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arXiv:1506.03703 · Physical Review B

Spin-wave approach for entanglement entropies of the J1−J2 Heisenberg antiferromagnet on the square lattice

Nicolas Laflorencie, David J. Luitz, Fabien Alet

Published Sep 11, 2015• 32 citations
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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 ${J}_{1}\ensuremath{-}{J}_{2}$ 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, $\ensuremath{\pi}/2$ corners lead to additional (negative) logarithmic corrections with a prefactor ${l}_{q}^{c}$ 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 ${J}_{1}\ensuremath{-}{J}_{2}$ antiferromagnet, from the frustrated side $0<{J}_{2}/{J}_{1}<1/2$ where the area law term is maximal, to the strongly ferromagnetic regime $\ensuremath{-}{J}_{2}/{J}_{1}\ensuremath{\gg}1$ with a purely logarithmic growth ${S}_{q}=\frac{{n}_{G}}{2}lnN$, thus recovering the mean-field limit for a subsystem of $N$ sites. Finally, a universal subleading constant term ${\ensuremath{\gamma}}_{q}^{\mathrm{ord}}$ 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 ${\ensuremath{\gamma}}_{q}^{\text{ord}}$ a regular part and a singular component, explaining the discrepancy of the linear scaling term for fixed width vs fixed aspect ratio subsystems.

DOI
10.1103/physrevb.92.115126

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We explore the R\'enyi entanglement entropies of a one-dimensional (line) subsystem of length LLL embedded in a two-dimensional $L\ifmmode\times\else\texttimes\fi{}L$ square lattice for quantum spin models whose ground state breaks a continuous symmetry in the thermodynamic limit. We first study the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model (with antiferromagnetic nearest-neighbor J1>0{J}_{1}>0J1​>0 and ferromagnetic second-neighbor couplings ${J}_{2}\ensuremath{\le}0)$, breaking SU(2) symmetry in the ground state, using quantum Monte Carlo simulations. The signature of SU(2)-symmetry breaking on finite-size systems, ranging from L=4L=4L=4 up to L=40L=40L=40, clearly appears as a universal additive logarithmic correction to the R\'enyi entanglement entropies: lqlnL{l}_{q}lnLlq​lnL with ${l}_{q}\ensuremath{\simeq}1$, independent of the R\'enyi index and values of J2{J}_{2}J2​. We then confirm this result using a spin-wave analysis (with restored spin rotational symmetry) on the same ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ model, as well as on the XY model, with a U(1)U(1)U(1) broken symmetry in the ground state. The high-precision spin-wave calculations are performed on finite lattices up to ${10}^{5}\ifmmode\times\else\texttimes\fi{}{10}^{5}$ sites, allowing us to explore further nonuniversal finite-size corrections. Our results fully agree with the prediction lq=nG/2{l}_{q}={n}_{G}/2lq​=nG​/2, where nG{n}_{G}nG​ is the number of Goldstone modes, by Metlitski and Grover (arXiv:1112.5166).
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