Entanglement Entropy Fluctuations in Quantum Ising Chains

The Ising chains in a transverse magnetic field of constant strength (h=1) and with the spin interaction value \lambda are considered. In the case of infinitely long chain, exact analytical expressions are found for the second central moment (dispersion) of the entropy operator S^\hat=-ln\rho with reduced density matrix \rho which corresponds to a semi-infinite part of the model in the ground state. It is shown that in the vicinity of a critical point \lambda_c=1, the entanglement entropy fluctuation \Delta S (square root of dispersion) diverges as \Delts S\sim[ln(1/|1-\lambda|)]^{1/2}. Taking into account the known behavior of the entanglement entropy S, this leads to that the value of relative entanglement fluctuation \delta S=(\Delta S)/S vanishes at the critical point, i.e. in fact a state with nonfluctuating entanglement is realized.