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Australian Journal of Physics

Hamiltonian Moments and the Lanczos Method in the Heisenberg Model

Lloyd C. L. Hollenberg, Christopher J Griffin

Published Dec 1, 1992• 3 citations
Publisher

Abstract

The connected Hamiltonian moments (Hn)c of the one-dimensional s = ~ Heisenberg model with respect to the Neel state are calculated up to n = 17. Subsequent calculation of the Hamiltonian moments (Hn) allows the first nine iterations of the Lanczos diagonalisation to be determined analytically for arbitrary lattice size.
DOI
10.1071/ph920717

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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.
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.
For pt.I see ibid., vol.14, p.241-57 (1981). Two efficient methods for finding the low-lying states of Hamiltonians on finite lattices are described. The first involves constructing a finite representation of the Hamiltonian using strong-coupling eigenstates, while the second is based on the Lanczos recursion method. The methods are used to determine the mass gap of the O(2) and O(3) Heisenberg Hamiltonians in (1+1) dimensions for a sequence of finite chains. The critical behaviour of the infinite chain is then analysed by extrapolating the finite-lattice estimates using finite-size scaling. A remarkably sensitive test is developed for the presence of a phase transition. For the O(2) model data this test yields strong evidence for a phase transition with the weak-coupling phase massless, while, in the O(3) case the test supports, although more weakly, the absence of any transition.
The classical formalism of the Moment Problem has been combined with a cumulant approach and applied to the extensive many-body problem. This has yielded many new exact results for many-body systems in the thermodynamic limit - for the ground state energy, for excited state gaps, and for arbitrary ground state averages. The method applies to any extensive Hamiltonian system, for any phase or symmetry arising in the model, whether on a lattice or in the continuum, and for any dimensionality. The theorems are of a nonperturbative nature with respect to any couplings occuring in the model.
In this thesis we take a look at the one-dimensional Heisenberg model. The Heisenberg model is the most used model to study the magnetic properties of materials and is of paramount importance for understanding magnets. First we sketch a brief derivation of the model and discuss its strengths and weaknesses. Afterwards we develop a transformation of the Heisenberg model to a model of spinless fermions, which allows for an easy solution of the Jz = 0 model. After solving the model for Jz = 0, we solve the model for general Jz. The results we find are very complicated and it is hard to calculate anything explicitly. So we move to to the thermodynamic limit to calculate the ground state energy for Jz = 1. After that we have found a solution for big system sizes, we use exact diagonalisation to solve the Heisenberg model. We do this for system sizes up too 12 particles. With our results we check how accurate some expansions for smaller systems sizes are. We find some accurate expansions for small ∆ and ∆ = 1. We also look at a energy gap of the system for ∆ > 1. We find how this gap behaves as a function of ∆.
The Heisenberg-Ising Hamiltonian has been exactly solved us- ing the Lanczos method, for chain of up to N = 28 spins. In the whole anisotropic region, the mass gap displays a dominant exponential behavior with size. The extrapolated ground state is doubly degenerate, and presents long-range order with dominant components of the Neel type. Quantum fluctuations become more important as long as the anisotropy is reduced, leading to a disordered singlet ground state for the infinite system at the isotropic point.