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

arXiv:1503.01094 · Physical Review B

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

David J. Luitz, Xavier Plat, Fabien Alet, Nicolas Laflorencie

Published Apr 27, 2015• 27 citations
arXivHTMLPDF

Abstract

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).

DOI
10.1103/physrevb.91.155145

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

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

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

Improved scaling of the entanglement entropy of quantum antiferromagnetic Heisenberg systems

Zehui Deng, Lu Liu, Wenan Guo +1 more

In this paper, we derive corrections to the subleading logarithmic term of the entanglement entropy in systems with spontaneous broken continuous symmetry. Using quantum Monte Carlo simulations, we show that the improved scaling formula leads to much better estimations of the number of Goldstone modes in the two-dimensional square lattice spin-1/2 Heisenberg model and bilayer spin-1/2 Heisenberg model in systems of rather small sizes, compared with previous results. In addition, the universal geometry-dependent finite constant in the entanglement entropy scaling is also obtained in good agreement with the theoretical value.

arXiv:2306.01554•Physical review. B./Physical review. B•Sep 26, 2023•16 citations

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

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 $S_{n>1} \simeq \mathcal{O}(N)+ \frac{N_\text{NG}}{4}\frac{n}{n-1} \log{N}$ 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.

•Jan 1, 2016•0 citations

Quantum Monte Carlo detection of SU(2) symmetry breaking in the participation entropies of line subsystems

David J. Luitz, Nicolas Laflorencie

Using quantum Monte Carlo simulations, we compute the participation (Shannon-Rényi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length L L embedded in two-dimensional ( L\times L L × L ) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional ( L\times L\times L L × L × L ) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term l_q\ln L l q ln L in the Rényi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa . We also study the dependence of the log prefactor l_q l q on the Rényi index q q for which a transition is detected at q_c\simeq 1 q c ≃ 1 .

arXiv:1612.06338•SciPost Physics•Mar 23, 2017•6 citations
We explore the R'enyi entanglement entropies of a one-dimensional (line) subsystem of length LLL embedded in a two-dimensional L\ifmmode×\else\texttimes\fiLL\ifmmode\times\else\texttimes\fi{}LL\ifmmode×\else\texttimes\fiL square lattice for quantum spin models whose ground state breaks a continuous symmetry in the thermodynamic limit. We first study the J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ Heisenberg model (with antiferromagnetic nearest-neighbor J1>0{J}_{1}>0J1​>0 and ferromagnetic second-neighbor couplings J2\ensuremath≤0){J}_{2}\ensuremath{\le}0)J2​\ensuremath≤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 lq\ensuremath≃1{l}_{q}\ensuremath{\simeq}1lq​\ensuremath≃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 J1\ensuremath-J2{J}_{1}\text{\ensuremath{-}}{J}_{2}J1​\ensuremath-J2​ 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 105\ifmmode×\else\texttimes\fi105{10}^{5}\ifmmode\times\else\texttimes\fi{}{10}^{5}105\ifmmode×\else\texttimes\fi105 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).
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.
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.
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 this paper, we derive corrections to the subleading logarithmic term of the entanglement entropy in systems with spontaneous broken continuous symmetry. Using quantum Monte Carlo simulations, we show that the improved scaling formula leads to much better estimations of the number of Goldstone modes in the two-dimensional square lattice spin-1/2 Heisenberg model and bilayer spin-1/2 Heisenberg model in systems of rather small sizes, compared with previous results. In addition, the universal geometry-dependent finite constant in the entanglement entropy scaling is also obtained in good agreement with the theoretical value.
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.
Using quantum Monte Carlo simulations, we compute the participation (Shannon-Rényi) entropies for groundstate wave functions of Heisenberg antiferromagnets for one-dimensional (line) subsystems of length L L embedded in two-dimensional ( L\times L L × L ) square lattices. We also study the line entropy at finite temperature, i.e. of the diagonal elements of the density matrix, for three-dimensional ( L\times L\times L L × L × L ) cubic lattices. The breaking of SU(2) symmetry is clearly captured by a universal logarithmic scaling term l_q\ln L l q ln L in the Rényi entropies, in good agreement with the recent field-theory results of Misguish, Pasquier and Oshikawa . We also study the dependence of the log prefactor l_q l q on the Rényi index q q for which a transition is detected at q_c\simeq 1 q c ≃ 1 .