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:0912.4733 · Journal of Physics A Mathematical and Theoretical

Exact results for the bipartite entanglement entropy of the AKLT spin-1 chain

Scott D Geraedts, Erik Schwartz Sørensen

Published Apr 16, 2010• 10 citations
arXivHTMLPDF

Abstract

We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.

DOI
10.1088/1751-8113/43/18/185304

Related papers

Entanglement in a Valence-Bond Solid State

Heng Fan, V. E. Korepin, Vwani Roychowdhury

We study entanglement in a valence-bond solid state, which describes the ground state of an Affleck-Kennedy-Lieb-Tasaki quantum spin chain, consisting of bulk spin-1's and two spin-1/2's at the ends. We characterize entanglement between various subsystems of the ground state by mostly calculating the entropy of one of the subsystems; when appropriate, we evaluate concurrences as well. We show that the reduced density matrix of a continuous block of bulk spins is independent of the size of the chain and the location of the block relative to the ends. Moreover, we show that the entanglement of the block with the rest of the sites approaches a constant value exponentially fast, as the size of the block increases. We also calculate the entanglement of (i) any two bulk spins with the rest, and (ii) the end spin-1/2's (together and separately) with the rest of the ground state.

arXiv:quant-ph/0406067•Physical Review Letters•Nov 22, 2004•134 citations

Exact valence bond entanglement entropy and probability distribution in the XXX spin chain and the Potts model

Jesper Lykke Jacobsen, Hubert Saleur

By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We find in particular that for the XXX spin chain, the number N_c of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.

•Nov 21, 2007•2 citations

Entanglement in low dimensional systems

Raul A. Santos

In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages

•Jul 23, 2013•0 citations

Emergent infinite-randomness fixed points from the extensive random bipartitions of the spin-1 Affleck-Kennedy-Lieb-Tasaki topological state

Min Lu, Wen-Jia Rao, Rajesh Narayanan +2 more

Quantum entanglement under an extensive bipartition can reveal the critical boundary theory of a topological phase in a parameter space. In this study we demonstrate that the infinite-randomness fixed point for spin-$\frac{1}{2}$ degrees of freedom can emerge from an extensive random bipartition of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. The nested entanglement entropy of the ground state of the reduced density matrix exhibits a logarithmic scaling with an effective central charge $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}=0.72\ifmmode\pm\else\textpm\fi{}0.02\ensuremath{\approx}ln2$. We further discuss, in the language of bulk quantum entanglement, how to understand all phase boundaries and the surrounding Griffiths phases for the antiferromagnetic Heisenberg spin-1 chain with quenched disorder and dimerization.

arXiv:1502.02095•Physical review. B./Physical review. B•Dec 23, 2016•1 citation

Bipartite entanglement and entropic boundary law in lattice spin systems

Alioscia Hamma, Radu Ionicioiu, Paolo Zanardi A

We investigate bipartite entanglement in spin-$1∕2$ systems on a generic lattice. For states that are an equal superposition of elements of a group $G$ of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems $A$ and $B$. These states are stabilized by the group $G$. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus $\mathfrak{g}$. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem $A$ and is equal to the perimeter (up to an additive constant) when $A$ is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.

arXiv:quant-ph/0409073•Physical Review A•Feb 22, 2005•296 citations

Exact Valence Bond Entanglement Entropy and Probability Distribution in the XXX Spin Chain and the Potts Model

Jesper Lykke Jacobsen, Hubert Saleur

We determine exactly the probability distribution of the number ${N}_{c}$ of valence bonds connecting a subsystem of length $L\ensuremath{\gg}1$ to the rest of the system in the ground state of the $XXX$ antiferromagnetic spin chain. This provides, in particular, the asymptotic behavior of the valence-bond entanglement entropy ${S}_{\mathrm{VB}}=⟨{N}_{c}⟩\mathrm{ln}2=\frac{4\mathrm{ln}2}{{\ensuremath{\pi}}^{2}}\mathrm{ln}L$ disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to $\frac{1}{3}\mathrm{ln}L$. Our results generalize to the $Q$-state Potts model.

arXiv:0711.3391•Physical Review Letters•Feb 29, 2008•39 citations
We study the bipartite entanglement between a subsystem of size l and the rest of the system of total size L as it occurs in a spin-1 Affleck–Kennedy–Lieb–Tasaki (AKLT) chain subject to open boundary conditions. In this case, the ground-state manifold is four-fold degenerate and there is strong dependence on the parity of the number of spins, L . We present exact analytical results for the von Neumann entanglement entropy, as a function of both the size of the subsystem, l , and the total system size, L , for all four degenerate ground states for both odd and even L . In the large l , L limits the entanglement entropy approaches ln (2) for the S z T = ±1 while it approaches twice that value, 2ln (2), for the S z T = 0 states. In all cases, it is found that this constant is approached exponentially fast defining a length scale ξ = 1/ln (3) equal to the known bulk correlation length.
We study entanglement in a valence-bond solid state, which describes the ground state of an Affleck-Kennedy-Lieb-Tasaki quantum spin chain, consisting of bulk spin-1's and two spin-1/2's at the ends. We characterize entanglement between various subsystems of the ground state by mostly calculating the entropy of one of the subsystems; when appropriate, we evaluate concurrences as well. We show that the reduced density matrix of a continuous block of bulk spins is independent of the size of the chain and the location of the block relative to the ends. Moreover, we show that the entanglement of the block with the rest of the sites approaches a constant value exponentially fast, as the size of the block increases. We also calculate the entanglement of (i) any two bulk spins with the rest, and (ii) the end spin-1/2's (together and separately) with the rest of the ground state.
By relating the ground state of Temperley-Lieb hamiltonians to partition functions of 2D statistical mechanics systems on a half plane, and using a boundary Coulomb gas formalism, we obtain in closed form the valence bond entanglement entropy as well as the valence bond probability distribution in these ground states. We find in particular that for the XXX spin chain, the number N_c of valence bonds connecting a subsystem of size L to the outside goes, in the thermodynamic limit, as = (4/pi^2) ln L, disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to 1/(3 ln 2) ln L. Our results generalize to the Q-state Potts model.
In the ground state of gapped systems, the entanglement entropy of a subsystem A scales with the length of the boundary of A. This observation suggests that the entanglement properties of the sub- system can be described in terms of degrees of freedom living in the boundary of A. We will discuss the the connection between en- tanglement properties and effective boundary descriptions in spin systems in one and two dimensions. In one dimension we present analytic results for the spin S = 1, Affleck-Kennedy-Lieb-Tasaki (AKLT) ground state entanglement, characterized by negativity and entanglement spectrum. We also discuss a generalization of the AKLT model, based on the quantum group Uq (sl(2)) for gen- eral integer spin S. In two dimensions, we study two spin systems whose ground state can be written in terms of tensor product states of bond dimension two, the AKLT model in the hexagonal lattice and the Ising projected entangled pair state (Ising PEPS) in the square lattice. We show how the reduced density matrix of a parti- tion is associated with a thermal state of a one dimensional model along the boundary of that partition. We also present arguments supporting this correspondence for arbitrary gapped systems. Fi- nally we discuss the behavior of this boundary theory when the original two dimensional model is tuned through a quantum phase transition,126 pages
Quantum entanglement under an extensive bipartition can reveal the critical boundary theory of a topological phase in a parameter space. In this study we demonstrate that the infinite-randomness fixed point for spin-12\frac{1}{2}21​ degrees of freedom can emerge from an extensive random bipartition of the spin-1 Affleck-Kennedy-Lieb-Tasaki chain. The nested entanglement entropy of the ground state of the reduced density matrix exhibits a logarithmic scaling with an effective central charge $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{c}=0.72\ifmmode\pm\else\textpm\fi{}0.02\ensuremath{\approx}ln2$. We further discuss, in the language of bulk quantum entanglement, how to understand all phase boundaries and the surrounding Griffiths phases for the antiferromagnetic Heisenberg spin-1 chain with quenched disorder and dimerization.
We investigate bipartite entanglement in spin-1∕21∕21∕2 systems on a generic lattice. For states that are an equal superposition of elements of a group GGG of spin flips acting on the fully polarized state ${\ensuremath{\mid}0⟩}^{\ensuremath{\bigotimes}n}$, we find that the von Neumann entropy depends only on the boundary between the two subsystems AAA and BBB. These states are stabilized by the group GGG. A physical realization of such states is given by the ground state manifold of the Kitaev's model on a Riemann surface of genus g\mathfrak{g}g. For a square lattice, we find that the entropy of entanglement is bounded from above and below by functions linear in the perimeter of the subsystem AAA and is equal to the perimeter (up to an additive constant) when AAA is convex. The entropy of entanglement is shown to be related to the topological order of this model. Finally, we find that some of the ground states are absolutely entangled, i.e., no partition has zero entanglement. We also provide several examples for the square lattice.
We determine exactly the probability distribution of the number Nc{N}_{c}Nc​ of valence bonds connecting a subsystem of length $L\ensuremath{\gg}1$ to the rest of the system in the ground state of the XXXXXXXXX antiferromagnetic spin chain. This provides, in particular, the asymptotic behavior of the valence-bond entanglement entropy ${S}_{\mathrm{VB}}=⟨{N}_{c}⟩\mathrm{ln}2=\frac{4\mathrm{ln}2}{{\ensuremath{\pi}}^{2}}\mathrm{ln}L$ disproving a recent conjecture that this should be related with the von Neumann entropy, and thus equal to 13lnL\frac{1}{3}\mathrm{ln}L31​lnL. Our results generalize to the QQQ-state Potts model.