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We calculate the entanglement and the universal boundary entropy (BE) in thecritical quantum spin chains, such as the transverse field Ising chain and theXXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). Wedetermine the boundary universality class that an ADBMF induces. In particular,we show that the induced boundary conformal field theory (BCFT) depends on thepoint on the Bloch sphere where the boundary magnetic field directs. We showthat the classification of the directions boils down to the simple fact thatthe boundary field breaks the bulk symmetry or does not. We present a procedureto estimate the universal BE, based on the finite-size corrections of theentanglement entropy, that apply to the ADBMF. To calculate the universal BE inthe XXZ chain, we use the density matrix renormalization group (DMRG). Thetransverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation isnot a quadratic free fermion Hamiltonian. We map this model to a quadratic freefermion chain by introducing two extra ancillary spins coupled to the mainchain at the boundaries, which makes the problem {\it{integrable}}. Theeigenstates of the transverse field XY chain can be obtained by properprojection in the enlarged chain. Using this mapping, we are able to calculatethe entanglement entropy of the transverse field XY chain using the usualcorrelation matrix technique up to relatively large sizes.

Entanglement and boundary entropy in quantum spin chains with arbitrary direction of the boundary magnetic fields