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.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.

arXiv:2303.16732Physical review. B./Physical review. BDec 4, 20233 citations

Cluster approach to the spinless fermion model

Dieter Ihle

physica status solidi (b)Oct 1, 19831 citation

Introduction to Renormalization Group and Ward Identities in Critical Phenomena and in Fermi and Bose Liquids

S. Caprara

We review some of the main applications of the renormalization‐group technique in condensed matter physics. The first relevant example is the description of critical phenomena. Here perturbation theory is affected by singularities, which are a consequence of the long‐ranged character of the dominant fluctuations when approaching criticality. The use of renormalization group allows to sum up these singularities into a power‐law behavior of the physical quantities, which is experimentally observed near a continuous phase transition. The second example is provided by the description of the physical properties of interacting Fermi and Bose systems. Here perturbation theory is affected by infrared divergences within stable liquid phases, due to the presence of massless excitations, in reduced dimensionality. However, the condition of stability of the system implies exact cancellations among the singular contributions, controlled by additional Ward identities, which must be considered besides the standard Ward identities related to the conservation of the total particle and spin density. The combined use of renormalization group and these new Ward identities allows for the closure of the renormalization‐group equations, leading to the description of the asymptotic behavior of the system at low energies.We review some of the main applications of the renormalization‐group technique in condensed matter physics. The first relevant example is the description of critical phenomena. Here perturbation theory is affected by singularities, which are a consequence of the long‐ranged character of the dominant fluctuations when approaching criticality. The use of renormalization group allows to sum up these singularities into a power‐law behavior of the physical quantities, which is experimentally observed near a continuous phase transition. The second example is provided by the description of the physical properties of interacting Fermi and Bose systems. Here perturbation theory is affected by infrared divergences within stable liquid phases, due to the presence of massless excitations, in reduced dimensionality. However, the condition of stability of the system implies exact cancellations among the singular contributions, controlled by additional Ward identities, which must be considered besides the standard Ward identities related to the conservation of the total particle and spin density. The combined use of renormalization group and these new Ward identities allows for the closure of the renormalization‐group equations, leading to the description of the asymptotic behavior of the system at low energies.

AIP conference proceedingsJan 1, 20023 citations

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.