General relation between entanglement and fluctuations in one dimension

In one dimension very general results from conformal field theory and exact calculations for quantum spin chains have established universal scaling properties of the entanglement entropy between two parts of a critical system. Using both analytical and numerical methods, we show that if particle number or spin is conserved, fluctuations in a subsystem obey identical scaling as a function of subsystem size in one dimension. We investigate the effects of boundaries and subleading corrections for critical spin and bosonic chains.

Figure 1

(Color online) Results from exact diagonalization for the entanglement entropy (squares) and spin fluctuations (circles) for a spin-1/2 XX chain with PBCs (upper symbols) and OBCs (lower symbols, inset shows zoom around the chain center), $L=100$. For PBCs solid lines are the analytical results in Eqs. (9, 10). For OBCs, solid lines are the analytical result [Eq. (11)] for the fluctuations and a fit to Eq. (12) for the entropy with $c\simeq 0.997$ and $s_1\simeq 1.050$ (the exact values are $c=1$ and $s_1\simeq 1.047$). The fluctuations are scaled by 4.

Figure 2

(Color online) DMRG results (circles) for the spin fluctuations in a spin-1/2 XXZ chain with PBCs, $L=100$. Only $\Delta =-0.2–0.9$ are shown (from top to bottom, in 0.1 increments). Solid lines are fits to Eq. (14). (Inset) Fitted Luttinger parameter $K$ for $\left|\Delta \right|\le 0.9$ (dots) with the solid line showing the Bethe-ansatz solution. The first and last ten sites were dropped for fitting purposes.