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Physical Review

Fermi Surface and Some Simple Equilibrium Properties of a System of Interacting Fermions

J. M. Luttinger

Published Aug 15, 1960• 1,323 citations
Publisher

Abstract

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).

DOI
10.1103/physrev.119.1153

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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).
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.
An exactly soluble model of a one-dimensional many-fermion system is discussed. The model has a fairly realistic interaction between pairs of fermions. An exact calculation of the momentum distribution in the ground state is given. It is shown that there is no discontinuity in the momentum distribution in this model at the Fermi surface, but that the momentum distribution has infinite slope there. Comparison with the results of perturbation theory for the same model is also presented, and it is shown that, for this case at least, the perturbation and exact answers behave qualitatively alike. Finally, the response of the system to external fields is also discussed.
An exactly soluble model of a one‐dimensional many‐fermion system is discussed. The model has a fairly realistic interaction between pairs of fermions. An exact calculation of the momentum distribution in the ground state is given. It is shown that there is no discontinuity in the momentum distribution in this model at the Fermi surface, but that the momentum distribution has infinite slope there. Comparison with the results of perturbation theory for the same model is also presented, and it is shown that, for this case at least, the perturbation and exact answers behave qualitatively alike. Finally, the response of the system to external fields is also discussed.
We derive rigorous bounds on the average momentum occupation numbers ⟨nkσ⟩\langle n_{\mathbf{k}σ}\rangle⟨nkσ​⟩ in the Hubbard and Kondo models in the ground state and at non-zero temperature (T>0T>0T>0) in the grand canonical ensemble. For the Hubbard model with T>0T>0T>0 our bound proves that, when interaction strength ≪kBT≪\ll k_B T\ll≪kB​T≪ Fermi energy, ⟨nkσ⟩\langle n_{\mathbf{k}σ}\rangle⟨nkσ​⟩ is guaranteed to be close to its value in a low temperature free fermion system. For the Kondo model with any T>0T>0T>0 our bound proves that ⟨nkσ⟩\langle n_{\mathbf{k}σ}\rangle⟨nkσ​⟩ tends to its non-interacting value in the infinite volume limit. In the ground state case our bounds instead show that ⟨nkσ⟩\langle n_{\mathbf{k}σ}\rangle⟨nkσ​⟩ approaches its non-interacting value as k\mathbf{k}k moves away from a certain surface in momentum space. For the Hubbard model at half-filling on a bipartite lattice, this surface coincides with the non-interacting Fermi surface. In the Supplemental Material we extend our results to some generalized versions of the Hubbard and Kondo models. Our proofs use the Fermi statistics of the particles in a fundamental way.
In periodic systems of interacting electrons, Fermi and Luttinger surfaces refer to the locations within the Brillouin zone of poles and zeros, respectively, of the single-particle Green's function at zero energy and temperature. Such difference in analytic properties underlies the emergence of well-defined quasiparticles close to a Fermi surface, in contrast to their supposed non-existence close to a Luttinger surface, where the single-particle density-of-states vanishes at zero energy. We here show that, contrary to such common belief, coherent `quasiparticles' do exist also approaching a Luttinger surface in compressible interacting electron systems. Thermodynamic and dynamic properties of such 'quasiparticles' are just those of conventional ones. For instance, they yield well defined quantum oscillations in Luttinger's surface and linear in temperature specific heat, which is striking given the vanishing density of states of physical electrons, but actually not uncommon in strongly correlated materials.
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 ${C}_{k}{(x\ensuremath{-}\ensuremath{\mu})}^{2}$ (Ck>0{C}_{k}>0Ck​>0) for xxx very near $\ensuremath{\mu}$, is proved.