Correlation length and unusual corrections to entanglement entropy

We study analytically the corrections to the leading terms in the Rényi entropy of a massive lattice theory, showing significant deviations from naive expectations. In particular, we show that finite size and finite mass effects give rise to different contributions (with different exponents) and thus violate a simple scaling argument. In the specific, we look at the entanglement entropy of a bipartite $XYZ$ spin-$1/2$ chain in its ground state. When the system is divided into two semi-infinite half-chains, we have an analytical expression of the Rényi entropy as a function of a single mass parameter. In the scaling limit, we show that the entropy as a function of the correlation length formally coincides with that of a bulk Ising model. This should be compared with the fact that, at criticality, the model is described by a $c=1$ conformal field theory and the corrections to the entropy due to finite size effects show exponents depending on the compactification radius of the theory. We will argue that there is no contradiction between these statements. If the lattice spacing is retained finite, the relation between the mass parameter and the correlation length generates new subleading terms in the entropy, whose form is path dependent in phase space and whose interpretation within a field theory is not available yet. These contributions arise as a consequence of the existence of stable bound states and are thus a distinctive feature of truly interacting theories, such as the $XYZ$ chain.

Figure 1

Phase diagram of the $XYZ$ model in the $(J_z,J_y)$ plane. The blue solid lines—$J_z=\pm 1$, $|J_y|\le 1$; $J_y=\pm 1$, $|J_z|\le 1$; and $J_z=\pm J_y$, $|J_z|\ge 1$—correspond to the critical phase of a rotated $XXZ$ chain. Out of the four “tricritical” points, $C_{1,2}$ are conformal and $E_{1,2}$ are not. The area within the red rectangle is the portion of the phase diagram we study in this article, and it is completely equivalent to the principal regime of the eight-vertex model. The yellow and green lines are the curves of constant $l$ and $\mu$, respectively, in the $(J_z,J_y)$ plane, according to the parametrization (11).

Figure 2

Curves of constant correlation length of the $XYZ$ model in the ($J_z,J_y$) plane. Regions of similar colors correspond to the same correlation length values and red colors are associated to higher values of $\xi$. It is worth stressing that the correlation length does not show any essential critical behavior, while the block entropy shows it in proximity of points $E_1$ and $E_2$.

Figure 3

Curves of constant entropy of the $XYZ$ model in the ($J_z,J_y$) plane. Regions of similar colors have similar entropy values and the line where colors change are the lines of constant entropy. The brighter is the color, the bigger the entropy.

Figure 4

Three different ways to approach the critical line: along the $\mu$-constant lines (blue line), which is referred to as renormalization group flow in the text; along straight lines (red line); and along lines that approach the criticality with zero derivative (green line).