Absence of Luttinger’s Theorem due to Zeros in the Single-Particle Green Function

The Luttinger-Ward functional Φ [G], which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function G, is found to be ill-defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy Σ[G] ∝ δΦ[G]/δG is not a single-valued functional of G: in addition to the physical solution for Σ [G], there exists at least one qualitatively distinct unphysical branch. This result is demonstrated for several models: the Hubbard atom, the Anderson impurity model, and the full two-dimensional Hubbard model. Despite this pathology, the skeleton Feynman diagrammatic series for Σ in terms of G is found to converge at least for moderately low temperatures. However, at strong interactions, its convergence is to the unphysical branch. This reveals a new scenario of breaking down of diagrammatic expansions. In contrast, the bare series in terms of the non-interacting Green's function G0 converges to the correct physical branch of Σ in all cases currently accessible by diagrammatic Monte Carlo. Besides their conceptual importance, these observations have important implications for techniques based on the explicit summation of diagrammatic series.

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“…in Refs. [34][35][36]. In the latter, it was found that δΣ[G]/δG diverges at certain discrete points in the parameter space.…”

Section: The Use Of Functionals ω[G] φ[G] σ[G]mentioning

confidence: 98%

Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

2015

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“…Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1 − ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et A fundamental property of a Landau Fermi liquid is captured by Luttinger's theorem [1], according to which the volume occupied by the Fermi sea, appropriately defined, remains invariant when the interaction is switched on, so long as no phase boundary is crossed. A violation of this theorem signifies non-Fermi liquid behavior, which has motivated investigations [2][3][4][5][6][7] of its applicability for various strongly correlated systems, such as high-temperature superconductors. This Letter investigates the Luttinger theorem for an exotic emergent Fermi sea.…”

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confidence: 99%

Luttinger Theorem for the Strongly Correlated Fermi Liquid of Composite Fermions

2015

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While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1 − ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.

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“…This feature is correct in the atomic limit of the Hubbard model 7 , but in general the electron self energy should depend on the momentum whenever t/U is non-zero. Nevertheless, since we are mainly interested in the Mott insulating state in which t/U is a small parameter, the momentum-dependence of the electron self energy should be very weak in the Mott insulating state, which has been discussed in previous studies [31][32][33][34][35][36] . As a result, the predicted re-entry of the non-interacting QPIs at a critical bias voltage should be robust even as the momentum dependence of the electron self energy is considered.…”

Section: As a Result T (ω) Can Be Evaluated Frommentioning

confidence: 99%

Characterizing featureless Mott insulating state by quasiparticle interference: A dynamical mean field theory view

Mukherjee

Lee

2015

Phys. Rev. B

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The quasiparticle interferences (QPIs) of the featureless Mott insulators are investigated by a T -matrix formalism implemented with the dynamical mean-field theory (T -DMFT). In the Mott insulating state, due to the singularity at zero frequency in the real part of the electron self energy (ReΣ(ω) ∼ η/ω) predicted by DMFT, where η can be considered as the 'order parameter' for the Mott insulating state, QPIs are completely washed out at small bias voltages. However, a further analysis shows that ReΣ(ω) serves as an energy-dependent chemical potential shift. As a result, the effective bias voltage seen by the system is eV ′ = eV − ReΣ(eV ), which leads to a critical bias voltage eVc ∼ √ η satisfying eV ′ = 0 if and only if η is non-zero. Consequently, the same QPI patterns produced by the non-interacting Fermi surfaces appears at this critical bias voltage eVc in the Mott insulating state. We propose that this re-entry of non-interacting QPI patterns at eVc could serve as an experimental signature of the Mott insulating state, and the 'order parameter' can be experimentally measured as η ∼ (eVc)2 .

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Absence of Luttinger’s Theorem due to Zeros in the Single-Particle Green Function

Cited by 65 publications

References 23 publications

Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

Nonexistence of the Luttinger-Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models

Luttinger Theorem for the Strongly Correlated Fermi Liquid of Composite Fermions

Characterizing featureless Mott insulating state by quasiparticle interference: A dynamical mean field theory view

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