Spectral Gap and Exponential Decay of Correlations

Abstract

We study the relation between the spectral gap above the ground state and the decay of the correlations in the ground state in quantum spin and fermion systems with short-range interactions on a wide class of lattices. We prove that, if two observables anticommute with each other at large distance, then the nonvanishing spectral gap implies exponential decay of the corresponding correlation. When two observables commute with each other at large distance, the connected correlation function decays exponentially under the gap assumption. If the observables behave as a vector under the U(1) rotation of a global symmetry of the system, we use previous results on the large distance decay of the correlation function to show the stronger statement that the correlation function itself, rather than just the connected correlation function, decays exponentially under the gap assumption on a lattice with a certain self-similarity in (fractal) dimensions D < 2. In particular, if the system is translationally invariant in one of the spatial directions, then this self-similarity condition is automatically satisfied. We also treat systems with long-range, power-law decaying interactions.

Making Almost Commuting Matrices Commute

“…The next theorem is a Lieb-Robinson bound for such finite range Hamiltonians, similar to those proven for many-body Hamiltonians [8][9][10][11]. This result is also similar to results on the decay of entries of smooth functions of matrices proven in [12,13].…”

Correlations and enlarged superconducting phase of t−J⊥ chains of ultracold molecules on optical lattices

“…Indeed, Hastings and Koma proved this [77] for general short-range interacting spin systems. In contrast, recent theoretical studies have found that long-range interacting systems can have algebraically decaying correlation functions despite the existence of a gap [62,63,65].…”

Lieb-Robinson Bounds and the Simulation of Time-Evolution of Local Observables in Lattice Systems

“…Theorem 4 (Clustering of correlations in unique ground states [31,43]). Let H ∈ A(H) be a local Hamiltonian with a unique ground state ψ and a spectral gap ∆E > 0 and X, Y ⊂ V .…”

“…Outside the space time cone defined by this speed, any signal is typically exponentially suppressed in the distance. The results of Lieb and Robinson, originally derived in the setting of translation invariant 1D spin systems with short range, or exponentially decaying interactions [38] have since been tightened [27,43] and extended to more general graphs [31,47] and to interactions decaying only polynomially with the distance, both, for spin systems [44] and fermionic systems [31] (see also Ref. [45] for a review).…”