We study the scaling of the (basis-dependent) Shannon entropy for two-dimensional quantum antiferromagnets with N\'eel long-range order. We use a massless free-field description of the gapless spin wave modes and phase space arguments to treat the fact that the finite-size ground state is rotationally symmetric, while there are degenerate physical ground states which break the symmetry. Our results show that the Shannon entropy (and its R\'enyi generalizations) possesses some universal logarithmic term proportional to the number
NNG of Nambu-Goldstone modes. In the case of a torus, we show that ${S}_{n>1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N+\frac{{N}_{\text{NG}}}{4}\frac{n}{n\ensuremath{-}1}lnN$ and ${S}_{1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N\ensuremath{-}\frac{{N}_{\text{NG}}}{4}lnN$, where
N is the total number of sites and
n the R\'enyi index. The result for
n>1 is in reasonable agreement with the quantum Monte Carlo results of Luitz et al. [Phys. Rev. Lett. 112, 057203 (2014)], and qualitatively similar to those obtained previously for the entanglement entropy. The Shannon entropy of a line subsystem (embedded in the two-dimensional system) is also considered. Finally, we present some density-matrix renormalization group (DMRG) calculations for a spin-
21 XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the $n=\ensuremath{\infty}$ R\'enyi entropy (also called $\ensuremath{-}ln{p}_{\mathrm{max}})$. They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case,
pmax is related to a noncompact free-boson partition function in dimension 1+1.
We explore the R\'enyi entanglement entropies of a one-dimensional (line) subsystem of length
L 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 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=4 up to
L=40, clearly appears as a universal additive logarithmic correction to the R\'enyi entanglement entropies:
lqlnL with ${l}_{q}\ensuremath{\simeq}1$, independent of the R\'enyi index and values of
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) 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, where
nG is the number of Goldstone modes, by Metlitski and Grover (arXiv:1112.5166).
We examine the entanglement properties of the spin-half Heisenberg model on the two-dimensional square-lattice bilayer based on quantum Monte Carlo calculations of the second R\'enyi entanglement entropy. In particular, we extract the dominant area-law contribution to the bipartite entanglement entropy that shows a nonmonotonous behavior upon increasing the interlayer exchange interaction: a local maximum in the area-law coefficient is located at the quantum critical point separating the antiferromagnetically ordered region from the disordered dimer-singlet regime. Furthermore, we consider subleading logarithmic corrections to the R\'enyi entanglement entropy scaling. Employing different subregion shapes, we isolate the logarithmic corner term from the logarithmic contribution due to Goldstone modes that is found to be enhanced in the limit of decoupled layers. At the quantum critical point, we estimate a contribution of
0.016(1) due to each ${90}^{\ensuremath{\circ}}$ corner. This corner term at the SU(2) quantum critical point deviates from the Gaussian theory value, while it compares well with recent numerical linked cluster calculations on the bilayer model.
We study entanglement properties of systems with spontaneously broken continuous symmetry. We find that in addition to the expected area law behavior, the entanglement entropy contains a subleading contribution which diverges logarithmically with the subsystem size in agreement with the Monte Carlo simulations of A. Kallin et. al. (Phys. Rev. B 84, 165134 (2011)). The coefficient of the logarithm is a universal number given simply by
NG(d−1)/2, where
NG is the number of Goldstone modes and
d is the spatial dimension. This term is present even when the subsystem boundary is straight and contains no corners, and its origin lies in the interplay of Goldstone modes and restoration of symmetry in a finite volume. We also compute the "low-energy" part of the entanglement spectrum and show that it has the same characteristic "tower of states" form as the physical low-energy spectrum obtained when a system with spontaneously broken continuous symmetry is placed in a finite volume.
In the face of mounting numerical evidence, Metlitski and Grover (arXiv:1112.5166) have given compelling analytical arguments that systems with spontaneous broken continuous symmetry contain a subleading contribution to the entanglement entropy that diverges logarithmically with system size. They predict that the coefficient of this log is a universal quantity that depends on the number of Goldstone modes. In this paper, we confirm the presence of this log term through quantum Monte Carlo calculations of the second R\'enyi entropy on the spin-1/2
XY model. Devising an algorithm to facilitate convergence of entropy data at extremely low temperatures, we demonstrate that the single Goldstone mode in the ground state can be identified through the coefficient of the log term. Furthermore, our simulation accuracy allows us to obtain an additional geometric constant additive to the R\'enyi entropy, that matches a predicted fully universal form obtained from a free bosonic field theory with no adjustable parameters.
We compute the von Neumann and generalized R\'enyi entanglement entropies in the ground-state of the spin-
1/2 antiferromagnetic Heisenberg model on the square lattice using the modified spin-wave theory for finite lattices. The addition of a staggered magnetic field to regularize the zero modes associated with symmetry breaking is shown to be essential for obtaining well-behaved values for the entanglement entropy. The von Neumann and R\'enyi entropies obey an area law with additive logarithmic corrections, and are in good quantitative agreement with numerical results from valence bond quantum Monte Carlo and density matrix renormalization group calculations. We also compute the spin fluctuations and observe a multiplicative logarithmic correction to the area law in excellent agreement with quantum Monte Carlo calculations.