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

Physical Review Letters

Topological Estimator of Block Entanglement for Heisenberg Antiferromagnets

Ravindra W. Chhajlany, Piotr Tomczak, Antoni Wójcik

Published Oct 18, 2007• 32 citations
Publisher

Abstract

We introduce a computable estimator of block entanglement entropy for many-body spin systems admitting total singlet ground states. Building on a simple geometrical interpretation of entanglement entropy of the so-called valence bond states, this estimator is defined as the average number of common singlets to two subsystems of spins. We show that our estimator possesses the characteristic scaling properties of the block entanglement entropy in critical and noncritical one-dimensional Heisenberg systems. We invoke this new measure to examine entanglement scaling in the two-dimensional Heisenberg model on a square lattice revealing an "area law" for the gapped phase and a logarithmic correction to this law in the gapless phase.

DOI
10.1103/physrevlett.99.167204

Related papers

Valence Bond Entanglement Entropy

Fabien Alet, Sylvain Capponi, Nicolas Laflorencie +1 more

We introduce for SU(2) quantum spin systems the valence bond entanglement entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a valence bond solid state and multiplicative logarithmic corrections for the Néel phase.

arXiv:cond-mat/0703027•Physical Review Letters•Sep 12, 2007•65 citations

Definitions of entanglement entropy of spin systems in the valence-bond basis

Yu‐Cheng Lin, Anders W. Sandvik

The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific Hamiltonians or using importance sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agree with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one-dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also N\'eel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions while still violating it in one dimension---both behaviors in accord with expectations for proper measures of entanglement entropy.

arXiv:1005.0821•Physical Review B•Dec 14, 2010•9 citations

Entanglement entropy for the valence bond solid phases of two-dimensional dimerized Heisenberg antiferromagnets

Leonardo S. G. Leite, R. L. Doretto

We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to ${L}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the N\'eel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.

arXiv:1902.04355•Physical review. B./Physical review. B•Jul 10, 2019•4 citations

Measuring Entanglement Entropy in Valence Bond Quantum Monte Carlo Simulations

Ann B. Kallin

In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.

•Dec 8, 2010•2 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

Entanglement entropy scaling in the bilayer Heisenberg spin system

Johannes Helmes, Stefan Weßel

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.

arXiv:1403.7395•Physical Review B•Jun 13, 2014•72 citations
We introduce a computable estimator of block entanglement entropy for many-body spin systems admitting total singlet ground states. Building on a simple geometrical interpretation of entanglement entropy of the so-called valence bond states, this estimator is defined as the average number of common singlets to two subsystems of spins. We show that our estimator possesses the characteristic scaling properties of the block entanglement entropy in critical and noncritical one-dimensional Heisenberg systems. We invoke this new measure to examine entanglement scaling in the two-dimensional Heisenberg model on a square lattice revealing an "area law" for the gapped phase and a logarithmic correction to this law in the gapless phase.
We introduce for SU(2) quantum spin systems the valence bond entanglement entropy as a counting of valence bond spin singlets shared by two subsystems. For a large class of antiferromagnetic systems, it can be calculated in all dimensions with quantum Monte Carlo simulations in the valence bond basis. We show numerically that this quantity displays all features of the von Neumann entanglement entropy for several one-dimensional systems. For two-dimensional Heisenberg models, we find a strict area law for a valence bond solid state and multiplicative logarithmic corrections for the Néel phase.
The valence-bond structure of spin-1/2 Heisenberg antiferromagnets is closely related to quantum entanglement. We investigate measures of entanglement entropy based on transition graphs, which characterize state overlaps in the overcomplete valence-bond basis. The transition graphs can be generated using projector Monte Carlo simulations of ground states of specific Hamiltonians or using importance sampling of valence-bond configurations of amplitude-product states. We consider definitions of entanglement entropy based on the bonds or loops shared by two subsystems (bipartite entanglement). Results for the bond-based definition agree with a previously studied definition using valence-bond wave functions (instead of the transition graphs, which involve two states). For the one-dimensional Heisenberg chain, with uniform or random coupling constants, the prefactor of the logarithmic divergence with the size of the smaller subsystem agrees with exact results. For the ground state of the two-dimensional Heisenberg model (and also N\'eel-ordered amplitude-product states), there is a similar multiplicative violation of the area law. In contrast, the loop-based entropy obeys the area law in two dimensions while still violating it in one dimension---both behaviors in accord with expectations for proper measures of entanglement entropy.
We calculate the bipartite von Neumann and second R\'enyi entanglement entropies of the ground states of spin-1/2 dimerized Heisenberg antiferromagnets on a square lattice. Two distinct dimerization patterns are considered: columnar and staggered. In both cases, we concentrate on the valence bond solid (VBS) phase and describe such a phase with the bond-operator representation. Within this formalism, the original spin Hamiltonian is mapped into an effective interacting boson model for the triplet excitations. We study the effective Hamiltonian at the harmonic approximation and determine the spectrum of the elementary triplet excitations. We then follow an analytical procedure, which is based on a modified spin-wave theory for finite systems and was originally employed to calculate the entanglement entropies of magnetic ordered phases, and calculate the entanglement entropies of the VBS ground states. In particular, we consider one-dimensional (line) subsystems within the square lattice, a choice that allows us to consider line subsystems with sizes up to ${L}^{\ensuremath{'}}\phantom{\rule{0.16em}{0ex}}=\phantom{\rule{0.16em}{0ex}}1000$. We combine such a procedure with the results of the bond-operator formalism at the harmonic level and show that, for both dimerized Heisenberg models, the entanglement entropies of the corresponding VBS ground states obey an area law as expected for gapped phases. For both columnar-dimer and staggered-dimer models, we also show that the entanglement entropies increase but they seem to not diverge as the dimerization decreases and the system approaches the N\'eel-VBS quantum phase transition. Finally, the entanglement spectra associated with the VBS ground states are presented.
In this thesis we examine methods for measuring entanglement entropy in spin-1/2 Heisenberg systems using quantum Monte Carlo in the valence bond basis. We begin by presenting the quantum Monte Carlo techniques used in this research. We then use these techniques to directly compare the recently proposed valence bond entanglement entropy to the standard definition of entanglement entropy: the von Neumann entanglement entropy. We find that the valence bond entanglement entropy does not give a bound on the von Neumann entanglement entropy, and that it exhibits a multiplicative logarithmic correction to the area law that is not present in the scaling of the von Neumann entanglement entropy. We then present a method to measure higher orders of the generalized Renyi entanglement entropies using valence bond quantum Monte Carlo, and show results for the second Renyi entropy. We find the results converge to the exact results for one dimensional Heisenberg spin-1/2 chains, and see that the scaling of the second Renyi entropy follows an area law in the two dimensional Heisenberg ground state.
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.
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)0.016(1)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.