Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Shannon-R\\'enyi entropy for Nambu-Goldstone modes in two dimensions

Grégoire Misguich, Vincent Pasquier, Masaki Oshikawa

Published Jan 1, 2016• 0 citations
OpenAlex

Abstract

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 $N_\text{NG}$ of Nambu-Goldstone modes. In the case of a torus, we show that $S_{n>1} \simeq \mathcal{O}(N)+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \log{N}$ and $S_1 \simeq \mathcal{O}(N) - \frac{N_\text{NG}}{4} \log{N}$, 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$\frac{1}{2}$ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the $n=\infty$ R\'enyi entropy (also called $-\log{p_{\rm max}}$). They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, $p_{\rm max}$ is related to a non-compact free-boson partition function in dimension 1+1.

Related papers

Finite-size scaling of the Shannon-Rényi entropy in two-dimensional systems with spontaneously broken continuous symmetry

Grégoire Misguich, Vincent Pasquier, Masaki Oshikawa

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 ${N}_{\text{NG}}$ 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-$\frac{1}{2}$ 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, ${p}_{\mathrm{max}}$ is related to a noncompact free-boson partition function in dimension 1+1.

arXiv:1607.02465•Physical review. B./Physical review. B•May 30, 2017•18 citations

Shannon-Rényi entropies and participation spectra across three-dimensional O(3) criticality

David J. Luitz, Fabien Alet, Nicolas Laflorencie

Universal features in the scalings of Shannon-R\'enyi entropies of many-body ground states are studied for interacting spin-$\frac{1}{2}$ systems across (2+1)-dimensional $O(3)$ critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line-shaped subsystems, SU(2) symmetry breaking on the N\'eel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-R\'enyi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-R\'enyi entropies for line-shaped subsystems of length $L$ embedded in $L\ifmmode\times\else\texttimes\fi{}L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant ${b}_{\ensuremath{\infty}}^{*\mathrm{line}}=0.41(1)$ emerges in the critical scaling of the line Shannon-R\'enyi entropy ${S}_{\ensuremath{\infty}}^{\mathrm{line}}$. This number appears to be universal for 3d $O(3)$ criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-$\frac{1}{2}$ Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular, the N\'eel ordering transition can be simply understood in the ${{S}^{z}}$ basis by a confinement mechanism of ferromagnetic domain walls.

arXiv:1402.4813•Physical Review B•Apr 7, 2014•53 citations

Universal logarithmic corrections to entanglement entropies in two dimensions with spontaneously broken continuous symmetries

David J. Luitz, Xavier Plat, Fabien Alet +1 more

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 ${J}_{1}>0$ 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: ${l}_{q}lnL$ with ${l}_{q}\ensuremath{\simeq}1$, independent of the R\'enyi index and values of ${J}_{2}$. 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 ${l}_{q}={n}_{G}/2$, where ${n}_{G}$ is the number of Goldstone modes, by Metlitski and Grover (arXiv:1112.5166).

arXiv:1503.01094•Physical Review B•Apr 27, 2015•27 citations

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

Nicolas Laflorencie, David J. Luitz, Fabien Alet

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.

arXiv:1506.03703•Physical Review B•Sep 11, 2015•32 citations

Detecting Goldstone modes with entanglement entropy

Bohdan Kulchytskyy, C. M. Herdman, Stephen Inglis +1 more

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.

arXiv:1502.01722•Physical Review B•Sep 25, 2015•40 citations

Entanglement entropy of the two-dimensional Heisenberg antiferromagnet

Hao Song, Nicolas Laflorencie, Stephan Rachel +1 more

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.

arXiv:1103.1636•Physical Review B•Jun 21, 2011•96 citations
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 NNGN_\text{NG}NNG​ of Nambu-Goldstone modes. In the case of a torus, we show that Sn>1≃O(N)+NNG4nn−1log⁡NS_{n>1} \simeq \mathcal{O}(N)+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \log{N}Sn>1​≃O(N)+4NNG​​n−1n​logN and S1≃O(N)−NNG4log⁡NS_1 \simeq \mathcal{O}(N) - \frac{N_\text{NG}}{4} \log{N}S1​≃O(N)−4NNG​​logN, where NNN is the total number of sites and nnn the R'enyi index. The result for n>1n>1n>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 spin12\frac{1}{2}21​ XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the n=∞n=\inftyn=∞ R'enyi entropy (also called −log⁡pmax-\log{p_{\rm max}}−logpmax​). They also reveal some universal dependence on the cylinder aspect ratio, in good agreement with the fact that, in that case, pmaxp_{\rm max}pmax​ is related to a non-compact free-boson partition function in dimension 1+1.
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{N}_{\text{NG}}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 NNN is the total number of sites and nnn the R\'enyi index. The result for n>1n>1n>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-12\frac{1}{2}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{p}_{\mathrm{max}}pmax​ is related to a noncompact free-boson partition function in dimension 1+1.
Universal features in the scalings of Shannon-R\'enyi entropies of many-body ground states are studied for interacting spin-12\frac{1}{2}21​ systems across (2+1)-dimensional O(3)O(3)O(3) critical points, using quantum Monte Carlo simulations on dimerized and plaquettized Heisenberg models on the square lattice. Considering both full systems and line-shaped subsystems, SU(2) symmetry breaking on the N\'eel ordered side of the transition is characterized by the presence of a logarithmic term in the scaling of Shannon-R\'enyi entropies, which is absent in the disordered gapped phase. Such a difference in the scalings allows to capture the quantum critical point using Shannon-R\'enyi entropies for line-shaped subsystems of length LLL embedded in $L\ifmmode\times\else\texttimes\fi{}L$ tori, as the smaller subsystem entropies are numerically accessible to much higher precision than for the full system. Most interestingly, at the quantum phase transition an additive subleading constant ${b}_{\ensuremath{\infty}}^{*\mathrm{line}}=0.41(1)$ emerges in the critical scaling of the line Shannon-R\'enyi entropy ${S}_{\ensuremath{\infty}}^{\mathrm{line}}$. This number appears to be universal for 3d O(3)O(3)O(3) criticality, as confirmed for the finite-temperature transition in the 3d antiferromagnetic spin-12\frac{1}{2}21​ Heisenberg model. Additionally, the phases and phase transition can be detected in several features of the participation spectrum, consisting of the diagonal elements of the reduced density matrix of the line subsystem. In particular, the N\'eel ordering transition can be simply understood in the Sz{{S}^{z}}Sz basis by a confinement mechanism of ferromagnetic domain walls.
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).
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 nG=2{n}_{G}=2nG​=2 Nambu-Goldstone modes give additive logarithmic corrections with a prefactor nG/2{n}_{G}/2nG​/2 independent of the R\'enyi index. On the other hand, $\ensuremath{\pi}/2$ corners lead to additional (negative) logarithmic corrections with a prefactor lqc{l}_{q}^{c}lqc​ which does depend on both nG{n}_{G}nG​ and the R\'enyi index qqq, in good agreement with scalar field theory predictions. By varying the second neighbor coupling J2{J}_{2}J2​ 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<J2/J1<1/20<{J}_{2}/{J}_{1}<1/20<J2​/J1​<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 Sq=nG2lnN{S}_{q}=\frac{{n}_{G}}{2}lnNSq​=2nG​​lnN, thus recovering the mean-field limit for a subsystem of NNN 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-SSS 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.
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 XYXYXY 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-1/21/21/2 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.