Entanglement Entropy in Critical Harmonic Chains with Even Dynamical Exponents

We study the behavior of the entanglement entropy in a chain of coupled harmonic oscillators at the critical regime and in the absence of conformal symmetry. We consider a specific class of the so-called ``squared'' interactions [1], namely interactions leading to the dispersion $\omega_k = (2 Sin (k/2))^z$ with even dynamical exponent, $z$, in which up to the $z^{th}$ nearest neighbors are coupled. Similar to the conformally symmetric case, we find a logarithmic scaling for the entanglement entropy, with a coefficient that can be calculated analytically and depends only on $z$. \\[4pt] [1] M. B. Plenio, J. Eisert, J. Drei{\ss}ig, and M. Cramer, Phys. Rev. Lett. {\bf94}, 060503 (2005)