arXiv subjects

Behnam Farid

Publications and source records attributed to Behnam Farid.

At least 19 recordsLinked to original sources

arXiv2021

Many-body perturbation expansions without diagrams. I. Normal states

On the basis of an exact perturbational expression for the interacting one-particle Green function $G$ corresponding to bosons / fermions in terms of the bare interaction potential $v$ and permanents / determinants of the non-interacting one-particle Green function $G_0$, we deduce four recursive perturbation expansions for the self-energy $Σ$. With $W$ denoting the dynamic screened interaction potential, these perturbation expansions are identical to those of $Σ$ in terms of (i) self-energy diagrams and $(v, G_0)$, (ii) $G$-skeleton self-energy diagrams and $(v, G)$, (iii) $W$-skeleton self-energy diagrams and $(W, G_0)$, and (iv) $G$- and $W$-skeleton self-energy diagrams and $(W,G)$. For the calculation of $W$, we rely on a similar exact perturbational expression for the interacting two-particle Green function $G_2$ as for $G$. From this expression, we deduce four recursive perturbation expansions for the polarization function $P$, necessary for the calculation of $W$, that are similar to those for $Σ$ specified above. The correlation functions considered in this paper may be corresponding to ground states, and thermal ensemble of states. For thermal ensemble of states, we consider both the imaginary-time formalism of Matsubara, and the real-time formalism of thermo-field dynamics (TFD). The latter is advantageous for the direct calculation of dynamic correlation functions. In an appendix, we apply the formalisms presented in this paper to the Hubbard Hamiltonian for spin-$\tfrac{1}{2}$ fermions on a lattice in arbitrary $d$ spatial dimensions. In two further appendices, we present methods and short programs for determining the $ν$th-order diagrams corresponding to the perturbation expansions of $G$ in terms of $(v,G_0)$, and $Σ$ in terms of $(v,G_0)$ and $(v,G)$ on the basis of the cycle decompositions of the elements of the symmetric group $S_{2ν}$.

cond-mat.str-el↗

arXiv2021

On the Luttinger-Ward functional and the convergence of skeleton diagrammatic series expansion of the self-energy for Hubbard-like models

We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy $Σ(\mathbf{k};z)$ specific to uniform ground states (ensemble of states) of interacting fermion systems in terms of skeleton self-energy diagrams and the interacting Green function $G(\mathbf{k};z)$. Utilising a link between the latter series expansion and the classical moment problem (of the Hamburger type), along with the associated continued-fraction expansion, we reaffirm our earlier observation (2007) that for lattice models of fermions interacting through short-range two-body potentials (i.e. for Hubbard-like models) this series is uniformly convergent for almost all wave vectors $\mathbf{k}$ and complex energies $z$. The limit of this series is unique. We inquire into the reasons underlying the contrary observation by Kozik et al. (2015) regarding skeleton-diagrammatic perturbation series expansion of $Σ$. In doing so, we make a number of observations of general interest. Chief amongst these, we observe that contrary to general belief thermal correlation functions calculated in the energy domain (i.e. over the set of the relevant Matsubara frequencies) are generally unreliable, in contrast to those calculated in the imaginary-time domain. In an appendix we present a symbolic computational formalism and short programs for singling out topologically distinct proper self-energy diagrams that are algebraically equivalent up to determinate multiplicative constants. This complements the work presented in our previous publication (2019). [Abridged abstract]

cond-mat.str-el↗

arXiv2014

Comment on "Breaking the theoretical scaling limit for predicting quasi-particle energies: The stochastic GW approach", by Daniel Neuhauser et al., arXiv:1402.5035v1

We show that the recently-introduced formalism by Neuhauser et al. for the calculation of the quasi-particle energies of electronic systems within the framework of the GW approximation of the self-energy operator, named the `stochastic GW approach' and empirically shown to have a linear-scaling arithmetic complexity for increasing number of electrons, suffers from two fundamental shortcomings that cannot be overcome while maintaining the present empirical linear-scaling property of the approach.

cond-mat.mes-hall↗

arXiv2013

Some rigorous results concerning the uniform metallic ground states of single-band Hamiltonians in arbitrary dimensions

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]

cond-mat.str-el↗

arXiv2012

Comment on "Absence of Luttinger's Theorem", by Kiaran B. Dave, Philip W. Phillips and Charles L. Kane, arXiv:1207.4201

In this Comment, we first present general arguments showing that the absence of the Luttinger theorem (LT) for the SU(N) model of Dave, Phillips and Kane (DPK) is rooted in the non-uniqueness of the ground state (GS) of this model for 0 < n < N, where n denotes the number of particles in the GS; the validity of the Luttinger theorem for n = N/2, when N even, is accidental, a consequence of particle-hole symmetry. Consequently, by supplementing the Hamiltonian of the SU(N) model with a perturbation Hamiltonian that removes the GS degeneracy, the LT is to apply also for the SU(N) model in the limit of the coupling constant, λ, of this perturbation approaching zero, where the limit λ--> 0 is clearly to be taken subsequent to taking the zero-temperature limit of the thermal single-particle Green function in the expression for the Luttinger number N_L. We explicitly establish the validity of this statement for the case of N=4. The details of the relevant calculations being distinctly transparent, one can readily convince oneself that our observation is valid for arbitrary N. It follows that the issues raised by DPK, such as non-existence of the Luttinger-Ward functional and "breakdown of the elemental particle picture in strongly correlated electron matter", are all inessential to the observed failure of the LT. As regards the singularity of the self-energy Σ(ω) on the real ω-axis, observed by DPK, we demonstrate that this also is a direct consequence of the non-uniqueness of the GS of the SU(N) model for 0 < n < N. In the light of the above observations, we are in a position to state that to this date no case has come to light indicative of the failure of the LT under the conditions for which it has been deduced. [Abridged Abstract]

cond-mat.str-el↗

arXiv2009

Comment on "Breakdown of the Luttinger sum rule within the Mott-Hubbard insulator", by J. Kokalj and P. Prelovsek [Phys. Rev. B 78, 153103 (2008), arXiv:arXiv:0803.4468]

On the basis of an analysis of the numerical results corresponding to the half-filled 1D t-t'-V model on some finite lattices, Kokalj and Prelovsek (KP) have in a recent paper [Phys. Rev. B 78, 153103 (2008), arXiv:arXiv:0803.4468] concluded that the Luttinger theorem (LT) does not apply for the Mott-Hubbard (MH) insulating phase of this model (i.e. for V >> t) in the thermodynamic limit; KP even suggested, incorrectly, that failure of the LT were apparent for a half-filled finite system consisting of N=26 lattice sites. By employing a simple model for the self-energy Sigma of a MH state, we show that the finite-size-scaling approach of the type utilised by KP is not reliable for the system sizes considered by KP. On the basis of the equivalence of the model under consideration (at half-filling and for t'/t << 1) and the XXZ spin-chain Hamiltonian for SU(2) spins, we further show that for V > V_c(t,t') the system under consideration has a charge-density-wave (CDW) ground state (GS) in the thermodynamic limit, corresponding to a doubling of the unit cell in comparison with that specific to the underlying lattice. Although this GS is also insulating, its spectral gap is due to the broken translational symmetry of the GS; it is not a correlation-induced MH gap. The LT is therefore a priori valid for this GS. This fact establishes that the conclusion by KP is indeed erroneous. Finally, we present a heuristic argument due to Volovik that sheds light on the mechanism underlying the robustness of the LT. In an appendix, we present the details of the calculation of the single-particle Green function of the broken-symmetry GS of the model under consideration by means of bosonization and in terms of the form factors of a class of soliton-generating fields pertaining to the quantum sine-Gordon Hamiltonian. [Shortened abstract]

cond-mat.str-el↗

arXiv2009

Comment on "Violation of the Luttinger sum rule within the Hubbard model on a triangular lattice", by J. Kokalj and P. Prelovsek [Eur. Phys. J. B 63, 431 (2008), arXiv:arXiv:0709.0263]

Using the first-order series expansion of the function G(k;mu), in powers of mu' = mu - U/2, pertaining to the insulating ground state of a single-band Hubbard Hamiltonian at half-filling, Kokalj and Prelovsek have in a recent paper [Eur. Phys. J. B 63, 431 (2008), arXiv:arXiv:0709.0263] reported breakdown of the Luttinger theorem for the specific case where the lattice on which the Hubbard Hamiltonian is defined is a two-dimensional triangular lattice, for which the ground state is not invariant under particle-hole transformation. Here G(k;mu) is the single-particle Green function G(k;e) evaluated at e = mu, the zero-temperature limit of the chemical potential corresponding to half-filling, and U the on-site interaction energy. In this Comment we demonstrate that unless mu' = 0 (to be strictly distinguished from mu' small but non-vanishing), any finite-order series expansion for G(k;mu) in powers of mu' in general falsely signals breakdown of the Luttinger theorem. The violation of this theorem as asserted by Kokalj and Prelovsek is therefore an artifact of their first-order calculation.

cond-mat.str-el↗

arXiv2007

Reply to ``Comment on `On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions','' arXiv:0711.3093v1, by A. Rosch

We reply to the Comment by Achim Rosch [1] (arXiv:0711.3093v1) who challenges our finding in Ref. [2] (arXiv:0711.0952v1) with regard to the validity of the Luttinger theorem in the cases of Mott insulating N-particle ground states (even) when the chemical potential used in applying this theorem coincides with the zero-temperature limit of the chemical potential satisfying the equation of state corresponding to N particles. Rosch further argues that the strong-coupling expression for the single-particle Green function presented in Ref. [3] (arXiv:cond-mat/0602656v2) and analyzed in Ref. [2] does not imply destruction of the Mott insulating state at half-filling as a result of an arbitrary weak hopping contribution that breaks particle-hole symmetry and therefore suggests that our conclusion in Ref. [2] to the contrary were incorrect. Here we show the shortcomings of Rosch's arguments.

cond-mat.str-el↗

arXiv2007

On the Luttinger theorem concerning number of particles in the ground states of systems of interacting fermions

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.

cond-mat.str-el↗

arXiv2005

Composite fermions close to the one-half filling of the lowest Landau level revisited

By strictly adhering to the microscopic theory of composite fermions for the Landau-level filling fractions nu_e = p/(2 p + 1), we reproduce, with remarkable accuracy, the surface-acoustic-wave (SAW)-based experimental results by Willett and co-workers concerning two-dimensional electron systems with nu_e close to 1/2. Our results imply that the electron band mass m_b, as distinct from the composite fermion mass m_*, must undergo a substantial increase under the conditions corresponding to nu_e approximately equal to 1/2. In view of the relatively low aerial electronic densities n_e to which the underlying SAW experiments correspond, our finding conforms with the experimental results by Shashkin et al. [Phys. Rev. B 66, 073303 (2002)], concerning two-dimensional electrons in silicon, that signal sharp increase in m_b for n_e decreasing below approximately 2 x 10^{11} cm^{-2}. We further establish that a finite mean-free path l_0 is essential for the observed linearity of the longitudinal conductivity sigma_{xx}(q) as deduced from the SAW velocity shifts.

cond-mat.mes-hall↗

arXiv2004

Comment on ``Quasiparticle Anisotropy and Pseudogap Formation from the Weak-Coupling Renormalization Group Point of View''

In their recent Letter (Phys. Rev. Lett., Vol. 93, 106406 (2004)), Katanin and Kampf reported numerical results for the self-energy Sigma(k;e), at real values of e, of the single-band Hubbard Hamiltonian in two space dimensions, obtained through employing the functional renormalization-group (fRG) formalism at the one-loop level. Several of the results by Katanin and Kampf are in full conformity with the exact formal results reported earlier by the present author. This, as we shall elaborate in this contribution, strengthens one's confidence in the reliability of the fRG in dealing with models of strongly-correlated fermions.

cond-mat.str-el↗

arXiv2004

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

Recent data from angle-resolved photoemission experiments published by Zhou et al. [Nature, Vol. 423, 398 (2003)] concerning a number of hole-doped copper-oxide-based high-temperature superconductors reveal that in the nodal directions of the underlying square Brillouin zones (i.e. the directions along which the d-wave superconducting gap is vanishing) the Fermi velocities for some finite range of k inside the Fermi sea and away from the nodal Fermi wavevector k_F are to within an experimental uncertainty of approximately 20% the same both in all the compounds investigated and over a wide range of doping concentrations and that, in line with earlier experimental observations, at some characteristic wavevector k_* away from k_F the Fermi velocities undergo a sudden change, with this change (roughly speaking, a finite discontinuity) being the greatest (smallest) in the case of underdoped (overdoped) compounds. In this paper we present a rigorous analysis concerning the implications of these observations. [Short abstract]

cond-mat.supr-con↗

arXiv2004

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

We demonstrate that the skeleton of the Fermi surface S_{F;s} pertaining to a uniform metallic ground state (corresponding to fermions with spin index s) is determined by the Hartree-Fock contribution to the dynamic self-energy. The Fermi surface S_{F;s} consists of all points which in addition to satisfying the quasi-particle equation in terms of the Hartree-Fock self-energy, fulfill the equation S_{s}(k) = 0, where S_{s}(k) is defined in the main text; the set of k points which satisfy the Hartree-Fock quasi-particle equation but fail to satisfy S_{s}(k) = 0, constitute the pseudo-gap region of the putative Fermi surface of the interacting system. We consider the behaviour of the ground-state momentum-distribution function n_{s}(k) for k in the vicinity of S_{F;s} and show that whereas for the uniform metallic ground states of the conventional Hubbard Hamiltonian n_{s}(k) is greater/less than 0.5 for k approaching S_{F;s} from inside/outside the Fermi sea, for interactions of non-zero range these inequalities can be violated (without thereby contravening the condition of the non-negativity of the possible jump in n_{s}(k) on k crossing S_{F;s} from directly inside to directly outside the Fermi sea). 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. [Short abstract]

cond-mat.str-el↗

arXiv2003

The non-Fermi-liquid nature of the metallic states of the Hubbard Hamiltonian

We present a formalism which enables us to express, for arbitrary d, the behaviour of the momentum-distribution function n_{sigma}(k) pertaining to uniform metallic ground states of the single-band Hubbard Hamiltonian H close to S_{F;sigma} (the Fermi surface of the fermions with spin index sigma, sigma = uparrow,downarrow) 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 of n_{sigma}(k) for k infinitesimally inside and outside the Fermi sea pertaining to fermions with spin index sigma and consequently the range of variation of the zero-temperature limit of n_{sigma}(k) for k on S_{F;sigma}. On the basis of some available accurate numerical results for n_{sigma}(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 H for d=2 cannot be Fermi-liquid, nor can they in general be purely Luttinger- or marginal-Fermi liquids. We further rigorously identify the pseudo-gap phenomenon, or `truncation' of the Fermi surface, clearly observed in the normal states of under-doped copper-oxide superconductors, as corresponding to a line of resonance energies located below the Fermi energy, with a concomitant suppression to zero of the jump in n_{sigma}(k) over the `truncated' parts of the Fermi surface. [Short Abstract]

cond-mat.str-el↗

arXiv2002

Dynamical correlation functions expressed in terms of many-particle ground-state wavefunction; the dynamical self-energy operator

We present the explicit expressions for the (regularized) terms in the large-epsilon asymptotic series of in particular the self-energy operator pertaining to arbitrary systems of interacting spin-s fermions in d spatial dimensions and deduce a variety of exact results concerning the general properties of this operator. We give especial attention to systems of spin-1/2 fermions in d=3 interacting through the Coulomb potential. [Short Abstract]

cond-mat.str-el↗

arXiv2000

A Luttinger's theorem revisited

For uniform systems of spin-less fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential v(r-r'), a celebrated theorem due to Luttinger (1961) states that under the_assumption_ of validity of the many-body perturbation theory the self-energy Sigma(k;epsilon), with 0 <,= k <,~ 3 k_F (where k_F stands for the Fermi wavenumber), satisfies the following universal asymptotic relation as epsilon approaches the Fermi energy epsilon_F: Im[Sigma(k;epsilon)] ~ -,+ alpha_k (epsilon-epsilon_F)^2, epsilon >,< epsilon_F, with alpha_k >,= 0. As this is, by definition, specific to self-energies of Landau Fermi-liquid systems, treatment of non-Fermi-liquid systems are therefore thought to lie outside the domain of applicability of the many-body perturbation theory; that, for these systems, the many-body perturbation theory should_necessarily_ break down. We demonstrate that Im[Sigma(k;epsilon)] ~ -,+ alpha_k (epsilon-epsilon_F)^2, epsilon >,< epsilon_F, is_implicit_ in Luttinger's proof and that, for d > 1, in principle nothing prohibits a non-Fermi-liquid-type (and, in particular Luttinger-liquid-type) Sigma(k;epsilon) from being obtained within the framework of the many-body perturbation theory. We in addition indicate how seemingly innocuous Taylor expansions of the self-energy with respect to k, epsilon or both amount to tacitly assuming that the metallic system under consideration is a Fermi liquid, whether the self-energy is calculated perturbatively or otherwise. Proofs that a certain metallic system is in a Fermi-liquid state, based on such expansions, are therefore tautologies.

cond-mat.str-el↗

arXiv2000

The Haldane bosonisation scheme and metallic states of interacting fermions in d spatial dimensions

We consider the Haldane bosonisation scheme in d spatial dimensions as applied to a realistic model of interacting fermions in d=2 and unequivocally demonstrate failure of this scheme in d > 1, specifically in d=2. In addition to tracing back this failure to its origin, we show that {\sl nothing} as regards the true metallic state of the model under consideration is known with any degree of certainty.

cond-mat.str-el↗

arXiv2000

On Fermi systems with strong forward scattering in d spatial dimensions

We consider a specific generalisation to spatial dimensions d greater than one of a formalism based upon conservation laws and the associated Ward identities, that exactly solves the one-dimensional Luttinger model, and expose its inadequacies for d > 1. We conclude that findings arrived at through application of this generalised formalism concerning systems of fermions with strong forward scattering, in particular that d=1 would be the cross-over dimension from Luttinger- to Fermi-liquid state, are open to question.

cond-mat.str-el↗