Using the recently developed method of recurrence relations, we have studied the time- and frequency-dependent behavior of the two-dimensional electronic system at long wavelengths. Several families of autocorrelation functions including the relaxation and memory functions are obtained exactly. In addition, other linear-response quantities, e.g., the density-density response function and dynamic structure factor, are obtained. This work contains a detailed application of the method of recurrence relations.
Using the recently developed method of recurrence relations, we have studied the time- and frequency-dependent behavior of the two-dimensional electronic system at long wavelengths. Several families of autocorrelation functions including the relaxation and memory functions are obtained exactly. In addition, other linear-response quantities, e.g., the density-density response function and dynamic structure factor, are obtained. This work contains a detailed application of the method of recurrence relations.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. In particular, dynamic correlation functions appear in the fluctuation-dissipation theorem, where the response of a many-body system to an external perturbation is given in terms of the relaxation function of the unperturbed system, provided the disturbance is small. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numericallly exact diagonalizaton method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations. The method of recurrence relations relies on the solution to the coefficients of a continued fraction for the Laplace transformed relaxation function. The calculation of those coefficients becomes very involved and, only a few cases offer exact solution. We shall concentrate our efforts on the cases where extrapolation schemes must be used to obtain solutions for long times (or low frequency) regimes. We also cover numerical work based on the exact diagonalization of finite sized systems. The numerical work provides some thermodynamically exact results and identifies some difficulties intrinsic to the method of recurrence relations.
We review various theoretical methods that have been used in recent years to calculate dynamical correlation functions of many-body systems. Time-dependent correlation functions and their associated frequency spectral densities are the quantities of interest, for they play a central role in both the theoretical and experimental understanding of dynamic properties. The calculation of the relaxation function is rather difficult in most cases of interest, except for a few examples where exact analytic expressions are allowed. For most of systems of interest approximation schemes must be used. The method of recurrence relation has, at its foundation, the solution of Heisenberg equation of motion of an operator in a many-body interacting system. Insights have been gained from theorems that were discovered with that method. For instance, the absence of pure exponential behavior for the relaxation functions of any Hamiltonian system. The method of recurrence relations was used in quantum systems such as dense electron gas, transverse Ising model, Heisenberg model, XY model, Heisenberg model with Dzyaloshinskii-Moriya interactions, as well as classical harmonic oscillator chains. Effects of disorder were considered in some of those systems. In the cases where analytical solutions were not feasible, approximation schemes were used, but are highly model-dependent. Another important approach is the numerically exact diagonalization method. It is used in finite-sized systems, which sometimes provides very reliable information of the dynamics at the infinite-size limit. In this work, we discuss the most relevant applications of the method of recurrence relations and numerical calculations based on exact diagonalizations.
The method of recurrence relations allows one to calculate time dependent correlation functions from first principles. It originates from the Kubo scalar product which realizes abstract Hilbert space of dynamical variables. For this realized space, we show that there exists a unique orthogonalization process by recurrence relation. The method of recurrence relations exploits geometric properties of the structure of the realized Hilbert space such as the dimensionality and shape. These geometric properties depend on models as well as static constraints, e.g., temperature, wave vector, size. This method has been applied to several physical models including the homogeneous dense electron gas, the spin-1/2 XY and transverse Ising models, the spin van der Waals model and others. For these models, we have obtained the relaxation function, response function and memory function.
A formalism based on the generalized Langevin equation is used to obtain the dynamic response \ensuremath{\chi}(k,\ensuremath{\omega}) and relaxation function \ensuremath{\Xi}(k,t) of a dense electron gas in spatial dimension D for the low-momentum-transfer regime k\ensuremath{\ll}kF. Our results agree with known results for D=1, 2, and 3. In one dimension, the relaxation process is purely oscillatory. For D=4, we find a temporal scaling behavior, i.e., self-similarity of relaxation functions at different hierarchy levels. And in infinite dimensions, the system shows a Gaussian relaxation. The formalism is also extended to apply to an interacting system.
The dynamic structure function S(k,\ensuremath{\omega}) and the dielectric function \ensuremath{\epsilon}(k,\ensuremath{\omega}) of the two-dimensional electron gas is calculated phenomenologically at rs=1 and 5 in the memory-function formalism. The density-density response function \ensuremath{\chi}(k,\ensuremath{\omega}) is expressed in terms of that of a noninteracting system and an effective potential. The interaction part of the second-order memory function that determines the effective potential is approximated by a Gaussian in time domain. We use a consistent set of highly accurate ground-state static properties as input, coming from a Green's-function Monte Carlo calculation. The calculated S(k,\ensuremath{\omega}) shows a single-broad-peak structure at intermediate k, which is from the case in three dimensions.
The correlation energy per particle (expressed in rydbergs) of a two-dimensional electron gas is calculated by summing the ring diagrams. For high densities, this is found to be of the form C+Drslnrs+O(rs), where rs is a dimensionless parameter such that ${r}_{s}^{\ensuremath{-}2}$ is proportional to the number of electrons per unit area. The value of D is -0.172 and that of C is -0.38 \ifmmode\pm\else\textpm\fi{} 0.04, the latter involves a numerical evaluation. By the same methods, we have also calculated the difference in total energy of the nonmagnetic and ferromagnetic states and we find for rs>2.3 the system may be ferromagnetic. We have also obtained some exact relationships between the pair correlation function at zero separation and the asymptotic behavior of the structure factor and the momentum distribution function. These are of interest in relation to scattering experiments.