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arXiv:2303.16732 · Physical review. B./Physical review. B

Dimensional reduction of the Luttinger-Ward functional for spin-degenerate D -dimensional electron gases

Dmitry Miserev, Jelena Klinovaja, Daniel Loss

Published Dec 4, 2023• 3 citations
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Abstract

We consider an isotropic spin-degenerate interacting uniform $D$-dimensional electron gas (DDEG) with $D>1$ within the Luttinger-Ward (LW) formalism. We derive the asymptotically exact semiclassical/infrared limit of the LW functional at large distances, $r\ensuremath{\gg}{\ensuremath{\lambda}}_{F}$, and large times, $\ensuremath{\tau}\ensuremath{\gg}1/{E}_{F}$, where ${\ensuremath{\lambda}}_{F}$ and ${E}_{F}$ are the Fermi wavelength and the Fermi energy, respectively. The LW functional is represented by skeleton diagrams, each skeleton diagram consists of appropriately connected dressed fermion loops. First, we prove that every $D$-dimensional skeleton diagram consisting of a single fermion loop is reduced to a one-dimensional (1D) fermion loop with the same diagrammatic structure, which justifies the name dimensional reduction. This statement, combined with the fermion loop cancellation theorem (FLCT), agrees with results of multidimensional bosonization. Here we show that the backscattering and the spectral curvature, both explicitly violate the FLCT and both are irrelevant for a 1DEG, become relevant at $D>1$ and $D>2$, respectively. The reason for this is a strong infrared divergence of the skeleton diagrams containing multiple fermion loops at $D>1$. These diagrams, which are omitted within the multidimensional bosonization approaches, account for the noncollinear scattering processes. Thus, the dimensional reduction provides the framework to go beyond predictions of the multidimensional bosonization. A simple diagrammatic structure of the reduced LW functional is another advantage of our approach. The dimensional reduction technique is also applicable to the thermodynamic potential and various approximations, from perturbation theory to self-consistent approaches.

DOI
10.1103/physrevb.108.235116

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We consider an isotropic spin-degenerate interacting uniform DDD-dimensional electron gas (DDEG) with D>1D>1D>1 within the Luttinger-Ward (LW) formalism. We derive the asymptotically exact semiclassical/infrared limit of the LW functional at large distances, r\ensuremath≫\ensuremathλFr\ensuremath{\gg}{\ensuremath{\lambda}}_{F}r\ensuremath≫\ensuremathλF​, and large times, \ensuremathτ\ensuremath≫1/EF\ensuremath{\tau}\ensuremath{\gg}1/{E}_{F}\ensuremathτ\ensuremath≫1/EF​, where \ensuremathλF{\ensuremath{\lambda}}_{F}\ensuremathλF​ and EF{E}_{F}EF​ are the Fermi wavelength and the Fermi energy, respectively. The LW functional is represented by skeleton diagrams, each skeleton diagram consists of appropriately connected dressed fermion loops. First, we prove that every DDD-dimensional skeleton diagram consisting of a single fermion loop is reduced to a one-dimensional (1D) fermion loop with the same diagrammatic structure, which justifies the name dimensional reduction. This statement, combined with the fermion loop cancellation theorem (FLCT), agrees with results of multidimensional bosonization. Here we show that the backscattering and the spectral curvature, both explicitly violate the FLCT and both are irrelevant for a 1DEG, become relevant at D>1D>1D>1 and D>2D>2D>2, respectively. The reason for this is a strong infrared divergence of the skeleton diagrams containing multiple fermion loops at D>1D>1D>1. These diagrams, which are omitted within the multidimensional bosonization approaches, account for the noncollinear scattering processes. Thus, the dimensional reduction provides the framework to go beyond predictions of the multidimensional bosonization. A simple diagrammatic structure of the reduced LW functional is another advantage of our approach. The dimensional reduction technique is also applicable to the thermodynamic potential and various approximations, from perturbation theory to self-consistent approaches.
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We consider a number of questions regarding the Luttinger-Ward functional and the many-body perturbation series expansion of the proper self-energy Σ(k;z)Σ(\mathbf{k};z)Σ(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(k;z)G(\mathbf{k};z)G(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 k\mathbf{k}k and complex energies zzz. 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 ΣΣΣ. 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]
For a system of correlated electrons, the Luttinger-Ward functional provides a link between static thermodynamic quantities on the one hand and single-particle excitations on the other. The functional is useful to derive several general properties of the system as well as for the formulation of thermodynamically consistent approximations. Its original construction, however, is perturbative as it is based on the weak-coupling skeleton-diagram expansion. Here, it is shown that the Luttinger-Ward functional can be derived within a general functional-integral approach. This alternative and non-perturbative approach stresses the fact that the Luttinger-Ward functional is universal for a large class of models.
For a system of correlated electrons, the Luttinger-Ward functional provides a link between static thermodynamic quantities on the one hand and single-particle excitations on the other.The functional is useful in deriving several general properties of the system as well as in formulating the thermodynamically consistent approximations.Its original construction, however, is perturbative as it is based on the weak-coupling skeleton-diagram expansion.Here, it is shown that the Luttinger-Ward functional can be derived within a general functional-integral approach.This alternative and non-perturbative approach stresses the fact that the Luttinger-Ward functional is universal for a large class of models.
The Luttinger-Ward functional $\mathrm{\ensuremath{\Phi}}[\mathbf{G}]$, which expresses the thermodynamic grand potential in terms of the interacting single-particle Green's function G\mathbf{G}G, is found to be ill defined for fermionic models with the Hubbard on-site interaction. In particular, we show that the self-energy $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]\ensuremath{\propto}\ensuremath{\delta}\mathrm{\ensuremath{\Phi}}[\mathbf{G}]/\ensuremath{\delta}\mathbf{G}$ is not a single-valued functional of G\mathbf{G}G: in addition to the physical solution for $\mathbf{\ensuremath{\Sigma}}[\mathbf{G}]$, 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 $\mathbf{\ensuremath{\Sigma}}$ in terms of G\mathbf{G}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 noninteracting Green's function G0{\mathbf{G}}_{0}G0​ converges to the correct physical branch of $\mathbf{\ensuremath{\Sigma}}$ in all cases currently accessible by diagrammatic Monte Carlo calculations. In addition to their conceptual importance, these observations have important implications for techniques based on the explicit summation of the diagrammatic series.
We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model (i.e., one-dimensional fermions with linear energy dispersion and interactions involving only small momentum transfers) emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integrodifferential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.