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 .
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 .
Universal features in the scalings of Shannon-R\'enyi entropies of many-body ground states are studied for interacting spin-21 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-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 basis by a confinement mechanism of ferromagnetic domain walls.
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).
Shannon-Rényi entropies and associated participation spectra quantify how much a many-body wave-function is localized in a given configuration basis. Using these tools, we present an analysis of the ground-state wave functions of various quantum spin systems in one and two dimensions. General ideas and a review of the current status of this field are first given, with a particular emphasis on universal subleading terms characterizing different quantum phases of matter, and associated transitions. We highlight the connection with the related entanglement entropies and spectra when this is possible. In a second part, new results are presented for the participation spectra of interacting spin models, mostly based on quantum Monte Carlo simulations, but also using perturbation theory in some cases. For full antiferromagnetic systems, participation spectra are analyzed in terms of ferromagnetic domain walls which experience a repulsive pairwise interaction. This confinement potential is either linear for long-range Néel order, or logarithmic for quasi-long-range order. The case of subsystems is also analyzed in great detail for a 2d dimerized Heisenberg model undergoing a quantum phase transition between a gapped paramagnet and a Néel phase. Participation spectra of line shaped (1d) sub-systems are quantitatively compared with finite temperature participation spectra of ansatz effective boundary (1d) entanglement Hamiltonians. While short-range models describe almost perfectly the gapped side, the Néel regime is best compared using long-range effective Hamiltonians. Spectral comparisons performed using Kullback-Leibler divergences, a tool potentially useful for entanglement spectra, provide a quantitative way to identify both the best boundary entanglement Hamiltonian and temperature.
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 Sn>1≃O(N)+4NNGn−1nlogN and S1≃O(N)−4NNGlogN, 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 spin21 XY model on the square lattice in a cylinder geometry. These numerical data confirm our findings for logarithmic terms in the n=∞ R\'enyi entropy (also called −logpmax). 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 non-compact free-boson partition function in dimension 1+1.
Computing the subleading logarithmic term in the entanglement entropy (EE) of (2+1)d quantum many-body systems remains a significant challenge, despite its central role in revealing universal information about quantum states and quantum critical points (QCPs). Building on recent algorithmic advances that enable the stable calculation of EE as an exponential observable~\cite{zhouIncremental2024,zhangIntegral2024,liaoExtracting2024}, we develop a {\it bubble basis} projector quantum Monte Carlo (QMC) algorithm to precisely and efficiently compute the universal corner of EE at QCPs in a (2+1)d square-lattice transverse-field Ising model augmented with a four-body interaction. Turning on this interaction allows us to trace an Ising critical line, reaching the tricritical point, and then a line of first-order phase transition. In (2+1)d, the tricritical point is described by the Gaussian theory, where a theoretical calculation of the corner logarithmic term in the 2nd Rényi entropy term is available~\cite{UniversalCasini2007}. Our QMC results are in quantitative agreement with this theoretical value, providing a highly nontrivial benchmark of the algorithm. Furthermore, we also study the Rényi EE at the Ising critical line and on the first-order transition line, obtaining results consistent with theoretical expectations. These findings establish the long-sought connection between the universal values of an exactly solvable limit and those of a strongly correlated regime at (2+1)d.