Finite temperature scaling of multipartite entanglement in the critical 1-D spin ½ antiferromagnetic Heisenberg chain

Scaling laws of the bipartite entanglement entropy at 1+1-D conformally invariant quantum critical points motivated several developments in identifying signatures of critical behaviour in many body systems and their underlying field theories. Less is known about the critical behaviour of multipartite entanglement, which describes higher order infactorability of the many body density matrix. However, multipartite entanglement can be quantified through the Quantum Fisher information (QFI), an entanglement witness that can also be related to dynamical response functions. We show that the QFI can be expressed as the static structure factor, plus a correction that vanishes as T ? 0. Therefore, in systems with a divergent static structure factor, we can deduce finite temperature scaling of multipartite entanglement without integrating contributions to the QFI from arbitrarily high energy dynamical response. This result is useful when the critical theory has at most marginally relevant operators, where the real-space RG scaling theory of the QFI degenerates. The critical antiferromagnetic Heisenberg model is such a theory, and we show that the QFI in the Heisenberg chain diverges as log(ß)^3/2, the known scaling of the longitudinal static structure factor in this system. We use MPS simulations and calculations of the QFI from conformal field theory to demonstrate this scaling hypothesis. Our results imply that the Heisenberg chain hosts highly entangled states at experimentally accessible temperatures.