Entanglement and Density Matrix of a Block of Spins in AKLT Model

Abstract

We study a 1-dimensional AKLT spin chain, consisting of spins S in the bulk and S/2 at both ends. The unique ground state of this AKLT model is described by the Valence-Bond-Solid (VBS) state. We investigate the density matrix of a contiguous block of bulk spins in this ground state. It is shown that the density matrix is a projector onto a subspace of dimension (S + 1)2 . This subspace is described by nonzero eigenvalues and corresponding eigenvectors of the density matrix. We prove that for large block the von Neumann entropy coincides with Renyi entropy and is equal to ln (S + 1) 2 .

Entanglement spectra of Heisenberg ladders of higher spin

“…Here we have introduced two further types of singlet states, are the components of a two-rung triplet state composed of a triplet on rung i and a quintuplet on rung j. Summing up, the second-order correction to the ground state can be written as 19) where the first line on the r.h.s results from the second term in Eq. (A.3), and all other contributions stem from the second line there.…”

Spin-spin correlation functions of the q-valence-bond-solid state of an integer spin model

“…For isotropic higher-spin models, the spin-spin correlation functions [20] and the entanglement entropy [21,22] have been calculated by utilizing the spin coherent representation and the properties of Legendre polynomials. For the q-deformed spin-1 model, spin-spin correlation functions were evaluated [7,8,9] from the matrix product representation.…”

Block spin density matrix of the inhomogeneous AKLT model

“…The entanglement of quantum states, particularly related with spin systems has been attracting a great deal of interest, see for example [5], [15], [18], [24], [26], [28], [32], [37], [43], [46], [53], [58], [64], [65], [70]. Quantum entanglement can be quantified by the von Neumann entropy and Renyi entropy of a subsystem, as discussed in [18], [21], [22], [36], [38], [66], [71]. An area law of the von Neumann entropy in harmonic lattice systems has been proposed and studied in [13], [14], [58].…”

“…The density matrix of a contiguous block of bulk spins (we call it the density matrix later for short) of the homogeneous AKLT model has been studied extensively in [18], [23], [38], [41], [64], [71]. It contains information of all correlation functions [6], [37], [38], [71].…”

“…It contains information of all correlation functions [6], [37], [38], [71]. Moreover, the density matrix was shown [18], [38] to be independent of the size of the chain and the location of the block relative to the ends.…”