The exact relation between the entanglement entropies of the
XY and quantum Ising chains with free and fixed boundary conditions
The entanglement entropies of
XY chains and quantum Ising chains (QICs) with fixed boundary conditions are studied here. Three kinds of boundary conditions (BCs) are considered: fixed up--up or down--down (the spins at both ends are aligned in the same direction), fixed up--down or down--up (the spins at the two ends are aligned in opposite directions), and fixed--free (the spin at one end is aligned, and the other end is free). It is shown that i) the entanglement entropy of an
XY chain with a fixed--free BC is the sum of those of QICs with a fixed--free BC and with a free--free BC; ii) the entanglement entropy of an
XY chain with a fixed up--up boundary condition is the sum of those of QICs with a fixed up--up BC and with a free--free BC; and iii) the entanglement entropy of an
XY chain with a fixed up--down BC is the sum of that of a QIC with a fixed up--up BC and that of the first excited state of a QIC with a free--free BC.