Entanglement entropies of the $J_1 - J_2$ Heisenberg antiferromagnet on the square lattice

Abstract

Using a modified spin-wave theory which artificially restores zero sublattice magnetization on finite lattices, we investigate the entanglement properties of the N\'eel ordered $J_1 - J_2$ Heisenberg antiferromagnet on the square lattice. Different kinds of subsystem geometries are studied, either corner-free (line, strip) or with sharp corners (square). Contributions from the $n_G=2$ Nambu-Goldstone modes give additive logarithmic corrections with a prefactor ${n_G}/{2}$ independent of the R\'enyi index. On the other hand, corners lead to additional (negative) logarithmic corrections with a prefactor $l^{c}_q$ which does depend on both $n_G$ and the R\'enyi index $q$, in good agreement with scalar field theory predictions. By varying the second neighbor coupling $J_2$ we also explore universality across the N\'eel ordered side of the phase diagram of the $J_1 - J_2$ antiferromagnet, from the frustrated side $0<J_2/J_1<1/2$ where the area law term is maximal, to the strongly ferromagnetic regime $-J_2/J_1\gg1$ with a purely logarithmic growth $S_q=\frac{n_G}{2}\ln N$, thus recovering the mean-field limit for a subsystem of $N$ sites. Finally, a universal subleading constant term $\gamma_q^{\rm ord}$ is extracted in the case of strip subsystems, and a direct relation is found (in the large-S limit) with the same constant extracted from free lattice systems. The singular limit of vanishing aspect ratios is also explored, where we identify for $\gamma_q^\text{ord}$ a regular part and a singular component, explaining the discrepancy of the linear scaling term for fixed width {\it{vs}} fixed aspect ratio subsystems.