Understanding non-Fermi liquids in dimensions higher than one remains one of the most formidable challenges in modern condensed matter physics. These systems, characterized by an abundance of gapless degrees of freedom and the absence of well-defined quasiparticles, defy conventional analytical frameworks. Inspired by recent work [Delacretaz et al. Phys. Rev. Res. 4, 033131 (2022)], we present a procedure for bosonizing Fermi surfaces that does not rely on the existence of sharp excitation and is thus directly applicable to non-Fermi liquids. Our method involves parametrizing the generalized fermionic distribution function through a bosonic field that describes frequency-dependent local variations of the chemical potential in momentum space. We propose an effective action that produces the collisionless quantum Boltzmann equation as its equation of motion and can be used for any dimension and Fermi surface of interest. Even at the quadratic order, this action reproduces nontrivial results obtainable only through involved analysis with alternative means. By offering an alternative method directly applicable to studying the low-energy physics of Fermi and non-Fermi liquids, our work potentially stands as an important building block in advancing the comprehension of strange metals and associated phenomena.
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (
qc/
kF)dF0 [1+
F0${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here
qc is the cutoff of the interaction in wave-vector space,
kF is the Fermi wave vector, \ensuremath{\mu} is the chemical potential,
F0 is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff
qc is small compared with
kF, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
In this work, we consider
N-fold degenerate
D-dimensional electron gas with spherical Fermi surface and arbitrary forward-scattering density-density interaction transferring small momentum compared to the Fermi momentum
kF. The dimensional reduction that is mathematically equivalent to the Haldane patch construction and similar multidimensional bosonization techniques, provides a natural map of two-point
D-dimensional correlation functions (fermion Green function, susceptibilities etc.) onto effective one-dimensional (1D) correlators with the same diagrammatic structure, which can be evaluated exactly within a 1D bosonizable (Gaussian) theory. We then apply this formalism to evaluate the fermion Green function, pair and charge/flavor susceptibilities, as well as the composite correlation functions for the case of a finite-range interaction, where the interaction range
Rs≫1/kF is large compared to the Fermi wavelength. First, we find that the single-particle spectral function remains Fermi-liquid-like which is fully consistent with the previous research. In contrast to the single-particle sector, the many-body channels are efficiently dressed by finite-range interactions, and this dressing is fully equivalent to the one-loop renormalization group (RG), which is also in line with previous multidimensional bosonization results. Within the forward-scattering model, stable long-range order is not possible, and relevant susceptibilities demonstrate singular power-law scaling with temperature
T at
T→0. The rest of the abstract is in the PDF.
We discuss an approach to higher-dimensional bosonization of interacting fermions based on a picture of fluctuating Fermi surface. Compared with the linearized ``constructive'' approach this method allows an account of the Fermi surface curvature due to non-Gaussian terms in the bosonized Lagrangian. On the basis of this description we propose a procedure of calculating density response functions beyond the random-phase approximation (RPA). We also formulate a bosonic theory of the compressible metal-like state at the half-filled lowest Landau level and check that in the Gaussian approximation it reproduces RPA results of the gauge theory by Halperin, Lee, and Read.
We bosonize the low energy excitations of Fermi liquids in any number of dimensions in the limit of long wavelengths. The bosons are a coherent superposition of electron-hole pairs and are related with the displacements of the Fermi surface in some arbitrary direction. A coherent-state path integral for the bosonized theory is derived and it is shown to represent histories of the shape of the Fermi surface. The Landau theory of Fermi liquids can be obtained from the formalism in the absence of nesting of the Fermi surface and singular interactions. We show that the Landau equation for sound waves is exact in the semiclassical approximation for the bosons.
The single-particle Green's function of an interacting Fermi system with dominant forward scattering is calculated by decoupling the interaction by means of a Hubbard-Stratonowich transformation involving a bosonic auxiliary field
ϕα. We obtain a higher dimensional generalization of the well-known one-dimensional bosonization result for the Green's function by first calculating the Green's function for a fixed configuration of the
ϕα-field and then averaging the resulting expression with respect to the probability distribution
P{ϕα}∝exp[−Seff{ϕα}], where
Seff{ϕα} is the effective action of the
ϕα-field. We emphasize the approximations inherent in the higher-dimensional bosonization approach and clarify its relation with diagrammatic perturbation theory.