Ground states of quantum many-body systems and exponential decay of correlations
Luca Capizzi (LPTMS)
Ground states of quantum many-body systems are known to have qualitatively distinct properties compared to eigenstates in the middle of the spectrum. Specifically, in the presence of a finite energy gap, while the latter display volume-law entanglement, the former show area-law entanglement; in other words, the ground-state entanglement entropy of a subregion scales with its area. As a consequence, these states can be efficiently simulated using state-of-the-art numerical tensor network methods, such as Matrix Product States (MPS). This fundamental result of modern quantum physics traces back to a series of mathematical theorems proven by M.B. Hastings at the beginning of this century [1-3]. In this seminar, based on Ref. [1], I will show how a gap in the energy spectrum implies an exponential decay of spatial correlations, a highly non-trivial result obtained via the Lieb-Robinson bound and energy filters. Furthermore, I will emphasize that having short-range correlations, a property that holds generically even in the middle of the energy spectrum [4], does not directly relate to area-law entanglement.
[1] M. B. Hastings, Locality in Quantum and Markov Dynamics on Lattices and Networks, Phys. Rev. Lett. 93, 140402 (2004).
[2] M. B. Hastings, An Area Law for One Dimensional Quantum Systems, JSTAT, P08024 (2007).
[3] M. B. Hastings, Entropy and entanglement in quantum ground states, Phys. Rev. B 76, 035114 (2007).
[4] Araki, H. Gibbs states of a one dimensional quantum lattice, Commun.Math. Phys. 14, 120–157 (1969).