Time-dependent behavior of one-dimensional many-fermion models: Comparison with two- and three-dimensional models

Abstract

Time-dependent behavior of one-dimensional (1D) many-fermion models is obtained by a method of recurrence relations. The Hilbert space of the density-fluctuation operator is two dimensional (2d), resulting in a time-independent generalized random force. The relevant Hilbert spaces of 2D and 3D many-fermion models, however, are infinite dimensional and the generalized random forces are consequently time dependent. The structure of these Hilbert spaces provides a picture of time-dependent behavior for 1D models which is fundamentally very different from that for 2D or 3D models.The Tomonaga model is perhaps the best known strictly one-dimensional (1D) many-fermion model. ' It is an exactly soluble model. The exact solutions for the Tomonaga model are not generalizable to higher dimensions. Hence this model is not so useful for understanding the importance of dimensionality in many-body problems. The standard electron gas model, when restricted to electron-hole scattering, is often referred to as the Sawada mode. ' The Sawada model is not exactly soluble, but one can show that the Sawada model in 1D reduces to the Tomonaga model if the electron-hole excitations are further confined to the vicinity of the Fermi surface. Hence one may regard the Sawada model as a generalizable version of the Tomonaga model.The time-dependent behavior of the Tomonaga or 1D Sawada model is basically very different from that of the Sawada model in 2D or 3D. If a dense electron gas in the ground state is slightly perturbed momentarily, the system will undergo a relaxation process. In 1D the relaxation process will be purely oscillatory. It will remain oscillatory even when the electron-hole interaction is removed, i.e. , when the system becomes an ideal, degenerate electron gas. This is because the 1D system has only one degree of freedom in momentum space whether there is an interaction or not. In 2D or 3D the system has infinitely many degrees of freedom and its relaxation process is richer, reflecting the two distinctly different single-particle and collective modes. 4 The different time-dependent behavior of the Sawada model in 1D and 2D or 3D is especially apparent if we study the time evolution in this model via the method of recurrence relations. ' Consider the density fluctuations at wave vector k, pk =~C p C&+k P where ck and ck are, respectively, the creation and annihilation operators at wave vector k. %e shaB confine our consideration to~k/kF~((1, where kF is the Fermi wave vector and A =1. The time-dependent behavior of this system can be completely determined by pk(t) and its generalized random force 5'(t). According to the method of recurrence relations, ps(t) and 9q(t) are given by(lb) 5"+ta"+t(t) = -a"(t) +a" t(t), 0~v~d -1, (2) where a I~O, a"=da"/dt, /s, "=(f",f")/(f" t, f" t) with 50~1. The inner product denotes the Kubo scalar product. We shall term 4" the vth recurrant. Also, {b"} satisfies exactly the same recurrence relation but starting with v=1 and b0=0. Hence dl does not appear in the recurrence relation for b...

Recent Advances in the Calculation of Dynamical Correlation Functions

“…Also, from RRII one obtains (da 0 (t)/dt)| 0 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [33][34][35][36], harmonic oscillator chains [37][38][39][40][41][42][43][44][45][46], many-particle systems [47][48][49][50], spin chains [51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66], plasmonic Dirac systems [67,68], dynamics of simple liquids [69,70], etc.…”

Recent advances in the calculation of dynamical correlation functions

“…Also, from RRII one obtains (da 0 (t)/dt)| 0 = 0, which precludes a pure time exponential as well as other functions that do not have zero derivative at t = 0. The method of recurrence relations have since been applied to a variety of problems, such as the electron gas [27,28,30,29], harmonic oscillator chains [31,32,33,34,35,36,37,38,39,40], many-particle systems [44,43,41,42], spin chains [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60], plasmonic Dirac systems [61,62], etc.…”

The topology of Hilbert spaces and the dynamics of molecular processes

“…(They are also realized in infinite systems, e.g. a one-dimensional homogeneous electron gas [63], higher lattice-dimensional spin models [l 11. ) Since these models yield &rite continued fractions, they do not at first appear interesting from the point of view of the analytic theory of continued fractions.…”

“…In the one-dimensional homogeneous electron gas at long wavelengths [63], d = 2. In the harmonic oscillator chain of N molecules [62], d = N + 1.…”