# Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms

@article{Nachtergaele2018QuasilocalityBF, title={Quasi-locality bounds for quantum lattice systems. I. Lieb-Robinson bounds, quasi-local maps, and spectral flow automorphisms}, author={Bruno Nachtergaele and Robert Sims and Amanda Young}, journal={Journal of Mathematical Physics}, year={2018} }

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…

## 72 Citations

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## References

SHOWING 1-10 OF 137 REFERENCES

### Locality of dynamics in general harmonic quantum systems

- Physics
- 2008

The Lieb-Robinson theorem states that locality is approximately preserved in the dynamics of quantum lattice systems. Whenever one has finite-dimensional constituents, observables evolving in time…

### Topological quantum order: Stability under local perturbations

- Mathematics
- 2010

We study zero-temperature stability of topological phases of matter under weak time-independent perturbations. Our results apply to quantum spin Hamiltonians that can be written as a sum of…

### Dynamics for QCD on an Infinite Lattice

- Mathematics
- 2015

We prove the existence of the dynamics automorphism group for Hamiltonian QCD on an infinite lattice in $${{\mathbb{R}}^3}$$R3, and this is done in a C*-algebraic context. The existence of ground…

### A Short Proof of Stability of Topological Order under Local Perturbations

- Mathematics
- 2010

Recently, the stability of certain topological phases of matter under weak perturbations was proven. Here, we present a short, alternate proof of the same result. We consider models of topological…

### The spectral gap for some spin chains with discrete symmetry breaking

- Mathematics
- 1996

We prove that for any finite set of generalized valence bond solid (GVBS) states of a quantum spin chain there exists a translation invariant finite-range Hamiltonian for which this set is the set of…

### Elementary excitations in gapped quantum spin systems.

- PhysicsPhysical review letters
- 2013

It is shown that for translation invariant systems, simultaneous eigenstates of energy and momentum with an eigenvalue that is separated from the rest of the spectrum in that momentum sector can be arbitrarily well approximated by building a momentum superposition of a local operator acting on the ground state.

### On the Stability of Charges in Infinite Quantum Spin Systems

- PhysicsCommunications in Mathematical Physics
- 2019

We consider a theory of superselection sectors for infinite quantum spin systems, describing charges that can be approximately localized in cone-like regions. The primary examples we have in mind are…

### Quasiadiabatic continuation of quantum states: The stability of topological ground-state degeneracy and emergent gauge invariance

- Physics
- 2005

We define for quantum many-body systems a quasiadiabatic continuation of quantum states. The continuation is valid when the Hamiltonian has a gap, or else has a sufficiently small low-energy density…

### Locality Estimates for Quantum Spin Systems

- Mathematics
- 2009

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 Bounds for Multi-Commutators and Applications to Response Theory

- Mathematics
- 2016

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…