On the (anisotropic) uniform metallic ground states of fermions interacting through arbitrary two-body potentials inddimensions

Abstract

We demonstrate that the skeleton of the Fermi surface Sf;σ pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree-Fock contribution Σ hf σ (k) to the dynamic self-energy Σσ(k; ε). That is to say, in order for k ∈ Sf;σ, it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied:where ε k stands for the underlying non-interacting energy dispersion and εf for the exact interacting Fermi energy. The Fermi surface Sf;σ consists of the set of k points which in addition to satisfying the above equation fulfilwhereThe set of k points which satisfy the first of the above two equations but fail to satisfy the second constitute the pseudogap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum-distribution function nσ(k) for k in the vicinity of Sf;σ and show that, whereas for the uniform metallic ground states of the conventional single-band Hubbard Hamiltonian, described in terms of an on-site interaction, nσ(k − f;σ ) ≥ 1 2 and nσ(k + f;σ ) ≤ 1 2 (here k − f;σ and k + f;σ denote vectors infinitesimally close to Sf;σ, located respectively inside and outside the underlying Fermi sea), for interactions of non-zero range these inequalities can be violated (without thereby contravening the stability condition nσ(k − f;σ ) ≥ nσ(k + f;σ )). This aspect is borne out by the nσ(k) pertaining to the normal states of for instance liquid 3 He (corresponding to a range of applied pressure) as determined by means of quantum Monte Carlo calculations. We further demonstrate that for Fermi-liquid metallic states of fermions interacting through interaction potentials of non-zero range (e.g. the Coulomb potential), the zero-temperature limit of nσ(k) does not need to be equal to 1 2 for k ∈ Sf;σ; this in strict contrast with the uniform Fermi-liquid metallic states of the single-band Hubbard Hamiltonian (if such states at all exist). This aspect should be taken into account while analyzing the nσ(k) deduced from the angle-resolved photoemission data concerning real materials. We discuss, in the light of the findings of the present work, the growing experimental evidence with regard to the 'frustration' of the kinetic energy of the charge carriers in the normal states of the copper-oxide-based high-temperature superconducting compounds.

On the break in the single-particle energy dispersions and the ‘universal’ nodal Fermi velocity in the high-temperature copper oxide superconductors

“…(23)) and ε + k;σ concerning k outside the Fermi sea. For cases where the two-particle interaction potential is the long-range Coulomb potential, in [10] it was however shown that no such strict relationship exists between ε respectively for cases in which either or both of ε < k;σ and ε > k;σ are discontinuous at k = k ⋆ ; according to our analysis ( § 3.4), these functions cannot both be continuous at k = k ⋆ .…”

“…The formalism concerning the measured energy dispersions that follows from our purely heuristic arguments turns out to coincide, in all details, with that developed by the present author in his recent investigations regarding the nature of the uniform metallic GSs of the conventional single-band Hubbard Hamiltonian [9] and more general single-band Hamiltonians [10] in which the particle-particle interaction can be of arbitrary range. The specific aspects of the formalism developed in [9,10] that greatly suited the aims in the aforementioned investigations are that for the indicated GSs it formally yields (i) the exact GS total energy, (ii) the exact Fermi energy, (iii) the exact Fermi surface and (iv) the exact momentum distribution function. The formalism has made possible [9,10] to make exact predictions concerning the values that the GS momentum distribution function n σ (k) can take for k in the close neighbourhoods of the underlying Fermi surface S f;σ .…”

“…For the normal uniform metallic GSs of the single-band Hubbard Hamiltonian, we have established that, so long as these are Fermi liquids, for k inside the Fermi sea and close to the Fermi surface, a single-particle energy dispersion ε < k;σ , which in [9,10] was attributed to some 'fictitious' particles and in this paper we reason to be the one measured through the ARPES, is up to a multiplicative k-independent constant equal to the energy dispersion ε k;σ of the Landau quasi-particles inside the Fermi sea (more precisely, the energy dispersion ε − k;σ as obtained from the quasi-particle equation in Eq. (3) below; see § 3.1); similarly for ε > k;σ (Eq.…”

“…The formalism has made possible [9,10] to make exact predictions concerning the values that the GS momentum distribution function n σ (k) can take for k in the close neighbourhoods of the underlying Fermi surface S f;σ . On the basis of this, it has been possible to expose [10] the significant influence that the range of the two-body interaction potential can have on n σ (k) and consequently on the partitioning of the GS total energy into 'kinetic' and 'interaction' parts. This observation led us to the conclusion [10] that the experimentally-inferred excess 'kinetic' energy in the normal states of the copper-oxide based high-temperature superconductors [11,12] (for other pertinent references see [10]) may be signifying the importance of the long range of the two-body interaction potential in determining other distinctive aspects of the normal metallic states of these compounds that mark them as unconventional metals [13].…”

“…The specific aspects of the formalism developed in [9,10] that greatly suited the aims in the aforementioned investigations are that for the indicated GSs it formally yields (i) the exact GS total energy, (ii) the exact Fermi energy, (iii) the exact Fermi surface and (iv) the exact momentum distribution function. The formalism has made possible [9,10] to make exact predictions concerning the values that the GS momentum distribution function n σ (k) can take for k in the close neighbourhoods of the underlying Fermi surface S f;σ . On the basis of this, it has been possible to expose [10] the significant influence that the range of the two-body interaction potential can have on n σ (k) and consequently on the partitioning of the GS total energy into 'kinetic' and 'interaction' parts.…”