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arXiv:cond-mat/0601264 · Journal of Physics Condensed Matter

Random phase approximation for the 1D anti-ferromagnetic Heisenberg model

Aziz Rabhi, Peter Schuck, João da Providência

Published Oct 27, 2006• 4 citations
arXivHTMLPDF

Abstract

The Hartree–Fock random phase approximation (HF-RPA) approach is applied to the one-dimensional (1D) anti-ferromagnetic Heisenberg model in the Jordan–Wigner representation. Somewhat contrary to expectation, this leads to reasonable results for spectral functions and sum rules in the symmetry-unbroken phase. In a preliminary application of self-consistent RPA to finite-size chains, strongly improved results are obtained.

DOI
10.1088/0953-8984/18/45/011

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The Hartree–Fock random phase approximation (HF-RPA) approach is applied to the one-dimensional (1D) anti-ferromagnetic Heisenberg model in the Jordan–Wigner representation. Somewhat contrary to expectation, this leads to reasonable results for spectral functions and sum rules in the symmetry-unbroken phase. In a preliminary application of self-consistent RPA to finite-size chains, strongly improved results are obtained.
The Hartree-Fock-RPA approach is applied to the 1D anti-ferromagnetic Heisenberg model in the Jordan-Wigner representation. Somewhat contrary to expectation, this leads to reasonable results for spectral functions and sum rules in the symmetry unbroken phase. In a preliminary application of Self-Consistent RPA to finite size chains strongly improved results are obtained. pacs: 75.40.Gb, 75.50.Ee, 75.10.Pq, 75.10.Jm The 1D Anti-Ferromagnetic Heisenberg Model (AFHM) [1] belongs to one of the most classic many body research fields. It was the first many-body model to be solved exactly by Bethe with his famous Bethe ansatz solution [2] and it has been a playground for testing and developing many body approaches ever since. In spite of tremendous progress in the understanding of many facets of the model, it continues to be a very active field of interest and research [3, 4]. In an attempt to apply extended RPA-like theories, as e.g. the recently developed Self-Consistent RPA (SCRPA) approach [5, 6], we found out, somewhat to our surprise, that even the standard RPA theory has so far not fully been developed in its application to the 1D-AFHM. This probably stems from the fact that the HF-RPA scheme is usually considered as unreliable in low dimensions [7]. However, it will be found that RPA shows interesting features even in this 1D-version of the model. On the other hand SCRPA shows very promising results in a number of cases [5] and in a first application for a small numbers of sites we here get very close agreement with results from exact diagonalization. We mostly consider the isotropic Heisenberg model with anti-ferromagnetic coupling, however some consideration will also be given to the anisotropic case, for instance to the
The four site and eight site 1D anti-ferromagnetic Heisenberg chains in the Jordan–Wigner representation are investigated within the standard Hartree–Fock and random phase approximation (RPA) approaches, both in the symmetry unbroken and in the symmetry broken phases. A translation invariant groundstate, obtained by the projection method as a linear combination of a symmetry-broken HF state and its image under reflection, is also considered, for each chain type. It is found that the projection method considerably improves the HF treatment for instance as far as the groundstate energy is concerned, but also with respect to the RPA energies. The results are furthermore confronted with the ones obtained within so-called SCRPA scheme.
Mapping spins to fermions via the Jordan-Wigner (JW) transformation can render mean-field (Hartree-Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering, because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-Algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytic orbital Hessian to diagnose HF stability and compute ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J1-J2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.
High Resolution Image Download MS PowerPoint Slide Mapping spins to fermions via the Jordan–Wigner (JW) transformation can render mean-field (Hartree–Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytical orbital Hessian to diagnose HF stability and compute the ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J 1 – J 2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.
In this article, we revisit the question of the validity of Hartree-Fock and random-phase approximations. We show that there is a connection between the two and while the RPA as it is known in much of the physics literature is of limited validity, there is a generalised sense in which the random phase approximation is of much wider applicability including to systems that do not possess Fermi surfaces. The main conclusion is that the Hartree-Fock approximation is a mean-field idea applied to the density operator, and the random-phase approximation is a mean-field idea applied to the number operator. The generalised RPA is used to compute single-particle properties such as momentum distribution and spectral functions. It is found that we have to go beyond the generalised RPA and include fluctuations in the momentum distribution in order to recover a nonzero imaginary part of the one-particle self energy, which is also explicitly computed, all this in any number of spatial dimensions and no bosonization is needed.
The ground-state energy and static and dynamic correlation functions are investigated in the inhomogeneous-Hartree-Fock (IHF) plus random-phase-approximation (RPA) approach applied to a one-dimensional spinless-fermion model showing self-trapped doping states at the mean-field level. Results are compared with those obtained using homogeneous HF and exact diagonalization. RPA fluctuations added to the generally inhomogeneous-HF ground state allow the computation of dynamical correlation functions that compare well with exact diagonalization results. The RPA correction to the ground-state energy agrees well with the exact results at strong- and weak-coupling limits. We also compare it with a related quasiboson approach. The instability towards self-trapped behavior is signaled by a RPA mode with frequency approaching zero.