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.
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-
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-
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.
In this paper we study a frustrated antiferromagnetic isotropic Heisenberg S=1 quantum spin chain H=${\ensuremath{\sum}}_{\mathrm{i}}$
SiSi+1+\ensuremath{\alpha}${\ensuremath{\sum}}_{\mathrm{i}}$
SiSi+2, using a variational ansatz starting from valence bond states and the density-matrix renormalization group. We find both methods to give results in very good qualitative and good quantitative agreement, which clarify the phase diagram as follows: At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{D}}$=0.284(1), there is a disorder point of the first kind, marking the onset of incommensurate spin-spin correlations in the chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{L}}$=0.3725(25) there is a Lifshitz point, at which the excitation spectrum is found to develop a particular doubly degenerate structure. These points are the quantum remnants of the transition from antiferromagnetic to spiral order in the classical frustrated chain. At ${\mathrm{\ensuremath{\alpha}}}_{\mathrm{T}}$=0.7444(6) there is a first-order phase transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase characterized by nonvanishing string order to a phase which can be understood as a next-nearest-neighbor generalization of the AKLT model. At the transition, the string order parameter shows (to numerical precision) a discontinuous jump of 0.085 to zero; the correlation length and the gap are both finite at the transition. The problem of edge states in open frustrated chains is discussed at length.
We study a frustrated Heisenberg
S=1 quantum spin chain with next-nearest neighbor (NNN) coupling $\ensuremath{\alpha}$ using a variational ansatz and the density matrix renormalization group. We find as quantum remnants of the phase transition in the classical chain a disorder point at ${\ensuremath{\alpha}}_{D}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.284(1)$ and a Lifshitz point at ${\ensuremath{\alpha}}_{L}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.3725(25)$. Our main finding is a first-order transition from an Affleck-Kennedy-Lieb-Tasaki (AKLT) phase to a NNN generalization of the AKLT phase at ${\ensuremath{\alpha}}_{T}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.7444(6)$. At the transition, string order jumps discontinuously by $\ensuremath{\approx}0.085$ to 0; correlation length and gap are finite.
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.
Inspired by the Affleck-Kennedy-Lieb-Tasaki (AKLT) model, we present exact solutions for a spin-1 chain with Kitaev-like couplings. We consider an expanded Kitaev model with bilinear and biquadratic terms. At an exactly solvable point, the Hamiltonian can be reexpressed as a sum of projection operators. Unlike the AKLT model where projectors act on total spin, we project the component of spin along the bond direction. This leads to exponential ground-state degeneracy, expressed in terms of fractionalized spin-1/2 objects. Each ground state can be expressed concisely as a matrix product state. We construct a phase diagram by varying the relative strength of bilinear and biquadratic terms. The fractionalized states provide a qualitative picture for the spin-1 Kitaev model, yielding approximate forms for the ground state and low-lying excitations.