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Physical Review A

Variational Multistate-Ket Derivation of the Heisenberg Spin Hamiltonian

Daniel Klein, David C. Foyt

Published Nov 1, 1973• 12 citations
Publisher

Abstract

A new variational derivation scheme for the Heisenberg exchange Hamiltonian is presented. A many-body variational Ansatz for the multistate ket appearing in the derivation is described and is applied to a simple model problem containing the qualitatively important different types of exchange contributions. Advantages of the present derivation over others are briefly indicated.

DOI
10.1103/physreva.8.2280

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A new variational derivation scheme for the Heisenberg exchange Hamiltonian is presented. A many-body variational Ansatz for the multistate ket appearing in the derivation is described and is applied to a simple model problem containing the qualitatively important different types of exchange contributions. Advantages of the present derivation over others are briefly indicated.
Variational methods are of fundamental importance and widely used in theoretical physics, especially for strongly interacting systems. In this work, we present a set of variational equations of state (VES) for pure states of an interacting quantum Hamiltonian. The VES can be expressed in terms of the variation of the density operators or static correlation functions. We derive the algebraic relationship between a known pure state density matrix and its variation, and obtain the VES by applying this relation to the averaged Heisenberg-equations-of-motion for the exact density matrix. Additionally, we provide a direct expression of the VES in terms of correlation functions to make it computable. We present three nontrivial applications of the VES: a perturbation calculation of correlation functions of the transverse field Ising model in arbitrary spatial dimensions, a study of a longitudinal field perturbation to the one-dimensional transverse field Ising model at the critical point and variational calculation of magnetization and ground state energy of the two-dimensional spin-1/2 Heisenberg model on a square lattice. For the second one, our results not only recover the scaling limit, but also indicate the possibility of continuous tuning of the critical exponents by adjusting the longitudinal fields differently from the scaling limit. For the Heisenberg model, we obtained results numerically comparable to established results with simple calculations. The VES approach provides a powerful and versatile tool for studying interacting quantum systems.
The cluster variation method for the cooperative phenomena proposed by Kikuchi and reformulated and generalized by Morita, is applied to the Heisenberg model with arbitrary spin and range of exchange. A general expression for the two-body reduced density matrix is obtained in the approximation in which the clusters of pairs of lattice sites are retained correctly. The constant-coupling approximation for the Heisenberg model of $S\ensuremath{\geqq}1$ is shown to be derived by satisfying the reducibility conditions ${{\mathrm{tr}}_{k\ensuremath{\rho}}}^{(2)}(j, k)={\ensuremath{\rho}}^{(1)}(j)$ only partly, requiring the consistency for the zeroth and first moments of Sjz{S}_{\mathrm{jz}}Sjz​ and ignoring the consistency for the second to 2Sth2S\mathrm{th}2Sth moments. A natural method of extending the constant-coupling approximation for the Heisenberg model to the cases with arbitrary spin and range of exchange is suggested.
The antiferromagnetic Heisenberg chain with axial anisotropy and applied magnetic field parallel to the axis, is probed through a new variational approach based on the Bethe–Ansatz solution at zero field. The variational ket is built by rotating the spins in the Bethe solution, with the angles and the value of the local magnetic moment as variational parameters. This procedure allows for a description of the various magnetic phases in the presence of a parallel field, such as ferromagnetic (F), antiferromagnetic (AF), and spin-flop (SF) phases. The approximation reduces to the exact Bethe solution for vanishing field. We compare with exact numerical results (Lanczos calculations for finite systems), and exact solutions when available.
One of the most important quantum-mechanical properties is the property of exchange. The ideas behind this arise from the fact that elementary quanta are indistinguishable in the real word, but they are completely distinguishable in our theory, where we use different labels for the different particles. The procedure by which we make our theory indistinguishable as well leads to a new “force,” called quantum-mechanical exchange. In this chapter, we develop just what exchange is, we introduce the Heisenberg spin-exchange model and prove the Marshall sign rule, and we solve the one-dimensional version of the model using the so-called Bethe ansatz. You will find it amazing to learn the quantum origin of magnetism and details about how we work with quantum systems that have many particles. Always remember that permanent magnets are one of the best examples of macroscopic quantum mechanics.
We develop a general quantum many-body theory in configuration space by extending the traditional coupled-cluster method to a variational formalism. A set of distribution functions are introduced to evaluate the expectation value of the Hamiltonian. An algebraic technique for calculations of these distribution functions via a set of self-consistent set of equations is given. As a demonstration, we apply this method to a quantum antiferromagnetic spin model. It is shown that one of the lowest-order approximations within this new technique agrees with the traditional coupled-cluster method. Comparison with the method of correlated basis functions is also made.