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Physical Review Letters

Variational approach to theXXZspin-1 linear chain: Elementary excitations and Haldane conjecture

G. Gómez-Santos

Published Aug 14, 1989• 100 citations
Publisher

Abstract

The XXZ spin-1 chain is studied by means of a physically motivated variational approach. An effective Hamiltonian for the dynamics of spin-0 defects in antiferromagnetic environments (domain walls) is obtained, and its ground-state properties are analyzed. An order-disorder transition is found with critical behavior corresponding to the 2D Ising model universality class, in complete agreement with Haldane conjecture. The approximations involved in this variational scheme are carefully tested using known Monte Carlo results. In particular, excellent agreement with the recently calculated excited states at the Heisenberg (isotropic) point is found, clarifying the nature of these excitations.

DOI
10.1103/physrevlett.63.790

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Journal of Applied Physics•Apr 15, 1991•1 citation
The XXZ spin-1 chain is studied by means of a physically motivated variational approach. An effective Hamiltonian for the dynamics of spin-0 defects in antiferromagnetic environments (domain walls) is obtained, and its ground-state properties are analyzed. An order-disorder transition is found with critical behavior corresponding to the 2D Ising model universality class, in complete agreement with Haldane conjecture. The approximations involved in this variational scheme are carefully tested using known Monte Carlo results. In particular, excellent agreement with the recently calculated excited states at the Heisenberg (isotropic) point is found, clarifying the nature of these excitations.
A physically motivated approach is used to study the linear chain XXZ spin-1 Hamiltonian in the antiferromagnetic regime. A variational Hamiltonian is obtained restricting the basis set to states with local antiferromagnetic environments separated by sites with Sz=0 acting as domain walls. Ground-state properties of this variational Hamiltonian are obtained and, in particular, an order–disorder transition is found with critical behavior corresponding to the two-dimensional Ising model universality class, in agreement with Haldane conjecture. Excellent agreement with finite chain data concerning the position of Haldane’s point, ground-state energy, and elementary excitations is found.
The linear chain XXZ spin- (1)/(2) Hamiltonian is mapped onto a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The domain-wall Hamiltonian is solved within the simplest Hartree–Fock approximation with results in very good agreement with the exact solution in the antiferromagnetic regime. The location of ordered and disordered (critical) phases is correct and low-lying excitations are well described and easily interpreted in terms of domain walls. Our domain-wall version of the original Hamiltonian is shown to provide the natural framework for a physical understanding of this system in the antiferromagnetic regime.
The XXZ spin-1/2 Hamiltonian in the linear chain is exactly transformed into a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The ground state of the new Hamiltonian is studied with the Hartree-Fock approximation. This approximation is shown to become exact in the XY and extreme antiferromagnetic limits, with excellent results in between. Known results concerning the existence or nonexistence of long-range order and gap in the excitation spectrum are reproduced by our solution. Good quantitative agreement with representative properties of the exact solution is obtained. The domain-wall Hamiltonian is shown to be the natural framework for the understanding of the ground state and low-lying excited states of the XXZ system in the antiferromagnetic regime.
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
In a previous paper (Liu et al 2012 Phys. Rev. B 85 195144 ), a variational Monte Carlo method (based on Gutzwiller projected states) was generalized to S = 1 systems. This method provided very good trial ground states for the gapped phases of an S = 1 bilinear-biquadratic (BLBQ) Heisenberg chain. In this paper, we extend the approach to study the low-lying elementary excitations in S = 1 chains. We calculate the one-magnon and two-magnon excitation spectra of the BLBQ Heisenberg chain and the results agree very well with recent data in the literature. In our approach, the difference of the excitation spectrum between the Haldane phase and the dimer phase (such as the even/odd size effect) can be understood from their different topologies of the corresponding mean field theory. We especially study the Takhtajan–Babujian critical point. Despite the fact that the 'elementary excitations' are spin-1 magnons, which are different from the spin-1/2 spinons in Bethe solution, we show that the excitation spectrum, critical exponent ( ) and central charge ( c = 1.45) calculated from our theory agree well with the Bethe ansatz solution and conformal field theory predictions.
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.