Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that, with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance r after a time t decays as exp ( v t − r ) , where v is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on r d sites in d spatial dimensions. Perturbation theory and cluster expansion methods suggest that, at short times, these volume-filling operators are suppressed as exp ( − r d ) . We confirm this intuition, showing that, for r > v t , the volume-filling operator is suppressed by exp ( − ( r − v t ) d / ( v t ) d − 1 ) . This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance ε for any finite time t : as ε becomes sufficiently small, only ε − O ( t d − 1 ) resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the “solvable (Ising) point” in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
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 shows that the speed of propagating information in a nonrelativistic quantum lattice system is bounded by a finite velocity, which entails the clustering of correlations. In this paper, we extend the Lieb-Robinson bound to quantum systems at finite temperature by calculating the dynamical correlation function at nonzero temperature for systems whose interactions are, respectively, short range, exponentially decaying, and long range. We introduce a simple way of counting the clusters in a cluster expansion by using the combinatoric generating functions of graphs. Limitations and possible applications of the obtained bound are also discussed.
We derive an extension of the Lieb--Robinson bound that accounts for memory effects in quantum lattice systems evolving under discrete-time non-Markovian dynamics generated by local interactions with finite-dimensional memory subsystems associated with each bond. The global evolution is unitary on an extended Hilbert space
Hext=⨂i=1NHi⊗⨂bondsHM(i), while non-Markovian behavior, arising from repeated coupling to the same bond memory at successive time steps, leads to temporal correlations in the system. In this work, “finite memory” refers specifically to memory subsystems of fixed finite Hilbert space dimension whose reduced dynamics satisfy an exponential mixing condition in operator norm, meaning that correlation functions decay exponentially over time, with a characteristic timescale
τ. More precisely, the memory channels satisfy
∥TMs(X)−Tr(X)σM∥≤ce−s/τ∥X∥, for all memory observables
X, where
σM is a stationary state, and
c>0 depends only on the memory dynamics. For a one-dimensional lattice with nearest-neighbor interactions, we prove that the connected correlation function, defined as
Ci,j(t):=supAi=Ai†,Bj=Bj†∥Ai∥≤1,∥Bj∥≤1∣Tr[(Ai⊗Bj)ρ(t)]−Tr[Aiρi(t)]Tr[Bjρj(t)]∣, satisfies an exponential bound of the form
Ci,j(t)≤C0exp[−ξd(i,j)−veff(τ)t], where
d(i,j)=∣i−j∣ and
ξ>0 is independent of system size and time. The effective velocity
veff(τ) is explicitly bounded in terms of the local interaction strength and the memory mixing parameters, ensuring the system exhibits a finite light cone, meaning that information propagation is confined to a region growing linearly with time, with a well-defined maximum speed. In the limit
τ→∞, the bound no longer guarantees a finite propagation speed, reflecting the transition to a persistent memory regime where correlations do not decay exponentially. As an operational consequence, the entanglement fidelity between two-site reduced states
ρi,j(t) and a maximally entangled state satisfies
Fi,j(t)=⟨Φ+∣ρi,j(t)∣Φ+⟩≤21[1+Ci,j(t)]. These results extend Lieb--Robinson theory to a class of discrete-time quantum dynamics with memory subsystems associated with each bond, enabling resummation of multi-step memory effects into an effective correlation propagation kernel that governs the decay of temporal correlations. This establishes a quantitative relationship between the timescale of memory mixing and the effective speed of information propagation in the system.
This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.