2007/11/06 by Behnam Farid, Farid, Behnam
Chemistry · Physics and Astronomy · #Advanced Physical and Chemical Molecular Interactions #Advanced Thermodynamics and Statistical Mechanics #Cold Atom Physics and Bose-Einstein Condensates #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci) #Mathematical Physics (math-ph) #Scientific Research and Discoveries #Strongly Correlated Electrons (cond-mat.str-el) #Superconductivity (cond-mat.supr-con)
paper · pdf · doi:10.48550/arxiv.0711.0952
openalex publication_date 2007/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We analyze the original proof by Luttinger and Ward of the Luttinger theorem,\naccording to which for uniform ground states of systems of (interacting)\nfermions, which may be metallic or insulating, the number of k points\ncorresponding to non-negative values of Gs(k;mu) is equal to the total number\nof particles with spin index s in these ground states. Here Gs(k;mu) is the\nsingle-particle Green function of particles with spin index s at the chemical\npotential mu. For the cases where the two-body interaction potential is\nshort-range, and in particular for lattice models, we explicitly demonstrate\nthat this theorem is unconditionally valid, irrespective of the strength of the\nbare interaction potential. We arrive at this conclusion by amongst other\nthings demonstrating that the perturbation series expansion for self-energy in\nterms of skeleton diagrams, as encountered in the proof of the Luttinger-Ward\nidentity, is uniformly convergent for almost all momenta and energies. We\nfurther investigate the mechanisms underlying some reported instances of\nfailure of the Luttinger theorem. With one exception, for all the cases\nconsidered in this paper, we show that the apparent failures of the Luttinger\ntheorem can be attributed either to shortcomings of the employed\nsingle-particle Green functions or to misapplication of this theorem. The one\nexceptional case brings to light the possibility of a genuine failure of the\nLuttinger theorem for insulating ground states, which we show to be brought\nabout by a false limit that in principle can be reached on taking the\nzero-temperature limit without the value of mu coinciding with the\nzero-temperature limit of the chemical potential satisfying the equation of\nstate at finite temperatures; no such ambiguity can arise for metallic states.\n