Lieb-Robinson Bounds and the Exponential Clustering Theorem

2005

248 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Condensed Matter Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/64d1ae4ce9169302ae374b219f675814b11353b1

Citations in the corpus

• Spectral Gap and Exponential Decay of Correlations (2005, unknown): “The same bound for fermionic observables is obtained by replacing the commutator with the anticommutator in the left-hand side. For exponentially decaying interaction hX, the following bound is valid [6]: Theorem A.2 Let AX,BY be a pair of bosonic observables with the compact support, X,Y, respectiv”
• Pre-thermal phases of matter protected by time-translation symmetry (2016, unknown): “Now, since D is quasi-local, it must obey a LiebRobinson bound [99, 100], which implies that there exists a local operator ÔA′ supported on a finite region A ′ such that ‖ô(t1) − ÔA′‖ ≤ γ/8, where ô(t1) = e1 ôe−iDt1 .”
• Towards the fast scrambling conjecture (2011, unknown): “Hastings improved the original Lieb-Robinson technique so as to produce dimension-independent bounds [54] and Nachtergaele-Sims showed how to adapt it to general graphs [27].”
• Stability of Frustration-Free Hamiltonians (2011, unknown): “We make extensive use of Lieb-Robinson bounds [28], both for the evolution of operators according to the Hamiltonians under consideration [15,19,30–34,39,40], as well as Lieb-Robinson bounds on the quasi-adiabatic evolution, or spectral flow, of gapped eigenspaces [3,17,20].”
• Light-cone-like spreading of correlations in a quantum many-body system (2011, unknown): “For example, they enable a rigorous proof of a long-standing conjecture that linked the presence of a spectral gap in a lattice system to the exponential decay of correlations in the ground stat”
• Quasiparticle engineering and entanglement propagation in a quantum many-body system (2014, unknown): “These results, known as Lieb-Robinson bounds, have allowed various important theorems to be proven about systems with nearest-neighbour interactions, including restrictions on ground-state correlations [18, 19] and the time to create states for topological quantum computation [16].”
• Matrix product states and projected entangled pair states: Concepts, symmetries, theorems (2020, unknown): “A) and thus cannot be ground states of gapped Hamiltonians (Hastings and Koma, 2006; Nachtergaele and Sims, 2006).”
• A Short Proof of Stability of Topological Order under Local Perturbations (2010, unknown): “It will be applicable only to evolution under Hamiltonians H having a finite Lieb-Robinson velocity [4, 5, 6, 24], for example, Hamiltonians with exponentially decaying interactions.”