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

Abstract

We calculate the entanglement and the universal boundary entropy (BE) in the critical quantum spin chains, such as the transverse field Ising chain and the XXZ chain, with arbitrary direction of the boundary magnetic field (ADBMF). We determine the boundary universality class that an ADBMF induces. In particular, we show that the induced boundary conformal field theory (BCFT) depends on the point on the Bloch sphere where the boundary magnetic field directs. We show that the classification of the directions boils down to this simple fact that the boundary field breaks the bulk symmetry or not. We present a procedure to estimate the universal BE, based on the finitesize corrections of the entanglement entropy, that apply to ADBMF. To calculate the universal BE in the XXZ chain, we use density matrix renormalization group (DMRG). The transverse field XY chain with ADBMF after Jordan-Wigner (JW) transformation is not a quadratic free fermion Hamiltonian. We are able to map this model to a quadratic free fermion chain by introducing two extra ancillary spins coupled to the main chain at the boundaries, which makes the problem integrable. The eigenstates of the transverse field XY chain can be obtained by a proper projection in the enlarged chain. Using this mapping, we are able to calculate the entanglement entropy of the transverse field XY chain using the usual correlation matrix technique up to relatively large sizes.

Second Rényi entropy and annulus partition function for one-dimensional quantum critical systems with boundaries

“…We give here the organization of the paper. Section 2 provides a detailed derivation of the main result (1.8) and a non-trivial check that our calculation does recover the result of [63,9,67] for the S 2 Rényi entropy of an interval A touching the boundary. We also recover the known results for the Dirac fermion [66,83,84] and more generally the compact scalar field of [75].…”

“…The eigenvalues of M are simply related to the values of the entanglement spectrum, and thus one can calculate Rényi entropies for large sizes with an advantageous computational cost that scales as O(N ) with the number N of spins in the system. This method, known in the literature as Peschel's trick [91,92], has been employed in several works [93,67] for both free and periodic boundary conditions, and we refer to them for detailed explanations of the implementation.…”

“…The Peschel trick fails, however, for the case of fixed BC, where the above feature of the wavefunction no longer holds, as pointed out in [66]. There has 3 COMPARISON WITH NUMERICS AND FINITE-SIZE SCALING been progress, however, in adapting the trick to fixed BC, for the case of an interval touching the boundary [66][67][68]. Extending the technique to efficiently find the entanglement spectrum for an interval A that does not touch the boundary is still an open problem.…”

“…In the scaling limit, such an open critical system is described by a Boundary Conformal Field Theory (BCFT), with a well-established [59][60][61] correspondence between the chiral Virasoro representations and the conformal boundary conditions (BC) allowed by the theory. The case of an interval touching the boundary has been extensively studied (see [62] for a review) using either conformal field theory methods [9,39,[63][64][65] or exact free fermion methods [66][67][68], including symmetry-resolved entropies [69]. It has also been checked numerically using density-matrix renormalization group techniques [70][71][72][73].…”

Rényi entropies for one-dimensional quantum systems with mixed boundary conditions

“…In the scaling limit, such an open critical system is described by a Boundary Conformal Field Theory (BCFT), with a well understood [53][54][55][56] correspondence between the chiral Virasoro representations and the conformal boundary conditions allowed by the theory, and an algebra of boundary operators that interpolate between them. The more accessible setup of an interval touching one of two identical boundaries has been thoroughly analysed using either conformal field theory methods [2,52,[57][58][59][60] or exact free fermion techniques [61][62][63]. Such configurations are also well-handled numerically, through density-matrix renormalization group (DMRG) techniques [64][65][66][67].…”