Logarithmic divergence of the block entanglement entropy for the ferromagnetic Heisenberg model

that for critical (gapless) quantum system (for the XXZ model when ∆ belongs to the interval (−1, 1)) the entanglement entropy of a block of n spins diverges logarithmically as γ log2 n, while for non critical systems (∆ outside the above mentioned interval), it converges to a constant finite value [1, 2, 3]. This property was interpreted in the framework of conformal field theory [4] associated with the corresponding quantum phase transition and the prefactor γ of the logarithm related to the central charge of the theory c = 3γ (for the XXZ model this gives γ = 1/3). The aim of this Letter is to show that the entanglement entropy of a block of spins in the ground state of the antiferromagnetic XXZ model (1), at the point ∆ = −1 grows faster than for other critical points −1 < ∆ ≤ 1, namely as γ log2 n with the logarithmic prefactor 1 2 ≤ γ ≤ 1. Our approach uses the permutational invariance of the ground state of (1) at ∆ = −1, this allowing to compute the entanglement entropy exactly for blocks of arbitrary size and system of arbitrary length. To this regard we remark that by performing the transformation which overturns each second spin along the chain (we assume the length of the chain even) the Hamiltonian (1) for ∆ = −1 reduces to the isotropic Heisenberg ferromagnet (2). Since this transformation does not change the entropy of entanglement, one can compute the block entropy of the antiferromagnetic Heisenberg chain at ∆ = −1 directly from the one of the isotropic ferromagnetic model. It is worth noting that, in contrast with critical points −1 < ∆ ≤ 1, the point ∆ = −1 cannot be studied by means of conformal field theory since this point is not conformal invariant [4], the ground state being infinitely degenerated at ∆ = −1. [5] The paper is organized as follows. After introducing the model we formulate a theorem which gives the analytical expression of the eigenvalues of the reduced density matrix. Using this theorem we compute the entanglement entropy of a block of size n in the finite system of total length L for two specific choices of the ground state sector. Taking the limit of large subsystem sizes, we derive analytical expressions for the entanglement entropy S(n) of a block of spins of size n in the ferromagnetic ground state, both for n,L ≫ 1 and for n ≫ 1, L = ∞. As a result, we obtain that in the ground state sector with a fixed value of S the block entanglement entropy grows for large n, as S(n) = 1 2 log2 n(L−n) L , while in the ground state sector in which all the S components of the spin multiplet are equally weighted, S(n) = log2(n + 1) for arbitrary n and L. We consider the ferromagnetic Heisenberg model with nearest neighbor interaction,