Unphysical and physical solutions in many-body theories: from weak to strong correlation

Stan, Adrian; Romaniello, Pina; Rigamonti, Santiago; Reining, Lucia; Berger, J. A.

doi:10.1088/1367-2630/17/9/093045

Cited by 71 publications

(77 citation statements)

References 43 publications

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“…We therefore expect that vertex anomalies should also be found in this case, consistent with their interpretation of precursors of the Mott transition. We also note that the recent analysis of the Hedin's equations in the zero dimensional limit 32 provides further support to this speculation.…”

Section: Distributed Disordersupporting

confidence: 64%

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Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

et al. 2016

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We analyze the highly non-perturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory (DMFT) calculations at the two-particle level. By extending the results of Schäfer, et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. By comparing our numerical data for the Hubbard model with analytical calculations for exactly solvable systems of increasing complexity [disordered binary mixture (BM), Falicov-Kimball (FK) and atomic limit (AL)], we have (i) identified two different kinds of divergence lines; (ii) classified them in terms of the frequency-structure of the associated singular eigenvectors; (iii) investigated their relation to the emergence of multiple branches in the Luttinger-Ward functional. In this way, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, single energy scale ν * below which perturbation theory is no longer applicable, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the non-perturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.

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Section: Distributed Disordersupporting

confidence: 64%

“…Note that the same phase-diagram is also applicable to the frequency-localized divergences of the FK model. studies 24,26,32 at T = 0. Please note that W < W MIT = 1, which supports the physical interpretation 24,26 of the vertex divergences as a precursor of the MIT.…”

Section: The Irreducible Vertex γC For the Bmmentioning

confidence: 99%

Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

et al. 2016

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“…Their results were soon connected to earlier numerical evidence of possible pathologies in the diagrammatic approach by Schäfer and collaborators [6,7]. First investigations on an entirely analytical level were carried out by Stan and collaborators [8], and Rossi and Werner [9], who independently managed to qualitatively reproduce the results of Kozik et al using two analytically treatable toy models. Those models, however, only bear the algebraic structure of the original quantum mechanical problem and cannot be directly related to a Hamiltonian.…”

Section: Introductionmentioning

confidence: 86%

“…As mentioned in [8], for zero temperature, equilibrium Green's functions G, the Hohenberg-Kohn theorem, which connects the external potential vext to the corresponding (non-degenerate) ground-state wavefunction Ψ (link 1) and to the density n (link 2), guarantees that there is a oneto-one correspondence between vext and G, in the following way: if we know vext, we can build Ψ (link 1) and hence G (link 3); on the other hand, if we know G we then know n (link 4) and hence vext (links 2,1). Since such a correspondence {vext} ↔ {G} holds also in the non-interacting case {vext} ↔ {G0}, it follows that {G0} ↔ {G}.…”

Section: B On the Self-consistent Dyson Equationmentioning

confidence: 93%

Self-consistent Dyson equation and self-energy functionals: An analysis and illustration on the example of the Hubbard atom

et al. 2017

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Perturbation theory using self-consistent Green's functions is one of the most widely used approaches to study many-body effects in condensed matter. On the basis of general considerations and by performing analytical calculations for the specific example of the Hubbard atom, we discuss some key features of this approach. We show that when the domain of the functionals that are used to realize the map between the non-interacting and the interacting Green's functions is properly defined, there exists a class of self-energy functionals for which the self-consistent Dyson equation has only one solution, which is the physical one. We also show that manipulation of the perturbative expansion of the interacting Green's function may lead to a wrong self-energy as functional of the interacting Green's function, at least for some regions of the parameter space. These findings confirm and explain numerical results of Kozik et al. for the widely used skeleton series of Luttinger and Ward [Phys. Rev. Lett. 114, 156402]. Our study shows that it is important to distinguish between the maps between sets of functions and the functionals that realize those maps. We demonstrate that the self-consistent Green's functions approach itself is not problematic, whereas the functionals that are widely used may have a limited range of validity.

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“…The reason for this can be traced back to the occurrence of singularities in the generalized susceptibilities of these (secondary) channels. Such singularities are reflected in the corresponding divergencies of the two-particle irreducible vertex functions, recently discovered in the DMFT solution of the Hubbard and Falicov-Kimball models [48][49][50][51][52][53][54] . Here we extend the study of their origin and generalize earlier results 48 to DCA.…”

Section: -47mentioning

confidence: 98%

Parquet decomposition calculations of the electronic self-energy

et al. 2016

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The parquet decomposition of the self-energy into classes of diagrams, those associated with specific scattering processes, can be exploited for different scopes. In this work, the parquet decomposition is used to unravel the underlying physics of non-perturbative numerical calculations. We show the specific example of dynamical mean field theory (DMFT) and its cluster extensions (DCA) applied to the Hubbard model at half-filling and with hole doping: These techniques allow for a simultaneous determination of two-particle vertex functions and self-energies, and hence, for an essentially "exact" parquet decomposition at the single-site or at the cluster level. Our calculations show that the self-energies in the underdoped regime are dominated by spin scattering processes, consistent with the conclusions obtained by means of the fluctuation diagnostics approach [Phys. Rev. Lett. 114, 236402 (2015)]. However, differently from the latter approach, the parquet procedure displays important changes with increasing interaction: Even for relatively moderate couplings, well before the Mott transition, singularities appear in different terms, with the notable exception of the predominant spin-channel. We explain precisely how these singularities, which partly limit the utility of the parquet decomposition, and -more generally -of parquet-based algorithms, are never found in the fluctuation diagnostics procedure. Finally, by a more refined analysis, we link the occurrence of the parquet singularities in our calculations to a progressive suppression of charge fluctuations and the formation of an RVB state, which are typical hallmarks of a pseudogap state in DCA.

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Unphysical and physical solutions in many-body theories: from weak to strong correlation

Cited by 71 publications

References 43 publications

Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

Nonperturbative landscape of the Mott-Hubbard transition: Multiple divergence lines around the critical endpoint

Self-consistent Dyson equation and self-energy functionals: An analysis and illustration on the example of the Hubbard atom

Parquet decomposition calculations of the electronic self-energy

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