Entanglement Entropy in Quantum Spin Chains with Finite Range Interaction

Abstract

We study the entropy of entanglement of the ground state in a wide family of one-dimensional quantum spin chains whose interaction is of finite range and translation invariant. Such systems can be thought of as generalizations of the XY model. The chain is divided in two parts: one containing the first consecutive L spins; the second the remaining ones. In this setting the entropy of entanglement is the von Neumann entropy of either part. At the core of our computation is the explicit evaluation of the leading order term as L → ∞ of the determinant of a block-Toeplitz matrix with symbol

$$\Phi(z) = \left(\begin{array}{cc} i\lambda & g(z) \\ g^{-1}(z) & i \lambda \end{array}\right),$$

where g(z) is the square root of a rational function and g(1/z) = g ⁻¹(z). The asymptotics of such determinant is computed in terms of multi-dimensional theta-functions associated to a hyperelliptic curve ${\mathcal{L}}$ of genus g ≥ 1, which enter into the solution of a Riemann-Hilbert problem. Phase transitions for these systems are characterized by the branch points of ${\mathcal{L}}$ approaching the unit circle. In these circumstances the entropy diverges logarithmically. We also recover, as particular cases, the formulae for the entropy discovered by Jin and Korepin [14] for the XX model and Its, Jin and Korepin [12, 13] for the XY model.