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Physical review. B./Physical review. B

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

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

Published Jul 30, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1103/f573-rmbl

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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.
Characterizing S=3/2S=3/2S=3/2 AKLT Hamiltonian with Scanning Tunneling Spectroscopy
The 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 non-linear 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 an 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 asses 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 non-equilibrium IETS-STM to obtain the energies of all excited states, and determine thereby the magnitude of both linear and non-linear exchange interactions. Second, we explore how IETS can probe the in-gap excitations associated to edge spins. 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.
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.
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.
Affleck-Kennedy-Lieb-Tasaki (AKLT) states are an important class of many-body quantum states that are useful in quantum information processing, including measurement-based quantum computation in particular. Here we propose a general approach for constructing efficient verification protocols for AKLT states on arbitrary graphs with local spin measurements. Our verification protocols build on bond verification protocols and matching covers (including edge coloring) of the underlying graphs, which have a simple geometric and graphic picture. We also provide rigorous performance guarantee that is required for practical applications. With our approach, most AKLT states of wide interest, including those defined on one- and two-dimensional lattices, can be verified with a constant sample cost, which is independent of the system size and is dramatically more efficient than all previous approaches. As an illustration, we construct concrete verification protocols for AKLT states on various lattices and on arbitrary graphs up to five vertices.
Using exact diagonalization and tensor network techniques, we compute the gap for the Affleck-Kennedy-Lieb-Tasaki (AKLT) Hamiltonian in one and two spatial dimensions. Tensor network methods are used to extract physical properties directly in the thermodynamic limit, and we support these results using finite-size scalings from exact diagonalization. Studying the AKLT Hamiltonian perturbed by an external field, we show how to obtain an accurate value of the gap of the original AKLT Hamiltonian from the field value at which the ground state verifies e0<0{e}_{0}<0e0​<0, which is a quantum critical point. With the tensor network renormalization group methods we provide direct evidence of a finite gap in the thermodynamic limit for the AKLT models in the one-dimensional chain and two-dimensional hexagonal and square lattices. This method can be applied generally to Hamiltonians with rotational symmetry, and we also show results beyond the AKLT model.
The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond state on a hexagonal lattice was shown to be a universal resource state for measurement-based quantum computation (MBQC). Can AKLT states of higher spin magnitude support universal MBQC? We demonstrate that several hybrid two-dimensional AKLT states involving a mixture of spin-2 and other lower-spin entities, such as spin-3/2 and spin-1, are also universal for MBQC. This significantly expands universal resource states in the AKLT family. Even though frustration may be a hindrance to quantum computational universality, lattices can be modified to yield AKLT states that are universal. The family of AKLT states thus provides a versatile playground for quantum computation.