We study the entanglement entropy (EE) of the one-dimensional transverse-field quantum Ising chain (QIC) and the XX (isotropic XY) chain under various boundary conditions (BCs). We numerically calculate the EE for a system of size $L$ and a subsystem of size $l$ at the critical point. Five BCs are considered: periodic, free-free, fixed-fixed $(+,+)$ and $(+,-)$, and fixed-free. We also employ exact solutions and conformal field theory (CFT) predictions, which yield results highly consistent with our numerical data. In our numerical approach, we apply the Jordan-Wigner transformation to map the spin chain to a system of spinless fermions and compute their correlation functions, from which the EE is subsequently derived. The EE is calculated for 19 different subsystem fractions, fixing $l/L$ at values of $1/20,2/20,\cdots ,19/20$ for each case. All numerical computations are performed using quadruple precision to ensure accuracy. We observe even-odd oscillations in the EE of the XX chain under free-free, fixed-fixed $(+,+)$, fixed-fixed $(+,-)$, and fixed-free BCs. We then fit the calculated EE using finite-size scaling expansions in terms of $L^{-n}$ and $L^{-n}lnL$ for each corresponding case. The fitted parameters for the leading $lnL$ term and the constant term are found to coincide with CFT predictions to very high accuracy. For periodic BCs, the parameter for the $L^{-2}$ correction term as a function of $l/L$ is consistent with the conjecture by Calabrese et al. Furthermore, we conjecture the form of the $L^{-4}$ correction and analytically calculate its leading term using random matrix theory. For the XX chain with free-free BCs, we verify the relationship between the coefficient of the $L^{-1}$ term and $l/L$ extracted from the formula conjectured by M. Fagotti et al., [J. Stat. Mech. (2011) P01017] and we obtain conjectured closed-form expressions for the coefficients of the $L^{-2}, L^{-3}$, and $L^{-4}$ terms, which are supported by high-precision numerical fitting and partial analytical arguments. Utilizing the relationship between free-free and $(+,+)$ BCs, we determine the form of the coefficients for the $L^{-2}$ and $L^{-4}$ terms in the XX chain with $(+,+)$ BCs. Consequently, we obtain first four parts of the alternating term in the EE for the XX chain under free-free and $(+,+)$ BCs. For the EE of the first excited state in the QIC with free-free BCs, we conjecture the exact form of the $L^{-1}$ correction term, which coincides perfectly with our numerical results. For the QIC with fixed-free BCs, we calculate the perturbation expansion around $l/L=0$ and 1 for the correction term stemming from the Ising CFT. Finally, inspired by the resonating valence bond (RVB) picture introduced by N. Laflorencie et al., [Phys. Rev. Lett. 96, 100603 (2006)] we discuss the EE oscillations in the XX chain under the four aforementioned BCs qualitatively.

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• Quantum information theory
• Quantum phase transitions
• 1-dimensional spin chains
• Conformal field theory
• Exact solutions for many-body systems
• Finite-size scaling
• Ising model
• XY model