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Australian Journal of Physics

Staggered Magnetisation in the Heisenberg Antiferromagnet

Lloyd C. L. Hollenberg, Michael J Tomlinson

Published Apr 1, 1994• 7 citations
Publisher

Abstract

In the presence of a staggered magnetic field, the plaquette expansion of the Lanczos matrix elements are obtained for the antiferromagnetic 2D Heisenberg model up to order 1/Np (Np is the number of plaquettes on the lattice). The resulting approximate tri-diagonal form of the Hamiltonian is diagonalised for various values of the field strength in the -> 00 limit for the ground state energy density. From the behaviour of the ground-state energy density at weak fields, the staggered magnetisation at this order in the plaquette expansion is found to be 0·71 (in units where the Neel state staggered magnetisation is 1· 0).

DOI
10.1071/ph940137

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In the presence of a staggered magnetic field, the plaquette expansion of the Lanczos matrix elements are obtained for the antiferromagnetic 2D Heisenberg model up to order 1/Np (Np is the number of plaquettes on the lattice). The resulting approximate tri-diagonal form of the Hamiltonian is diagonalised for various values of the field strength in the -> 00 limit for the ground state energy density. From the behaviour of the ground-state energy density at weak fields, the staggered magnetisation at this order in the plaquette expansion is found to be 0·71 (in units where the Neel state staggered magnetisation is 1· 0).
The plaquette expansion of the Lanczos recursion method is applied to the two-dimensional antiferromagnetic Heisenberg model. Connected Hamiltonian moments 〈Hn{\mathit{H}}^{\mathit{n}}Hn〉c{\mathrm{〉}}_{\mathit{c}}〉c​ are calculated with respect to the N\'eel state up to n=6. The subsequent plaquette expansion of the Lanczos matrix in the number of plaquettes on the lattice, Np{\mathit{N}}_{\mathit{p}}Np​, is determined to order 1/Np{\mathit{N}}_{\mathit{p}}Np​. Diagonalizing the Lanczos matrix in this form gives a value of the energy density of -0.664 in the limit Np{\mathit{N}}_{\mathit{p}}Np​\ensuremath{\rightarrow}\ensuremath{\infty}, in good agreement with existing calculations.
We develop a formalism for calculating arbitrary expectation values for any extensive lattice Hamiltonian system using an analytic Lanczos expansion, or plaquette expansion, and a recently proved exact theorem for ground-state energies. The ground-state energy, staggered magnetization, and excited state gap of the two-dimensional anisotropic antiferromagnetic Heisenberg model are then calculated using this expansion for a range of anisotropy parameters, and compared to other moment-based techniques, such as the t expansion, and spin-wave theory, and series-expansion methods. We find that far from the isotropic point all moment methods give essentially very similar results, but near the isotropic point the plaquette expansion is generally better than the others.
The Lanczos method in operator form is applied to a general lattice Hamiltonian and expressions for the first few Lanczos matrices in terms of the connected Hamiltonian moments 〈Hn{\mathit{H}}^{\mathit{n}}Hn〉c{\mathrm{〉}}_{\mathit{c}}〉c​ and the number of plaquettes, Np{\mathit{N}}_{\mathit{p}}Np​, are obtained. Expansions in 1/Np{\mathit{N}}_{\mathit{p}}Np​ suggest a very simple general form for the first few terms in the 1/Np{\mathit{N}}_{\mathit{p}}Np​ expansions for all ${\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}$ and ${\mathrm{\ensuremath{\beta}}}_{\mathit{n}}$. For the one-dimensional Heisenberg spin chain it is demonstrated that the ground-state eigenvalue of the tridiagonal Lanczos matrix derived from this plaquette expansion approaches the true infinite lattice limit as the number of terms in the plaquette expansion is increased.
We apply the plaquette renormalization scheme of tensor network states [Phys. Rev. E 83, 056703 (2011)] to study the spin-1/2 frustrated Heisenberg J1{J}_{1}J1​-J2{J}_{2}J2​ model on an $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with LLL === 8, 16, and 32. By treating tensor elements as variational parameters, we obtain the ground states for different J2/J1{J}_{2}/{J}_{1}J2​/J1​ values, and investigate staggered magnetizations, nearest-neighbor spin-spin correlations, and plaquette order parameters. In addition to the well-known N\'eel order and collinear order at low and high J2/J1{J}_{2}/{J}_{1}J2​/J1​, we observe a plaquettelike order at ${J}_{2}/{J}_{1}\ensuremath{\approx}0.5$. A continuous transition between the N\'eel order and the plaquettelike order near ${J}_{2}^{{c}_{1}}\ensuremath{\approx}0.40{J}_{1}$ is observed. The collinear order emerges at ${J}_{2}^{{c}_{2}}\ensuremath{\approx}0.62{J}_{1}$ through a first-order phase transition.
The Heisenberg XXZ spin-(1/2) chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter and by the magnetic-field strength h . At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attractive potential between two adjacent kinks leading to their confinement into the bound states. The energy spectra of the resulting two-kink bound states are perturbatively calculated in the extreme anisotropic (Ising) limit to the first order in the inverse anisotropy constant , and also for any to the first order in the weak magnetic field h .
We study the problem of the existence of long-range staggered magnetic order in the antiferromagnetic Heisenberg model on the square lattice by a renormalization group analysis. We first map the Heisenberg model onto an approximate effective model, described by the Hamiltonian: H = ∑Lj=1S(j)S(j + 1) + ∑Lj=1( − 1)j‖J(j)‖S(0)S(j), where S(0) represents a single spin located in the center of the ring. We show that the Ji tend to non-zero values as one increases the block size by successive renormalizations, thus establishing the existence of long-range staggered magnetic order in the effective model. We discuss the implications of this result for the original Heisenberg model.