Entanglement Entropy for the Long-Range Ising Chain in a Transverse Field

We consider the Ising model in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance. We study both the phase diagram and the entanglement properties as a function of the exponent of the interaction. The phase diagram can be used as a guide for future experiments with trapped ions. We find two gapped phases, one dominated by the transverse field, exhibiting quasi-long-range order, and one dominated by the long-range interaction, with long-range Néel ordered ground states. We determine the location of the quantum critical points separating those two phases. We determine their critical exponents and central charges. In the phase with quasi-long-range order the ground states exhibit exotic corrections to the area law for the entanglement entropy coexisting with gapped entanglement spectra.

Figure 1

Phase diagram of the LITF from the entanglement entropy. The half chain entanglement entropy provides information about the phase diagram of 1 as a function of $\theta$ and $\alpha$. The background colors represent the value $S_{L/2}$ for a system of size $L=100$. The maximum of it, signals the vicinity of a phase transition. Extrapolating its position as a function of $L$, for $L=20,\dots ,100$, we locate the position of the transition in the thermodynamic limit ${\theta }_c^{\infty }$. The results are superimposed as solid black dots connected by a dashed line.

Figure 2

Violations to the area law. The entanglement entropy for a bipartition increases monotonically with the system size in the whole $z$ polarized phase. For $\alpha \le 1$ the entropy scales logarithmically with the size of the system (small inset). The prefactor in Eq. (5) is extracted by plotting $c/6=\frac{\Delta S}{\log L^{\prime }}$, with $\Delta S=S_{L/2}-S_{L_0/2}$ and $L^{\prime }=L/L_0$, $L_0=20$, $L=30,\dots ,100$ (we use as a reference size $L_0$ in order to eliminate the constant terms in the scaling).

Figure 3

Structure of the ES defined in Eq. (3) as a function of $\log (\theta )$, for $0\le \theta \le 0.36$, deep in the $z$ polarized phase. Left panel $\alpha =2$, the spectrum presents well separated scales reproducible by the lowest order in PT. Several eigenvalues belong to the same order in PT (that can be distinguished as parallel lines) contrary to what happens in the ITF. Right panel, the “nonperturbative” regime for $\alpha =0.3$, there is no clear scale separation, and no clear power law dependence of the eigenvalues with $\theta$. The eigenvalues tend to cluster to form bands separated by gaps.

Figure 4

Long-range universality class. From a mean field analysis for $\alpha >2.25$ the LR become irrelevant. Upper panel, $2{\Delta }_x^{\mathrm{LR}}$ as a function of $\alpha$. As expected close to $\alpha =2$ the exponent tends to its SR value $1/4$. Middle panel, $2{\Delta }_z^{\mathrm{LR}}$ does not coincide with the expected thermal exponent 2 due to sub-leading corrections. Lower panel. The central charge extracted from $S_{L/2}$, unexpectedly, is systematically larger than $1/2$.