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.
This work provides an overview of gapped quantum spin systems, including concepts, techniques, properties, and results. The basic framework and objects of interest for quantum spin systems are introduced, and the main ideas behind methods for proving spectral gaps for frustration-free models are outlined. After reviewing recent progress on several spectral gap conjectures, we discuss quasi-locality of the Heisenberg dynamics and its utility in proving properties of gapped quantum spin systems. Lieb-Robinson bounds have played a central role in establishing exponential decay of ground state correlations, an area law for one-dimensional systems, a many-body adiabatic theorem, and spectral gap stability. They also aided in the development of the quasi-adiabatic continuation, which is a useful for investigating gapped ground state phases, both of which 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.
We review the Lieb–Schultz–Mattis theorem and its variants, which are no-go theorems that state that a quantum many-body system with certain conditions cannot have a locally-unique gapped ground state. We restrict ourselves to one-dimensional quantum spin systems and discuss both the generalized Lieb–Schultz–Mattis theorem for models with U(1) symmetry and the extended Lieb–Schultz–Mattis theorem for models with discrete symmetry. We also discuss the implication of the same arguments to systems on the infinite cylinder, both with the periodic boundary conditions and with the spiral boundary conditions. For models with U(1) symmetry, we here present a rearranged version of the original proof of Lieb, Schultz, and Mattis based on the twist operator. As the title suggests, we take a modern topological point of view and prove the generalized Lieb–Schultz–Mattis theorem by making use of a topological index (which coincides with the filling factor). By a topological index, we mean an index that characterizes a locally-unique gapped ground state and is invariant under continuous (or smooth) modification of the ground state. For models with discrete symmetry, we describe the basic idea of the most general proof based on the topological index introduced in the context of symmetry-protected topological phases. We start from background materials such as the classification of projective representations of the symmetry group. We also review the notion that we call a locally-unique gapped ground state of a quantum spin system on an infinite lattice and present basic theorems. This notion turns out to be natural and useful from the physicists’ point of view. We have tried to make the present chapter readable and almost self-contained. We only assume basic knowledge about quantum spin systems.
Lieb-Robinson bounds show that the speed of propagation of information under the Heisenberg dynamics in a wide class of nonrelativistic quantum lattice systems is essentially bounded. We review works 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 quasilocality 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 infinite-dimensional Hilbert space at each lattice site and Hamiltonians that may involve unbounded on-site contributions. These generalizations require replacing the operator norm topology with the strong operator topology in a number of basic results for the dynamics of quantum lattice systems. The main results in this paper form the basis for a detailed proof of the stability of gapped ground state phases of frustrationfree models satisfying a local topological quantum order condition, which we present in a sequel to this paper.