Correlation Decay in Fermionic Lattice Systems with Power-Law Interactions at Nonzero Temperature

We study correlations in fermionic lattice systems with long-range interactions in thermal equilibrium. We prove a bound on the correlation decay between anticommuting operators and generalize a long-range Lieb-Robinson-type bound. Our results show that in these systems of spatial dimension $D$ with, not necessarily translation invariant, two-site interactions decaying algebraically with the distance with an exponent $\alpha \ge 2D$, correlations between such operators decay at least algebraically to 0 with an exponent arbitrarily close to $\alpha$ at any nonzero temperature. Our bound is asymptotically tight, which we demonstrate by a high temperature expansion and by numerically analyzing density-density correlations in the one-dimensional quadratic (free, exactly solvable) Kitaev chain with long-range pairing.

Figure 1

Exponent $\nu$ as a function of the exponent of the interactions decay $\alpha$ extracted from the data for $L=2000$. The blue, orange, and green lines correspond to $\mu =0.5$, 1.0, 1.5. The line styles correspond to the inverse temperatures $\beta =0.1,1.0,\infty$. Exponents inside the shaded region are excluded by theorem 3 whenever $T>0$. For high temperatures and $\alpha <1$ finite size effects slightly distort the results; for large $\alpha$ the finite precision is the limiting factor.