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.
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.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
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.
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account. In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0. Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic second-magnon regime.
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 structure of the space-time-dependent spin-correlation functions for an anisotropic Heisenberg paramagnet with cylindrical symmetry at elevated temperature is analyzed in terms of a two-parameter Gaussian representation representation of the generalized diffusivity. The parameters are chosen so as to exactly preserve the frequency moments, i.e., ${{〈{\ensuremath{\omega}}^{2n}〉}_{\mathrm{K}}}^{\ensuremath{\alpha}\ensuremath{\alpha}}$ ($\ensuremath{\alpha}=z, x, or y$; n=0,1,2) of the longitudinal and the transverse spectral functions. These moments are calculated for arbitrary spin S, arbitrary range of the exchange interactions, and arbitrary dimensionality. The results for the correlation functions are worked out for one, two, and three dimensions. In the extreme transverse limit, the results are compared with the exactly soluble XY model for a one-dimensional spin-\textonehalf{} system with only nearest-neighbor (transverse) exchange. The agreement of the results is found to be satisfactory.
The frequency-dependent correlations ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})\ensuremath{\equiv}{〈{{S}_{0}}^{z}(t){{S}_{n}}^{z}({t}^{\ensuremath{'}})〉}_{(\ensuremath{\omega})}$, n=0,1,2,3,and4, and their inverse-lattice Fourier transforms $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$, have recently been exactly calculated by Carboni and Richards for finite linear chains consisting of N (N=6,7,8,9,and10) spins (S=21) in contact with a heat bath at temperature $T\ensuremath{\rightarrow}\ensuremath{\infty}$ and interacting via a nearest-neighbor isotropic Heisenberg exchange interaction. Carboni and Richards have used plausible extrapolation procedures to predict the corresponding correlations in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. As the extension of these exact calculations to two and three dimensions is of prohibitive difficulty, we have constructed an alternative theory, based upon a simple two-parameter Gaussian representation of the generalized diffusivity, for calculating these correlations for general spin and the dimensionality. To test the accuracy of this phenomenological theory (which is, however, free of any arbitrariness in the sense that the diffusivity is exactly specified by the second and the fourth frequency moments of the Fourier transform of the frequency-dependent correlation function, which are known at infinite temperature), we first compare our results for ${\ensuremath{\Phi}}_{n}(\ensuremath{\omega})$ with those given by Carboni and Richards for the one-dimensional spin-\textonehalf{} system and find the agreement to be excellent for n=0, good for n=1,2 and adequate for n=3,4. The comparison of the corresponding results for $\ensuremath{\Phi}(\mathbf{K},\ensuremath{\omega})$ reveals the agreement to be quite satisfactory for $|\mathbf{K}|\ensuremath{\lesssim}\frac{1}{2}\ensuremath{\pi}$ but only adequate for the higher-K range, i.e., $\frac{1}{2}\ensuremath{\pi}\ensuremath{\lesssim}\mathbf{K}\ensuremath{\lesssim}\ensuremath{\pi}$. Further support in favor of our phenomenological theory is obtained from a comparison with another set of available exact "computer experiment" results, whereby ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$, ${\ensuremath{\Phi}}_{0}(t)$, and ${\ensuremath{\Phi}}_{1}(t)$ are accurately known for a three-dimensional (simple-cubic) lattice of infinite spins, i.e., $S\ensuremath{\rightarrow}\ensuremath{\infty}$. (This is the limit in which the classical spin system studied by Windsor corresponds to the quantum spin system of interest to us.) The agreement of our result with Windsor's is excellent. The salient features of our results are as follows: (i) In two dimensions, the divergence of ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ for $\ensuremath{\omega}\ensuremath{\rightarrow}0$ becomes less sharp and disappears completely in three dimensions. (ii) The cutoff frequency, beyond which ${\ensuremath{\Phi}}_{0}(\ensuremath{\omega})$ is effectively zero, increases with the dimensionality, being in three dimensions about twice what it is in one dimension. (iii) There exists a system of reduced units, i.e., ${{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}(\ensuremath{\omega})\ensuremath{\rightarrow}{\ensuremath{\Phi}}_{n}(\ensuremath{\omega}){[S(S+1)]}^{\ensuremath{-}1}$, ${I}^{\ensuremath{'}}\ensuremath{\rightarrow}{[S(S+1)]}^{\frac{1}{2}}I$, and ${\ensuremath{\omega}}^{\ensuremath{'}}\ensuremath{\rightarrow}(\frac{\ensuremath{\omega}}{{I}^{\ensuremath{'}}})$, in which the function ${I}^{\ensuremath{'}}{{\ensuremath{\Phi}}_{n}}^{\ensuremath{'}}({\ensuremath{\omega}}^{\ensuremath{'}})$ is approximately the same for all spins, and the accuracy of this law of corresponding states seems to increase with the increase in the dimensionality.