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Physical Review

Analytic Properties of Single-Particle Propagators for Many-Fermion Systems

J. M. Luttinger

Published Feb 15, 1961• 360 citations
Publisher

Abstract

Certain general properties of single-particle propagators for a system of interacting fermions are derived. In addition, the properties of the proper self-energy part ${G}_{k}(\ensuremath{\zeta})$ which were used in previous work on the ground-state energy and on the Fermi surface are established. In particular, the fact that to all orders of perturbation theory in the interaction, $\mathrm{Im} {G}_{k}(x\ensuremath{-}i{0}^{+})$ behaves like ${C}_{k}{(x\ensuremath{-}\ensuremath{\mu})}^{2}$ (${C}_{k}>0$) for $x$ very near $\ensuremath{\mu}$, is proved.

DOI
10.1103/physrev.121.942

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Certain general properties of single-particle propagators for a system of interacting fermions are derived. In addition, the properties of the proper self-energy part Gk(\ensuremathζ){G}_{k}(\ensuremath{\zeta})Gk​(\ensuremathζ) which were used in previous work on the ground-state energy and on the Fermi surface are established. In particular, the fact that to all orders of perturbation theory in the interaction, ImGk(x\ensuremath−i0+)\mathrm{Im} {G}_{k}(x\ensuremath{-}i{0}^{+})ImGk​(x\ensuremath−i0+) behaves like Ck(x\ensuremath−\ensuremathμ)2{C}_{k}{(x\ensuremath{-}\ensuremath{\mu})}^{2}Ck​(x\ensuremath−\ensuremathμ)2 (Ck>0{C}_{k}>0Ck​>0) for xxx very near \ensuremathμ\ensuremath{\mu}\ensuremathμ, is proved.
We have examined the ground state of the many-fermion system, using perturbation theory, with particular attention to Brueckner's method. A study of the single-particle propagators in the ground state of the system sheds some light on the relation between the single-particle energies used, for example, in the optical model of nuclear reactions, and Brucekner's self-consistent energies. This leads naturally to the use of a renormalization factor to allow for propagation off the energy shell. It is possible to formulate the perturbation theory entirely in terms of graphs with no fermion self-energy parts, and this method is used to examine the approximations used for this problem, and to relate more closely the calculations of the ground-state energy and of other properties of the ground state.
Some analytic properties of the single-particle Green functions and proper self-energy parts for a system of interacting bosons at nonzero temperatures are derived. In particular, it is shown how to obtain the analytical continuations of simple functions of the proper self-energy parts, which can be obtained from perturbation theory at an infinite set of points along the imaginary energy axis, to the whole of the complex energy plane. This is useful in determining the poles of the Green functions.
It is shown that certain analytical properties of the propagators of many-fermion systems lead rigorously to the existence of sharp discontinuities of the momentum distribution at absolute zero. This discontinuity in the momentum distribution is used to define a Fermi surface for a system of interacting fermions. It is shown that the volume of this surface in momentum space is unaffected by the interaction. The same analytic properties are shown to lead, by direct statistical mechanical arguments, to simple expressions for the low-temperature heat capacity, the spin paramagnetism, and the compressibility of the system. These expressions are very analogous to the corresponding expressions for noninteracting particles. Finally, it is shown how the whole formalism may be generalized when an external periodic potential is present (band case).
We consider many-fermion systems with singular Fermi surfaces, which contain Van Hove points where the gradient of the band function k ↦ e(k) vanishes. In a previous paper, we have treated the case of spatial dimension d ≥ 3. In this paper, we focus on the more singular case d = 2 and establish properties of the fermionic self-energy to all orders in perturbation theory. We show that there is an asymmetry between the spatial and frequency derivatives of the self-energy. The derivative with respect to the Matsubara frequency diverges at the Van Hove points, but, surprisingly, the self-energy is C1 in the spatial momentum to all orders in perturbation theory, provided the Fermi surface is curved away from the Van Hove points. In a prototypical example, the second spatial derivative behaves similarly to the first frequency derivative. We discuss the physical significance of these findings.
Low-temperature properties of 3D interacting fermion systems are well described by Landau's theory of Fermi liquids; for reviews see, e.g. [183, 184]. It involves only excitations of the system around the Fermi surface on energy scales small compared to the Fermi energy. Excitations are well described by quasi-particles which are in one-to-one correspondence with the bare particles. The bare-particle interaction does not break the qualitative picture of non-interacting system, but renormalizes the dynamical characteristics (the effective mass and the pair interaction) of quasi-particles. Within the microscopic Green function formalism, the existence of quasi-particles is equivalent to assuming that the self-energy correction Σ( k , ω) is regular (has only short-range contributions in time and space) close to the Fermi surface. The lifetime of quasi-particles τ α(ε k −ε F ) −2 is long enough to consider them as well-defined eigenstates over long time-scales. The momentum occupation number of the bare particles exhibits a sharp discontinuity when crossing the Fermi momentum. The charge and spin degrees of freedom of quasi-particle always travel together. The Landau theory breaks down in 1D systems of interacting fermions which have a very specific Fermi surface consisting of two points ± k F . In such systems, the self-energy Σ( k , ω) no longer possesses the analytic properties required for introducing quasi-particles. In contrast to Fermi liquids, the momentum occupation number of the bare particles in the ground state is continuous at the Fermi momentum. Moreover, the charge and spin excitations are separated.
The ground-state energy and low-energy excitations of single-particle character of an infinite medium of fermions are discussed with the aid of time-dependent Green's functions, which are convenient generalizations of the exact particle correlation functions in the ground state of NNN particles. The power series development for the one- and two-particle functions, under the restriction to two-body forces, is derived and described by means of Feynman diagrams. The derivation of the linked-cluster expansion for the energy then follows immediately. The equivalence to previous versions is established. The one-particle function is examined in particular detail, and it is shown that the poles of its space-time Fourier transform studied as a function of the energy variable, for fixed momentum, determined the (N+1N+1N+1)-particle and ($N\ensuremath{-}1$)-particle excited states which have single-particle character. For a reasonable assumption about the full spectrum of excited states, it is found that for the interacting system, single-particle excitations with a real energy occur only at the Fermi momentum. It is pointed out that the corresponding energy, termed the perturbed Fermi energy, equals the binding energy per particle in the ground state of NNN particles for a saturating system at equilibrium density.It is shown, finally, that the entire structure of the theory may be carried over to the case of finite temperature, requiring only a redefinition of the Green's functions. The analogy is constructed from a discussion of the internal energy.