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arXiv:cond-mat/0304350 · The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

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

Behnam Farid

Published Jan 11, 2004• 5 citations
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Abstract

We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The 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 for k in the vicinity of 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, and (here and denote vectors infinitesimally close to , 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 ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (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 does not need to be equal to ½ for ; 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 analysing the 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.

DOI
10.1080/14786430310001613237

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We analyze the original proof by Luttinger and Ward of the Luttinger theorem, according to which for uniform ground states of systems of (interacting) fermions, which may be metallic or insulating, the number of k points corresponding to non-negative values of G_s(k;mu) is equal to the total number of particles with spin index s in these ground states. Here G_s(k;mu) is the single-particle Green function of particles with spin index s at the chemical potential mu. For the cases where the two-body interaction potential is short-range, and in particular for lattice models, we explicitly demonstrate that this theorem is unconditionally valid, irrespective of the strength of the bare interaction potential. We arrive at this conclusion by amongst other things demonstrating that the perturbation series expansion for self-energy in terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward identity, is uniformly convergent for almost all momenta and energies. We further investigate the mechanisms underlying some reported instances of failure of the Luttinger theorem. With one exception, for all the cases considered in this paper, we show that the apparent failures of the Luttinger theorem can be attributed either to shortcomings of the employed single-particle Green functions or to misapplication of this theorem. The one exceptional case brings to light the possibility of a genuine failure of the Luttinger theorem for insulating ground states, which we show to be brought about by a false limit that in principle can be reached on taking the zero-temperature limit without the value of mu coinciding with the zero-temperature limit of the chemical potential satisfying the equation of state at finite temperatures; no such ambiguity can arise for metallic states.

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We demonstrate that the skeleton of the Fermi surface pertaining to a uniform metallic ground state (corresponding to fermions with spin index σ) is determined by the Hartree–Fock contribution to the dynamic self-energy . That is to say, in order for , it is necessary (but for anisotropic ground states in general not sufficient) that the following equation be satisfied: where stands for the underlying non-interacting energy dispersion and ϵF for the exact interacting Fermi energy. The Fermi surface consists of the set of k points which in addition to satisfying the above equation fulfil where1 The 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 for k in the vicinity of 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, and (here and denote vectors infinitesimally close to , 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 ). This aspect is borne out by the pertaining to the normal states of for instance liquid 3He (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 does not need to be equal to ½ for ; 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 analysing the 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.
It has long since been argued that the metallic states of the single-band Hubbard Hamiltonian Ĥ in two spatial dimensions (i.e. for d = 2) should be non-Fermi liquid, a possibility that would lead to the understanding of the observed anomalous behaviour of the doped copper-oxide-based superconducting compounds in their normal metallic states. Here we present a formalism which enables us to express, for arbitrary d, the behaviour of the momentum-distribution function nσ(k) pertaining to uniform metallic ground states of Ĥ close to S F; σ (the Fermi surface of the fermions with spin index σ, σ = ↑, ↓) in terms of a small number of constant parameters which are bound to satisfy certain inequalities implied by the requirement of the stability of the ground state of the system. These inequalities restrict the range of variation in nσ(k) for k infinitesimally inside and outside the Fermi sea pertaining to fermions with spin index σ and consequently the range of variation in the zero-temperature limit of nσ(k) for k on S F; σ On the basis of some available accurate numerical results for nσ(k) pertaining to the Hubbard and the t-J Hamiltonian, we conclude that, at least in the strong-coupling regime, the metallic ground states of Ĥ for d = 2 cannot be Fermi-liquid nor can they in general be purely Luttinger or marginal Fermi liquids. We further rigorously identify the pseudogap phenomenon, or 'truncation' of the Fermi surface, clearly observed in the normal states of underdoped copper-oxide-based superconductors, as corresponding to a line of resonance energies (i.e. these energies strictly do not relate to quasiparticles) located below the Fermi energy, with a concomitant suppression to zero of the jump in nσ(k) over the 'truncated' parts of the Fermi surface. Our analyses make explicit the singular significance of the non-interacting energy dispersion ε k underlying Ĥ in determining the low-energy spectral and transport properties of the metallic ground states of Ĥ.
We reproduce and review some of the main results of three of our earlier papers, utilizing in doing so a considerably more transparent formalism than originally utilized. The most fundamental result to which we pay especial attention in this paper, is that the exact Fermi surface (FS) of the uniform metallic ground state (GS) of any single-band Hamiltonian, describing fermions, is a subset of the FS within the framework of the exact Hartree-Fock theory. We also review some of the physical implications of the latter result. Our considerations reveal that the interacting FS of a uniform metallic GS cannot be calculated exactly to order ν(ν\ge 2) in the coupling constant λof the interaction potential in terms of the self-energy calculated to order νin a non-self-consistent fashion. We show this to be interlinked with the failure of the Luttinger-Ward identity, and thus of the Luttinger theorem, for a self-energy that is not appropriately related to the single-particle Green function from which the FS is deduced. We further show that the same mechanism that embodies the Luttinger theorem within the framework of the exact theory, accounts for a non-trivial dependence of the exact self-energy on λthat cannot be captured within a non-self-consistent framework. We thus establish that the extant calculations that purportedly prove deformation of the interacting FS of the metallic GS of the single-band Hubbard Hamiltonian with respect to its Hartree-Fock counterpart at the second order in the on-site interaction energy U, are fundamentally deficient. In an appendix we show that the number-density distribution function, to be distinguished from the site-occupation distribution function, corresponding to the GS of the Hubbard Hamiltonian is not non-interacting v-representable, a fact established earlier numerically. [Abridged Abstract]
The ground state of the Hubbard model is studied within the constrained Hilbert space where no order parameter exists.The self-energy of electrons is decomposed into the single-site and multisite self-energies.The calculation of the single-site self-energy is mapped to a problem of self-consistently determining and solving the Anderson model.When an electron reservoir is explicitly considered, it is proved that the single-site self-energy is that of a normal Fermi liquid even if the multisite self-energy is anomalous.Thus, the ground state is a normal Fermi liquid in the supreme single-site approximation (S 3 A).In the strong-coupling regime, the Fermi liquid is stabilized by the Kondo effect in the S 3 A and is further stabilized by the Fock-type term of the superexchange interaction or the resonating-valence-bond (RVB) mechanism beyond the S 3 A. The stabilized Fermi liquid is frustrated as much as an RVB spin liquid in the Heisenberg model.It is a relevant unperturbed state that can be used to study a normal or anomalous Fermi liquid and an ordered state in the whole Hilbert space by Kondo lattice theory.Even if higher-order multisite terms than the Fock-type term are considered, the ground state cannot be a Mott insulator.It can be merely a gapless semiconductor even if the multisite self-energy is so anomalous that it is divergent at the chemical potential.A Mott insulator is only possible as a high temperature phase.
We have studied the pseudogap formation in the single-particle spectra of the half-filling two-dimensional Hubbard model. Using a Green's function with the one-loop self-energy correction of the spin and charge fluctuations, we have numerically calculated the self-energy, the spectral function, and the density of states in the weak-coupling regime at finite temperature. Pseudogap formations have been observed in both the density of states and the spectral function at the Fermi level. The pseudogap in the spectral function is explained by the non-Fermi-liquid-like nature of the self-energy. The anomalous behavior in the self-energy is caused by both the strong antiferromagnetic spin fluctuation and the nesting condition on the non-interacting Fermi surface. In the present approximation, we find a logarithmic singularity in the integrand of the imaginary part of the self-energy. Concerning the energy dependence of the spectral function and the self-energy, two theorems are proved. They give a necessary condition in the self-energy to produce the pseudogap at the Fermi level. The pseudogap in the spectral function is highly momentum dependent on the Fermi surface. It opens initially in the (±π,0)(\pm π,0)(±π,0), (0,±π)(0,\pm π)(0,±π) regions as the normal state pseudogap observed in the high-TcT_cTc​ superconductors and if the interaction is increased, it spreads to other Fermi surface sectors. The anisotropy of the pseudogap is produced by the low-energy enhancement of the spin excitation around Q=(π,π){\bf Q}=(π,π)Q=(π,π) and the flatness of the band dispersion around the saddle point.
The ground state of the Hubbard model is studied within the constrained Hilbert space where no order parameter exists. The self-energy of electrons is decomposed into the single-site and multisite self-energies. The calculation of the single-site self-energy is mapped to a problem of self-consistently determining and solving the Anderson model. When an electron reservoir is explicitly considered, it is proved that the single-site self-energy is that of a normal Fermi liquid even if the multisite self-energy is anomalous. Thus, the ground state is a normal Fermi liquid in the supreme single-site approximation (S3A). In the strong-coupling regime, the Fermi liquid is stabilized by the Kondo effect in the S3A and is further stabilized by the Fock-type term of the superexchange interaction or the resonating-valence-bond (RVB) mechanism beyond the S3A. The stabilized Fermi liquid is frustrated as much as an RVB spin liquid in the Heisenberg model. It is a relevant unperturbed state that can be used to study a normal or anomalous Fermi liquid and an ordered state in the whole Hilbert space by Kondo lattice theory. Even if higher-order multisite terms than the Fock-type term are considered, the ground state cannot be a Mott insulator. It can be merely a gapless semiconductor even if the multisite self-energy is so anomalous that it is divergent at the chemical potential. A Mott insulator is only possible as a high temperature phase.
We analyze the original proof by Luttinger and Ward of the Luttinger theorem, according to which for uniform ground states of systems of (interacting) fermions, which may be metallic or insulating, the number of k points corresponding to non-negative values of G_s(k;mu) is equal to the total number of particles with spin index s in these ground states. Here G_s(k;mu) is the single-particle Green function of particles with spin index s at the chemical potential mu. For the cases where the two-body interaction potential is short-range, and in particular for lattice models, we explicitly demonstrate that this theorem is unconditionally valid, irrespective of the strength of the bare interaction potential. We arrive at this conclusion by amongst other things demonstrating that the perturbation series expansion for self-energy in terms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward identity, is uniformly convergent for almost all momenta and energies. We further investigate the mechanisms underlying some reported instances of failure of the Luttinger theorem. With one exception, for all the cases considered in this paper, we show that the apparent failures of the Luttinger theorem can be attributed either to shortcomings of the employed single-particle Green functions or to misapplication of this theorem. The one exceptional case brings to light the possibility of a genuine failure of the Luttinger theorem for insulating ground states, which we show to be brought about by a false limit that in principle can be reached on taking the zero-temperature limit without the value of mu coinciding with the zero-temperature limit of the chemical potential satisfying the equation of state at finite temperatures; no such ambiguity can arise for metallic states.