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

Ian K. Affleck

Published Jun 29, 2022• 0 citations
Publisher

Abstract

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.

DOI
10.4171/90-1/1

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Existence of a Spectral Gap in the Affleck-Kennedy-Lieb-Tasaki Model on the Hexagonal Lattice

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The S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the S=3/2 AKLT model on the hexagonal lattice is also in a gapped phase has remained open, despite being a fundamental problem of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound Δ>0.006 on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two steps combining mathematical physics and high-precision computational physics. We first prove a mathematical finite-size criterion which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value. Then we verify the finite-size criterion numerically by performing state-of-the-art DMRG calculations on the subsystem.

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One-dimensional antiferromagnets have exotic disordered ground states. As was first argued by Haldane (1983), there is an excitation gap for integer, but not half-integer, spin. The authors review the arguments for this behaviour based on field-theory mappings, the Lieb-Schultz-Mattis theorem, exactly solvable models, finite-chain diagonalisation and real experiments.

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

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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.

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

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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.

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The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.

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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.

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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 S=1 Affleck-Kennedy-Lieb-Tasaki (AKLT) quantum spin chain was the first rigorous example of an isotropic spin system in the Haldane phase. The conjecture that the S=3/2 AKLT model on the hexagonal lattice is also in a gapped phase has remained open, despite being a fundamental problem of ongoing relevance to condensed-matter physics and quantum information theory. Here we confirm this conjecture by demonstrating the size-independent lower bound Δ>0.006 on the spectral gap of the hexagonal model with periodic boundary conditions in the thermodynamic limit. Our approach consists of two steps combining mathematical physics and high-precision computational physics. We first prove a mathematical finite-size criterion which gives an analytical, size-independent bound on the spectral gap if the gap of a particular cut-out subsystem of 36 spins exceeds a certain threshold value. Then we verify the finite-size criterion numerically by performing state-of-the-art DMRG calculations on the subsystem.
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.
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.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.
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.