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:1906.03758 · Physical review. B./Physical review. B

Low-energy dynamics of the Affleck-Kennedy-Lieb-Tasaki model in the one- and two-triplon basis

Jintae Kim, Rajarshi Pal, Jung Hoon Han

Published Oct 3, 2019• 1 citation
arXivHTMLPDF

Abstract

The elementary excitation in the antiferromagnetic spin-1 model known as the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian has been described alternatively as magnons or kinklike solitons (triplons). The latter, which we call the triplon throughout this paper, has been proven equivalent descriptions of the same magnon excitation and not an independent branch of excited states. On the other hand, no careful examination of multimagnon and multitriplon equivalence was made in the past. In this paper we prove that two-magnon and two-triplon states are also identical descriptions of the same excited states and furthermore that their energies break down as the sum of one-triplon energies exactly for the AKLT Hamiltonian. The statement holds despite the fact that the model is nonintegrable. Such magnon/triplon dichotomy is conjectured to hold for arbitrary $n$-magnon and $n$-triplon states. The one- and two-triplon states form orthogonal sets that can be used to span the low-energy Hilbert space. We construct an effective version of the AKLT Hamiltonian within such a subspace and work out the correction to the one-triplon energy gap that finds excellent agreement with the known exact value.

DOI
10.1103/physrevb.100.155104

Related papers

Dressed solitonic excitations for the AKLT Hamiltonian

Rainer Scharf, H -J Mikeska

In contrast to the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) isotropic spin-1 Hamiltonian its elementary excitations are not known exactly in general. Based on a simple variational ansatz we construct approximate excitations in the form of dressed hidden domain walls. This allows us to give an improved upper bound for the gap of the AKLT Hamiltonian which is only 1% larger than the best currently available numerical result. We compare the oneand two-soliton dispersion curves with low-lying excitations for a chain of 16 sites as given by Fath and Solyom and find excellent agreement. Spin and string correlations are calculated and found to agree in leading order with the corresponding correlations for the bare kinks. Finally the weight of the string-ordered part of the lowest excited state is calculated and compared with the exact numerical result.

Journal of Physics Condensed Matter•Jun 26, 1995•7 citations

Exact excited states of nonintegrable models

Sanjay Moudgalya, Stephan Rachel, Bogdan Andrei Bernevig +1 more

We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model, a celebrated nonintegrable model. The states thus obtained for the AKLT model can be interpreted as from one to an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong eigenstate thermalization hypothesis. We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another nonintegrable model, and extend our construction to the generalized integer spin AKLT models.

arXiv:1708.05021•Physical review. B./Physical review. B•Dec 27, 2018•297 citations

The Affleck–Kennedy–Lieb–Tasaki (AKLT) model

Ian K. Affleck

The Haldane conjecture (1983), which led to a Nobel prize for Duncan Haldane, was that half-integer spin chains are gapless while integer spin chains are gapped. Haldane made this observation by a mapping onto a non-linear $\sigma$-model at large spin. Gapless behavior for half-integer spin was proven rigorously by the Affleck–Lieb–Schultz–Mattis (ALSM) theorem. Gapped behavior for integer spin was demonstrated by an exactly solvable model known as Affleck–Kennedy–Lieb–Tasaki (AKLT) model (1987, 1988). In this review article, I will review the ALSM theorem and present results on the AKLT model. An important aspect of this model is that it has decoupled spin-$1/2$’s at the two ends with open boundary conditions. This was shown, by Kennedy (1990), to extend to the regular antiferromagnetic model. The first experimental demonstration of this behavior was in a paper by Hagiwara, Katsumata, Affleck, Halperin, and Renard (1990). I will review this experiment.

EMS Press eBooks•Jun 29, 2022•0 citations

Characterizing S = 3 2 Affleck-Kennedy-Lieb-Tasaki Hamiltonian with scanning tunneling spectroscopy

Mar Ferri-Cortés, J. C. G. Henriques, Joaquín Fernández-Rossier

The Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian is a particular instance of a general class of model Hamiltonians defined in lattices with coordination $z$ where each site hosts a spins $S=z/2$, interacting both with linear and nonlinear exchange couplings. In two dimensions, the AKLT model features a gap in the spectrum, and its ground state is a valence bond solid state; that is a universal resource for measurement based quantum computing, motivating the quest of physical systems that realize this Hamiltonian. Given a finite-size system described with a specific instance of this general class of models, we address the question of how to assess if such system is a realization of the AKLT model using inelastic tunnel spectroscopy implemented with scanning tunnel microscopy (IETS-STM). We propose two approaches. First, in the case of a dimer, we show how to leverage nonequilibrium IETS-STM to obtain the energies of all excited states, and determine thereby the magnitude of both linear and nonlinear exchange interactions. Second, we explore how IETS can probe the in-gap excitations associated with edge spins in a hexamer. In the AKLT limit, spins $S=3/2$ at the edge of the lattice have coordination 2, giving rise to $S=1/2$ dangling spins that can be probed with IETS. We propose a $S=1/2$ effective Hamiltonian to describe the interactions between these dangling spins in the neighborhood of the AKLT point, where their degeneracy lifted.

Physical review. B./Physical review. B•Jul 30, 2025•0 citations

Spin-1 chain with spin- 12 excitations in the bulk

Diptiman Sen, Naveen Surendran

We present a spin-1 chain with a Hamiltonian which has three exactly solvable ground states. Two of these are fully dimerized, analogous to the Majumdar-Ghosh (MG) states of a spin-$\frac{1}{2}$ chain, while the third is of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type. We use variational and numerical methods to study the low-energy excitations which interpolate between these ground states in different ways. In particular, there is a spin-$\frac{1}{2}$ excitation which interpolates between the MG and AKLT ground states; this is the lowest excitation of the system and it has a surprisingly small gap. We discuss generalizations of our model of spin fractionalization to higher spin chains and higher dimensions.

arXiv:cond-mat/0611697•Physical Review B•Mar 16, 2007•6 citations

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

Zenan Liu, Jun Li, Rui-Zhen Huang +3 more

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.

arXiv:2109.10901•Physical review. B./Physical review. B•Jan 18, 2022•15 citations
The elementary excitation in the antiferromagnetic spin-1 model known as the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian has been described alternatively as magnons or kinklike solitons (triplons). The latter, which we call the triplon throughout this paper, has been proven equivalent descriptions of the same magnon excitation and not an independent branch of excited states. On the other hand, no careful examination of multimagnon and multitriplon equivalence was made in the past. In this paper we prove that two-magnon and two-triplon states are also identical descriptions of the same excited states and furthermore that their energies break down as the sum of one-triplon energies exactly for the AKLT Hamiltonian. The statement holds despite the fact that the model is nonintegrable. Such magnon/triplon dichotomy is conjectured to hold for arbitrary nnn-magnon and nnn-triplon states. The one- and two-triplon states form orthogonal sets that can be used to span the low-energy Hilbert space. We construct an effective version of the AKLT Hamiltonian within such a subspace and work out the correction to the one-triplon energy gap that finds excellent agreement with the known exact value.
In contrast to the ground state of the Affleck-Kennedy-Lieb-Tasaki (AKLT) isotropic spin-1 Hamiltonian its elementary excitations are not known exactly in general. Based on a simple variational ansatz we construct approximate excitations in the form of dressed hidden domain walls. This allows us to give an improved upper bound for the gap of the AKLT Hamiltonian which is only 1% larger than the best currently available numerical result. We compare the oneand two-soliton dispersion curves with low-lying excitations for a chain of 16 sites as given by Fath and Solyom and find excellent agreement. Spin and string correlations are calculated and found to agree in leading order with the corresponding correlations for the bare kinks. Finally the weight of the string-ordered part of the lowest excited state is calculated and compared with the exact numerical result.
We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model, a celebrated nonintegrable model. The states thus obtained for the AKLT model can be interpreted as from one to an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong eigenstate thermalization hypothesis. We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another nonintegrable model, and extend our construction to the generalized integer spin AKLT models.
The Haldane conjecture (1983), which led to a Nobel prize for Duncan Haldane, was that half-integer spin chains are gapless while integer spin chains are gapped. Haldane made this observation by a mapping onto a non-linear σ\sigmaσ-model at large spin. Gapless behavior for half-integer spin was proven rigorously by the Affleck–Lieb–Schultz–Mattis (ALSM) theorem. Gapped behavior for integer spin was demonstrated by an exactly solvable model known as Affleck–Kennedy–Lieb–Tasaki (AKLT) model (1987, 1988). In this review article, I will review the ALSM theorem and present results on the AKLT model. An important aspect of this model is that it has decoupled spin-1/21/21/2’s at the two ends with open boundary conditions. This was shown, by Kennedy (1990), to extend to the regular antiferromagnetic model. The first experimental demonstration of this behavior was in a paper by Hagiwara, Katsumata, Affleck, Halperin, and Renard (1990). I will review this experiment.
The Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian is a particular instance of a general class of model Hamiltonians defined in lattices with coordination zzz where each site hosts a spins S=z/2S=z/2S=z/2, interacting both with linear and nonlinear exchange couplings. In two dimensions, the AKLT model features a gap in the spectrum, and its ground state is a valence bond solid state; that is a universal resource for measurement based quantum computing, motivating the quest of physical systems that realize this Hamiltonian. Given a finite-size system described with a specific instance of this general class of models, we address the question of how to assess if such system is a realization of the AKLT model using inelastic tunnel spectroscopy implemented with scanning tunnel microscopy (IETS-STM). We propose two approaches. First, in the case of a dimer, we show how to leverage nonequilibrium IETS-STM to obtain the energies of all excited states, and determine thereby the magnitude of both linear and nonlinear exchange interactions. Second, we explore how IETS can probe the in-gap excitations associated with edge spins in a hexamer. In the AKLT limit, spins S=3/2S=3/2S=3/2 at the edge of the lattice have coordination 2, giving rise to S=1/2S=1/2S=1/2 dangling spins that can be probed with IETS. We propose a S=1/2S=1/2S=1/2 effective Hamiltonian to describe the interactions between these dangling spins in the neighborhood of the AKLT point, where their degeneracy lifted.
We present a spin-1 chain with a Hamiltonian which has three exactly solvable ground states. Two of these are fully dimerized, analogous to the Majumdar-Ghosh (MG) states of a spin-12\frac{1}{2}21​ chain, while the third is of the Affleck-Kennedy-Lieb-Tasaki (AKLT) type. We use variational and numerical methods to study the low-energy excitations which interpolate between these ground states in different ways. In particular, there is a spin-12\frac{1}{2}21​ excitation which interpolates between the MG and AKLT ground states; this is the lowest excitation of the system and it has a surprisingly small gap. We discuss generalizations of our model of spin fractionalization to higher spin chains and higher dimensions.
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.