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Journal of Physics Condensed Matter

Accurate calculation of ground-state energies in an analytic Lanczos expansion

Nicholas S. Witte, Lloyd C. L. Hollenberg

Published Mar 3, 1997• 6 citations
Publisher

Abstract

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.

DOI
10.1088/0953-8984/9/9/016

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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.
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.
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.
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.
We study the ground-state properties and excitation spectrum of the Lieb-Liniger model, i.e. the one-dimensional Bose gas with repulsive contact interactions. We solve the Bethe-Ansatz equations in the thermodynamic limit by using an analytic method based on a series expansion on orthogonal polynomials developed in and push the expansion to an unprecedented order. By a careful analysis of the mathematical structure of the series expansion, we make a conjecture for the analytic exact result at zero temperature and show that the partially resummed expressions thereby obtained compete with accurate numerical calculations. This allows us to evaluate the density of quasi-momenta, the ground-state energy, the local two-body correlation function and Tan’s contact. Then, we study the two branches of the excitation spectrum. Using a general analysis of their properties and symmetries, we obtain novel analytical expressions at arbitrary interaction strength which are found to be extremely accurate in a wide range of intermediate to strong interactions.