A generalized random-phase approximation (RPA) for decoupling the double-time Green's functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.
A generalized random-phase approximation (RPA) for decoupling the double-time Green's functions (DTGF) equations of motion of spin systems is developed. The decoupling parameters are chosen by fixing the moments of both the commutator and anticommutator decoupled spectral functions, thus establishing a well-defined relationship between the approximation scheme and the fundamental properties of DTGF. It is shown that the standard RPA must be supplemented by a condition on correlations to provide a well-defined, complete approximation scheme. The procedure is demonstrated by application to the fully anisotropic S=1/2 Heisenberg model in a field.
The quantum-statistical mechanics of an anisotropic Heisenberg spin system is studied by a temperature-time-dependent Green's-function formalism. A consistent scheme of higher-order random-phase approximations (RPA) is developed and the second-order (2nd) RPA is examined in detail. It turns out that the 2nd RPA can be defined in two alternative versions, I and II, with equal a priori justification. These two versions of the 2nd RPA have then to be supplemented with either a dynamical or a kinematical sum rule, thus leading to four possible alternative descriptions of the problem. Each of these descriptions can in principle be used to determine the spectral functions of both the longitudinal and the transverse dynamical spin-correlation functions. However, upon detailed examination of the 2nd RPA, it is found that only one of these four possible descriptions satisfies all the various consistency requirements. At low temperatures, this description reproduces the spin-wave results. To facilitate analytical solutions, an approximate version of the 2nd RPA, called the modified (mod) RPA, is introduced which leads to a satisfactory expression for the longitudinal correlation function over the entire range of temperatures. Upon examination it is found that the mod RPA determines the longitudinal correlation to the same accuracy, for the case of the isotropic exchange, as the first RPA determines the transverse correlation function. In addition to this mod RPA version of the consistent 2nd RPA, there also appears to be another relatively satisfactory solution for the longitudinal correlation function which follows from one of the less consistent versions of the 2nd RPA. This is treated as a phenomenological result. The system thermodynamics is analyzed in the region of the transition temperature in terms of both the consistent version of the mod RPA, i.e., the I mod RPA, as well as the phenomenological representation, i.e., the II mod RPA, with results which in addition to being an improvement on those following from the first RPA are also free from the inherent inconsistencies of the first RPA. In conclusion, the related work of other authors is discussed and it is shown that all these works suffer from serious internal inconsistencies which render their results for the longitudinal correlation function completely unacceptable and erroneous.
Standard-basis-operator (SBO) Green's function equations of motion are developed in the random-phase approximation (RPA) for a general Hamiltonian characterizing interacting systems having discrete energy levels. It is shown that the RPA equations have the same algebraic structure as the chain-diagram equations in the perturbation theory of SBO Green's functions. The earlier problems of redundancy and failure to satisfy the monotopic restrictions in the calculation of autocorrelation functions are resolved. Ratios of RPA autocorrelation functions can always be determined from commutator equations from which thermal average energy-level occupation probabilities ${P}_{\ensuremath{\beta}}$ associated with state $|\ensuremath{\beta}〉$ are obtained. Using the properties of the SBO, a very simple derivation of an explicit formula for ${P}_{\ensuremath{\beta}}$ is given for an important class of Hamiltonians, which includes the Hubbard s-band model and the Heisenberg ferromagnet. This derivation completely avoids the use of complicated methods such as the moment generator used to solve equations coupling average moments of the spin operator ${\stackrel{^}{S}}_{Z}$ in ferromagnetic and antiferromagnetic systems.
We derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalize Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the N-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.
An approach for particle-hole correlation functions, based on the so-called self-consistent random-phase approximation is developed. This leads to a fully self-consistent RPA-like theory that satisfies the f-sum rule and several other theorems. As a first step, a simpler self-consistent approach, the renormalized RPA, is solved numerically in the one-dimensional Hubbard model. The charge and the longitudinal spin susceptibility, the momentum distribution, and several ground-state properties are calculated and compared with the exact results. Especially at half-filling, our approach provides quite promising results and matches the exact behavior apart from a general prefactor. The strong-coupling limit of our approach can be described analytically.
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
Generalization of the random-phase-approximation (RPA) theory of ferromagnetism for spins S=1/2, based on a refinement of the decoupling for a relevant three-spin Green's function, is presented. With the reasonable choice of two characteristic functions of the present theory---the virtual spectrum \ensuremath{\Delta}(\ensuremath{\omega}) and the decoupling factor \ensuremath{\kappa}(\ensuremath{\sigma}), where \ensuremath{\omega} is an excitation energy and \ensuremath{\sigma} is the relative magnetization---the theory provides a fair description of magnetic phenomena not only at temperature T\ensuremath{\rightarrow}0, in the vicinity of the Curie temperature TC, or at high T, but at all temperatures and external magnetic fields H. A relation between some parameters of the functions \ensuremath{\Delta}(\ensuremath{\omega}) and \ensuremath{\kappa}(\ensuremath{\sigma}), and the critical indices of the static scaling theory, has been established. The form of the universal scaling function W, following from our theory, has been found. For the power-law virtual spectrum \ensuremath{\Delta}\ensuremath{\propto}${\mathrm{\ensuremath{\omega}}}^{\mathit{s}}$ at \ensuremath{\omega}\ensuremath{\rightarrow}0, 0s1, we have found the nonanalytic increase of the magnetization versus the field at TTC:\ensuremath{\sigma}(T,H)-\ensuremath{\sigma}(T,0)\ensuremath{\propto}Hs. It reduces to the classic result of Holstein and Primakoff for the three-dimensional isotropic Heisenberg ferromagnet having s=1/2. Assuming this power-law spectrum at all energies up to some cutoff energy, we have obtained a rather good agreement of the present predictions for \ensuremath{\sigma}(T,H) with experimental data for Ni, for all temperatures and fields available. Some model virtual spectra are found to lead to interesting models of ferromagnetism, with the magnetization and the susceptibility functions given in terms of explicit analytic formulas.