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:2109.10901 · Physical review. B./Physical review. B

Bulk and edge dynamics of a two-dimensional Affleck-Kennedy-Lieb-Tasaki model

Zenan Liu, Jun Li, Rui-Zhen Huang, Jun Li, Zheng Yan, Dao‐Xin Yao

Published Jan 18, 2022• 15 citations
arXivHTMLPDF

Abstract

We study the dynamical properties of both bulk and edge spins of a two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model mainly by using the stochastic series expansion quantum Monte Carlo method with stochastic analytic continuation. In the deep AKLT phase, we obtain a spin spectrum with the flat band, which is a strong evidence for a localized state. Through the spectrum analysis, we see a clear continuous phase transition from the AKLT phase to the N\'eel phase in the model, and the energy gap becomes closed at the corresponding momentum point. In comparison with linear spin-wave theory, we find that there are strong interactions among magnons at high energies. With an open boundary condition, the gap of edge spins in the AKLT phase closes at both the $\mathrm{\ensuremath{\Gamma}}$ point and the $\ensuremath{\pi}$ point interestingly to emerge into a flat-band-like Luttinger liquid phase, which can be explained by symmetry and perturbation approximation. This paper helps us to better understand the completely different dynamical behaviors of bulk and edge spins in the symmetry protected topological phase.

DOI
10.1103/physrevb.105.014418

Related papers

Entanglement properties of the two-dimensional SU(3) Affleck-Kennedy-Lieb-Tasaki state

Olivier Gauthé, Didier Poilblanc

Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of $\text{SU}(3)$, relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating $\text{SU}(3)$ chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its $\text{SU}(2)$ analog. We discuss the SPT feature of the state.

arXiv:1708.00848•Physical review. B./Physical review. B•Sep 25, 2017•7 citations

Continuous Néel-VBS quantum phase transition in non-local one-dimensional systems with SO(3) symmetry

Chao‐Ming Jian, Yichen Xu, Xiao-Chuan Wu +1 more

One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Recently novel 1d physics was found numerically in systems with either spatially nonlocal interactions, or at the 1d boundary of 2d quantum critical points, and the critical fluctuation in the bulk also yields effective nonlocal interactions at the boundary. This work studies the edge states at the 1d boundary of 2d strongly interacting symmetry protected topological (SPT) states, when the bulk is driven to a disorder-order phase transition. We will take the 2d Affleck-Kennedy-Lieb-Tasaki (AKLT) state as an example, which is a SPT state protected by the SO(3) spin symmetry and spatial translation. We found that the original (1+1)d boundary conformal field theory of the AKLT state is unstable due to coupling to the boundary avatar of the bulk quantum critical fluctuations. When the bulk is fixed at the quantum critical point, within the accuracy of our expansion method, we find that by tuning one parameter at the boundary, there is a generic direct transition between the long range antiferromagnetic Néel order and the valence bond solid (VBS) order. This transition is very similar to the Néel-VBS transition recently found in numerical simulation of a spin-1/2 chain with nonlocal spatial interactions. Connections between our analytical studies and recent numerical results concerning the edge states of the 2d AKLT-like state at a bulk quantum phase transition will also be discussed.

arXiv:2004.07852•SciPost Physics•Feb 12, 2021•44 citations

Unconventional Surface Critical Behavior Induced by a Quantum Phase Transition from the Two-Dimensional Affleck-Kennedy-Lieb-Tasaki Phase to a Néel-Ordered Phase

Long Zhang, Fa Wang

A symmetry-protected topological phase has nontrivial surface states in the presence of certain symmetries, which can either be gapless or be degenerate. In this work, we study the physical consequence of such gapless surface states at the bulk quantum phase transition (QPT) that spontaneously breaks these symmetries. The two-dimensional Affleck-Kennedy-Lieb-Tasaki phase on a square lattice and its QPTs to Néel ordered phases are realized with the spin-1/2 Heisenberg model on a decorated square lattice. With large-scale quantum Monte Carlo simulations, we show that even though the bulk QPTs are governed by the conventional Landau phase transition theory, the gapless surface states induce unconventional universality classes of the surface critical behavior.

arXiv:1611.06477•Physical Review Letters•Feb 21, 2017•78 citations

Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice from t2g Orbitals

Maciej Koch-Janusz, Daniel I. Khomskii, Eran Sela

The two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a honeycomb lattice has been shown to be a universal resource for quantum computation. In this valence bond solid, however, the spin interactions involve higher powers of the Heisenberg coupling (S[over →](I)·S[over →](j))(n), making these states seemingly unrealistic on bipartite lattices, where one expects a simple antiferromagnetic order. We show that those interactions can be generated by orbital physics in multiorbital Mott insulators. We focus on t(2g) electrons on the honeycomb lattice and propose a physical realization of the spin-3/2 AKLT state. We find a phase transition from the AKLT to the Néel state on increasing Hund's rule coupling, which is confirmed by density matrix renormalization group simulations. An experimental signature of the AKLT state consists of protected, free S=1/2 spins on lattice vacancies, which may be detected in the spin susceptibility.

arXiv:1501.03512•Physical Review Letters•Jun 18, 2015•22 citations

Spin quantum Hall effects in featureless nonfractionalized spin-1 magnets

Yuan-Ming Lu, D. H. Lee

The Affleck-Kennedy-Lieb-Tasaki state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets that host symmetry-protected gapless excitations on the boundary. In this work, we show how to realize this type of featureless spin-1 state on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk, but they support gapless edge states protected by spin rotational symmetry along a certain direction, and they exhibit the spin quantum Hall effect. Using a fermion representation of integer spins, we show a concrete example of such spin-1 topological paramagnets on a kagome lattice, and we suggest a microscopic spin-1 Hamiltonian that may realize it.

arXiv:1212.0863•Physical Review B•May 27, 2014•22 citations

Quantum antiferromagnetic Heisenberg half-odd-integer spin model as the entanglement Hamiltonian of the integer-spin Affleck-Kennedy-Lieb-Tasaki states

Wen-Jia Rao, Guang-Ming Zhang, Kun Yang

Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-$S$ Haldane gapped phase, we can create an array of fractionalized spin-$S$/2 edge states with the super unit cell $l$ in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even $l$ corresponds to the quantum antiferromagnetic Heisenberg spin-$S$/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.

arXiv:1512.06550•Physical review. B./Physical review. B•Mar 15, 2016•5 citations
We study the dynamical properties of both bulk and edge spins of a two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model mainly by using the stochastic series expansion quantum Monte Carlo method with stochastic analytic continuation. In the deep AKLT phase, we obtain a spin spectrum with the flat band, which is a strong evidence for a localized state. Through the spectrum analysis, we see a clear continuous phase transition from the AKLT phase to the N'eel phase in the model, and the energy gap becomes closed at the corresponding momentum point. In comparison with linear spin-wave theory, we find that there are strong interactions among magnons at high energies. With an open boundary condition, the gap of edge spins in the AKLT phase closes at both the \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ point and the \ensuremathπ\ensuremath{\pi}\ensuremathπ point interestingly to emerge into a flat-band-like Luttinger liquid phase, which can be explained by symmetry and perturbation approximation. This paper helps us to better understand the completely different dynamical behaviors of bulk and edge spins in the symmetry protected topological phase.
Two-dimensional (spin-2) Affleck-Kennedy-Lieb-Tasaki (AKLT) type valence bond solids on a square lattice are known to be symmetry-protected topological (SPT) gapped spin liquids [S. Takayoshi, P. Pujol, and A. Tanaka Phys. Rev. B 94, 235159 (2016)]. Using the projected entangled pair state framework, we extend the construction of the AKLT state to the case of SU(3)\text{SU}(3)SU(3), relevant for cold atom systems. The entanglement spectrum is shown to be described by an alternating SU(3)\text{SU}(3)SU(3) chain of ``quarks'' and ``antiquarks'', subject to exponentially decaying (with distance) Heisenberg interactions, in close similarity with its SU(2)\text{SU}(2)SU(2) analog. We discuss the SPT feature of the state.
One dimensional (1d) interacting systems with local Hamiltonians can be studied with various well-developed analytical methods. Recently novel 1d physics was found numerically in systems with either spatially nonlocal interactions, or at the 1d boundary of 2d quantum critical points, and the critical fluctuation in the bulk also yields effective nonlocal interactions at the boundary. This work studies the edge states at the 1d boundary of 2d strongly interacting symmetry protected topological (SPT) states, when the bulk is driven to a disorder-order phase transition. We will take the 2d Affleck-Kennedy-Lieb-Tasaki (AKLT) state as an example, which is a SPT state protected by the SO(3) spin symmetry and spatial translation. We found that the original (1+1)d boundary conformal field theory of the AKLT state is unstable due to coupling to the boundary avatar of the bulk quantum critical fluctuations. When the bulk is fixed at the quantum critical point, within the accuracy of our expansion method, we find that by tuning one parameter at the boundary, there is a generic direct transition between the long range antiferromagnetic Néel order and the valence bond solid (VBS) order. This transition is very similar to the Néel-VBS transition recently found in numerical simulation of a spin-1/2 chain with nonlocal spatial interactions. Connections between our analytical studies and recent numerical results concerning the edge states of the 2d AKLT-like state at a bulk quantum phase transition will also be discussed.
A symmetry-protected topological phase has nontrivial surface states in the presence of certain symmetries, which can either be gapless or be degenerate. In this work, we study the physical consequence of such gapless surface states at the bulk quantum phase transition (QPT) that spontaneously breaks these symmetries. The two-dimensional Affleck-Kennedy-Lieb-Tasaki phase on a square lattice and its QPTs to Néel ordered phases are realized with the spin-1/2 Heisenberg model on a decorated square lattice. With large-scale quantum Monte Carlo simulations, we show that even though the bulk QPTs are governed by the conventional Landau phase transition theory, the gapless surface states induce unconventional universality classes of the surface critical behavior.
The two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) model on a honeycomb lattice has been shown to be a universal resource for quantum computation. In this valence bond solid, however, the spin interactions involve higher powers of the Heisenberg coupling (S[over →](I)·S[over →](j))(n), making these states seemingly unrealistic on bipartite lattices, where one expects a simple antiferromagnetic order. We show that those interactions can be generated by orbital physics in multiorbital Mott insulators. We focus on t(2g) electrons on the honeycomb lattice and propose a physical realization of the spin-3/2 AKLT state. We find a phase transition from the AKLT to the Néel state on increasing Hund's rule coupling, which is confirmed by density matrix renormalization group simulations. An experimental signature of the AKLT state consists of protected, free S=1/2 spins on lattice vacancies, which may be detected in the spin susceptibility.
The Affleck-Kennedy-Lieb-Tasaki state (or Haldane phase) in a spin-1 chain represents a large class of gapped topological paramagnets that host symmetry-protected gapless excitations on the boundary. In this work, we show how to realize this type of featureless spin-1 state on a generic two-dimensional lattice. These states have a gapped spectrum in the bulk, but they support gapless edge states protected by spin rotational symmetry along a certain direction, and they exhibit the spin quantum Hall effect. Using a fermion representation of integer spins, we show a concrete example of such spin-1 topological paramagnets on a kagome lattice, and we suggest a microscopic spin-1 Hamiltonian that may realize it.
Applying a symmetric bulk bipartition to the one-dimensional Affleck-Kennedy-Lieb-Tasaki valence-bond solid (VBS) states for the integer spin-SSS Haldane gapped phase, we can create an array of fractionalized spin-SSS/2 edge states with the super unit cell lll in the reduced bulk system, and the topological properties encoded in the VBS wave functions can be revealed. The entanglement Hamiltonian (EH) with even lll corresponds to the quantum antiferromagnetic Heisenberg spin-SSS/2 model. For the even integer spins, the EH still describes the Haldane gapped phase. For the odd integer spins, however, the EH just corresponds to the quantum antiferromagnetic Heisenberg half-odd integer-spin model with spinon excitations, characterizing the critical point separating the topological Haldane phase from the trivial gapped phase. Our results thus demonstrate that the topological bulk property not only determines its fractionalized edge states but also the quantum criticality associated with the topological phase, where the elementary excitations are precisely those fractionalized edge degrees of freedom confined in the bulk of the topological phase.