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arXiv:1607.02465 · Physical review. B./Physical review. B

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

Grégoire Misguich, Vincent Pasquier, Masaki Oshikawa

Published May 30, 2017• 18 citations
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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 ${S}_{n>1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N+\frac{{N}_{\text{NG}}}{4}\frac{n}{n\ensuremath{-}1}lnN$ 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.

DOI
10.1103/physrevb.95.195161

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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 Sn>1\ensuremath≃const.\ifmmode×\else\texttimes\fiN+NNG4nn\ensuremath−1lnN{S}_{n>1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N+\frac{{N}_{\text{NG}}}{4}\frac{n}{n\ensuremath{-}1}lnNSn>1​\ensuremath≃const.\ifmmode×\else\texttimes\fiN+4NNG​​n\ensuremath−1n​lnN and S1\ensuremath≃const.\ifmmode×\else\texttimes\fiN\ensuremath−NNG4lnN{S}_{1}\ensuremath{\simeq}\mathrm{const}.\ifmmode\times\else\texttimes\fi{}N\ensuremath{-}\frac{{N}_{\text{NG}}}{4}lnNS1​\ensuremath≃const.\ifmmode×\else\texttimes\fiN\ensuremath−4NNG​​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∞n=\ensuremath{\infty}n=\ensuremath∞ R'enyi entropy (also called \ensuremath−lnpmax)\ensuremath{-}ln{p}_{\mathrm{max}})\ensuremath−lnpmax​). 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.
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 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).
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.
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.
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 .
A generic scheme is proposed to investigate the entanglement entropy for a type of scale-invariant states, valid for orthonormal basis states in the ground state subspace of quantum many-body systems undergoing spontaneous symmetry breaking with type-B Goldstone modes in two spatial dimensions and beyond. It is argued that a contribution from the area law to the entanglement entropy is absent, since the closeness to the boundary between a subsystem and its environment is not well-defined, given that a permutation symmetry group with respect to the unit cells of degenerate ground state wave functions emerges. Three physical constraints imposed lead to a universal finite-system size scaling function in the dominant logarithmic contribution to the entanglement entropy. As a result, an abstract fractal underlying the ground state subspace is revealed, characterized by the fractal dimension. The latter in turn is identical to the number of type-B Goldstone modes for the orthonormal basis states. The prediction is numerically confirmed for the SU(2){\rm SU}(2)SU(2) spin-sss ferromagnetic Heisenberg model, the SU(2s+1){\rm SU}(2s+1)SU(2s+1) ferromagnetic model, and the staggered SU(3){\rm SU}(3)SU(3) spin-1 ferromagnetic biquadratic model.