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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Plaquette expansion in lattice Hamiltonian models

Lloyd C. L. Hollenberg

Published Feb 15, 1993• 42 citations
Publisher

Abstract

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 〈${\mathit{H}}^{\mathit{n}}$${\mathrm{〉}}_{\mathit{c}}$ and the number of plaquettes, ${\mathit{N}}_{\mathit{p}}$, are obtained. Expansions in 1/${\mathit{N}}_{\mathit{p}}$ suggest a very simple general form for the first few terms in the 1/${\mathit{N}}_{\mathit{p}}$ 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.

DOI
10.1103/physrevd.47.1640

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Australian Journal of Physics•Apr 1, 1994•7 citations
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 〈{\mathit{H}}^{\mathit{n}}$${\mathrm{〉}}_{\mathit{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 \ensuremathαn{\mathrm{\ensuremath{\alpha}}}_{\mathit{n}}\ensuremathαn​ and \ensuremathβn{\mathrm{\ensuremath{\beta}}}_{\mathit{n}}\ensuremathβn​. 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.
Employing a theorem on lower bounds on the zeros of orthogonal polynomials, the plaquette expansion to order 1/Np{\mathit{N}}_{\mathit{p}}Np​ of the tridiagonal Lanczos matrix elements is solved for the ground-state energy density in the infinite lattice limit. The resulting nonperturbative expression for the estimate of the energy density in terms of the connected coefficients to order 〈H4{\mathit{H}}^{4}H4〉c{\mathrm{〉}}_{\mathit{c}}〉c​ is completely general. This expression is applied to various Hamiltonian systems---the Heisenberg model in D dimensions and SU(2) and SU(3) lattice gauge theory in 3+1 dimensions. In all cases the analytic estimate to the energy density is not only a significant improvement on the trial state, but is typically accurate to a few percent. The energy density of the D-dimensional Heisenberg model is predicted to approach scrE0{\mathit{scrE}}_{0}scrE0​(N\'eel)-1/8 for large D. In the case of SU(2) and SU(3) the specific heat derived from the energy density peaks at the correct strong- to weak-coupling transition.
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.
An analysis of a general non-perturbative technique for calculating ground-state properties of extensive lattice many-body systems is presented, in order to extract accurate numerical values characterizing the ground-state spectrum. This technique, the plaquette expansion, employs an expansion about the thermodynamic limit of the coefficients that are generated by the Lanczos process. For the ground-state energy this error analysis, using theorems on the error bounds for the Lanczos method and the truncation in the plaquette expansion, allows for an accurate estimate when the approximation is taken to a given order. As an example we analyse the one-dimensional antiferromagnetic Heisenberg model, and find that the best ground-state energy density is within of the exact value, although the systematic error is . We also find, for this model, systematic improvement with each new order included in the expansion and have not observed any asymptotic tendencies. At equivalent orders of truncation we achieve far better results than for the other moment methods, such as the t-expansion or the connected-moment expansion.
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.
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).