal magnetic field. By "anomalous", we mean that the usual effective light cone defined by $|x|\leq v|t|$ is replaced by the region $|x|\leq v|t|^α$ for some $0<α<1$. In fact, we can characterize exactly the values of $α$ for which this holds as those exceeding the upper transport exponent $α_u^+$ of an appropriate one-body discrete Schrödinger operator. Previous study has produced a good amount of quantitative information on $α_u^+$.…
Instances of discrete quantum systems coupled to a continuum of oscillators are ubiquitous in physics. Often the continua are approximated by a discrete set of modes. We derive analytical error bounds on expectation values of system observables that have been time evolved under such discretised Hamiltonians. These bounds take on the form of a function of time and the number of discrete modes, where the discrete modes are chosen according to Gauss quadrature rules. The derivation makes use of tools from the field of Lieb-Robinson bounds and the theory of orthonormal polynominals.
The commutator between operators at different space and time has been a diagnostic for locality of unitary evolution. Most existing results are either for specific tractable (random) Hamiltonians(Out-of-Time-Order-Correlators calculations), or for worse case Hamiltonians (Lieb-Robinson-like bounds or OTOC bounds). In this work, we study commutators in typical Hamiltonians. Draw a sample from any zero-mean bounded independent random Hamiltonian ensemble, time-independent or Brownian, we formulate concentration bounds in the spectral norm and for the OTOC with arbitrary non-random state. Our bou…
Lieb-Robinson bounds with exponential-in-volume tails
本文提出了一种新的Lieb-Robinson界，用于改进许多体动力学的模拟资源和区分量子相。
Disordered Lieb-Robinson bounds in one dimension
本研究通过收紧Lieb-Robinson界，证明一维无序自旋链中弱链接抑制算子增长，确定正确动力学指数，并区分全局与子序列界的差异，对淬火动力学等应用有重要影响。
We consider free fermions living on lattices in arbitrary dimensions, where hopping amplitudes follow a power-law decay with respect to the distance. We focus on the regime where this power is larger than the spatial dimension (i.e., where the single particle energies are guaranteed to be bounded) for which we provide a comprehensive series of fundamental constraints on their equilibrium and nonequilibrium properties. First we derive a Lieb-Robinson bound which is optimal in the spatial tail. This bound then implies a clustering property with essentially the same power law for the Green's func…
Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of non-relativistic quantum lattice systems is essentially bounded. We review work of the past dozen years that has turned this fundamental result into a powerful tool for analyzing quantum lattice systems. We introduce a unified framework for a wide range of applications by studying quasi-locality properties of general classes of maps defined on the algebra of local observables of quantum lattice systems. We also consider a number of generalizations that include systems with an…
Stability of slow Hamiltonian dynamics from Lieb-Robinson bounds
本文严格证明了产生慢哈密顿动力学的局部自旋系统对一般性甚至含时局部扰动具有稳定性，其Lieb-Robinson界限由未微扰动力学和扰动两部分贡献组成，该非微扰结果对研究Many-Body-Localized系统稳定性及遍历系统中无序区域对动力学的减慢作用具有重要意义。
A quantum quench is the simplest protocol to investigate nonequilibrium many-body quantum dynamics. Previous studies on the entanglement properties of quenched quantum many-body systems mainly focus on the growth of entanglement entropy. Several rigorous results and phenomenological guiding principles have been established, such as the no-faster-than-linear entanglement growth generated by generic local Hamiltonians and the peculiar logarithmic growth for many-body localized systems. However, little is known about the dynamical behavior of the full entanglement spectrum, which is a refined cha…
The Lieb-Robinson (LR) bound rigorously shows that in quantum systems with short-range interactions, the maximum amount of information that travels beyond an effective "light cone" decays exponentially with distance from the light-cone front, which expands at finite velocity. Despite being a fundamental result, existing bounds are often extremely loose, limiting their applications. We introduce a method that dramatically and qualitatively improves LR bounds in models with finite-range interactions. Most prominently, in systems with a large local Hilbert space dimension $D$, our method gives an…
Enhanced Lieb-Robinson bounds for commuting long-range interactions
本文证明相互对易的长程相互作用在任意α>0下满足增强型Lieb-Robinson界t r^{-α}，将线性光锥阈值改进为α=1，并应用于基态关联和局部扰动局部影响原理。
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 o…
Symmetry-enhanced Lieb-Robinson bounds for a class of Bose-Hubbard type Hamiltonians
针对Bose-Hubbard型哈密顿量，利用平移对称性和局域多体排斥，将信息传播速度从加速改进为亚线性增长，并证明界的最优性。
Dissipation-Enabled Operator-Norm Locality Bounds for Bose-Hubbard Hamiltonians
本文证明局部耗散可恢复玻色-哈伯德系统的算子范数Lieb-Robinson界，通过正则化高占据位点实现几乎弹道式传播界，并推广至猫码耗散，支持局部绝热近似及高效量子模拟。
We consider many-body quantum systems on a finite lattice, where the Hilbert space is the tensor product of finite-dimensional Hilbert spaces associated with each site, and where the Hamiltonian of the system is a sum of local terms. We are interested in proving uniform bounds on various properties as the size of the lattice tends to infinity. An important case is when there is a spectral gap between the lowest state(s) and the rest of the spectrum which persists in this limit, corresponding to what physicists call a ``phase of matter". Here, the combination of elementary Fourier analysis with…
We prove an upper bound on the diffusivity of a general local and translation invariant quantum Markovian spin system: $D \leq D_0 + \left(α\, v_\text{LR} τ+ β\, ξ\right) v_\text{C}$. Here $v_\text{LR}$ is the Lieb-Robinson velocity, $v_\text{C}$ is a velocity defined by the current operator, $τ$ is the decoherence time, $ξ$ is the range of interactions, $D_0$ is a microscopically determined diffusivity and $α$ and $β$ are precisely defined dimensionless coefficients. The bound constrains quantum transport by quantities that can either be obtained from the microscopic interactions ($D_0, v_\te…
We review some recent results that express or rely on the locality properties of the dynamics of quantum spin systems. In particular, we present a slightly sharper version of the recently obtained Lieb-Robinson bound on the group velocity for such systems on a large class of metric graphs. Using this bound we provide expressions of the quasi-locality of the dynamics in various forms, present a proof of the Exponential Clustering Theorem, and discuss a multi-dimensional Lieb-Schultz-Mattis Theorem.
We provide a simple proof of the Lieb-Robinson bound and use it to prove the existence of the dynamics for interactions with polynomial decay. We then use our results to demonstrate that there is an upper bound on the rate at which correlations between observables with separated support can accumulate as a consequence of the dynamics.
We discuss the properties of two open quantum systems with a general class of irreversible quantum dynamics. First we study Lieb-Robinson bounds in a quantum lattice systems. The time-dependent generator of the dynamics of the system is of the Lindblad-Kossakowski type. This generator satisfies some suitable decay condition in space. We show that the dynamics with a such generator on a finite system is a well-defined quantum dynamics in a sense of a norm-continuous cocycle of unit preserving completely positive maps. After proving Lieb-Robinson bounds for a finite lattice systems we also show …
These lecture notes focus on the application of ideas of locality, in particular Lieb-Robinson bounds, to quantum many-body systems. We consider applications including correlation decay, topological order, a higher dimensional Lieb-Schultz-Mattis theorem, and a nonrelativistic Goldstone theorem. The emphasis is on trying to show the ideas behind the calculations. As a result, the proofs are only sketched with an emphasis on the intuitive ideas behind them, and in some cases we use techniques that give very slightly weaker bounds for simplicity. This is a preliminary version of the lecture note…
Speed limits and locality in many-body quantum dynamics
本文综述多体量子系统中 Lieb-Robinson 速度极限与局域性的数学理论、扩展及在模拟、纠缠和量子引力模型中的应用，并梳理开放问题与核心证明。
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 to…
Locality and error correction in quantum dynamics with measurement
本文将 Lieb-Robinson 定理推广到含测量的量子动力学，证明在 M 个局部区域测量结果已知时信息速度至多增强 M+1 倍，并据此约束量子计算、纠错和纠缠态制备的资源需求。
On propagation of information in quantum many-body systems
本文利用最大速度界证明一般晶格量子多体系统中信息传递、关联传播与态控制的最短时间界，并导出光锥近似及 Lieb-Robinson 型界，条件可涵盖长程相互作用。
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