A one-dimensional interacting Fermi system with weak-coupling parameters
g1(<0) and
g2 is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling ${g}_{1}^{\ensuremath{'}}$ varies, depending on whether the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$,
q=0 or $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$ is taken. For $\ensuremath{\omega}\ensuremath{\rightarrow}0$,
q=0, the invariant coupling ${g}_{1}^{\ensuremath{'}}$ scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}1.24$, and for $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$, it scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}0.716$. We also argue that the invariant coupling ${g}_{1}^{\ensuremath{'}}$ obtained from neither limit is appropriate for calculating the critical exponents of various response functions.
We formulate a momentum-shell renormalization-group (RG) procedure that can be used in theories containing both bosons and fermions with a Fermi surface. We focus on boson-fermion couplings that are nearly forward scattering, i.e., involving small momentum transfer $(\stackrel{P\vec}{q}\ensuremath{\approx}0)$ for the fermions. Special consideration is given to phase space constraints that result from the conservation of momentum and the imposition of ultraviolet cutoffs. For problems where the energy and momentum scale similarly (dynamic exponent
z=1), we show that more than one formalism can be used and they give equivalent results. When the energy and momentum must scale differently $(z\ensuremath{\ne}1)$, the procedures available are more limited but a consistent RG scheme can still be formulated. The approach is applicable to a variety of problems, such as itinerant-electron magnets and gauge fields interacting with fermions.
Several exact results are proved for two models of one-dimensional conductors, a model with transfer cutoff and spin- and charge-carrying interactions and the usual backscattering model with a bandwidth cutoff. A set of Ward identities is derived which connects the three-point charge vertex parts with single-particle Green's functions. These are combined with a modified renormalization-group scheme to prove the scale invariance of the couplings between charge densities in the absence of umklapp processes. The Ward identities are also used for the computation of the correlation functions in the charge sector of the system. Then some canonical transformations are introduced which allow one to relate various response functions among themselves. From these relations the phase boundaries of the ground state can be established. It is shown that the transfer cutoff and the backscattering models have the same phase diagram.
The Fermi gas model of one-dimensional conductors is reviewed. The exact solutions known for particular values of the coupling constants in a single chain problem (Tomonaga model, Luther-Emery model) are discussed. Renormalization group arguments are used to extend these solutions to arbitrary values of the couplings. The instabilities and possible ground states are studied by investigating the behaviour of the response functions. The relationship between this model and others is discussed and is used to obtain further information about the behaviour of the system. The model is generalized to a set of coupled chains to describe quasi-one-dimensional systems. The crossover from one-dimensional to three-dimensional behaviour and the type of ordering are discussed.
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the renormalization-group (RG) framework, in close analogy with the study of critical phenomena using ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ theory in four dimensions and the path-integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that eliminates a part of these modes, but preserves the action of the noninteracting system, is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A
d=1 warmup calculation involving a system of fermions shows how, in contrast to mean-field theory, which predicts a charge-density wave for arbitrarily weak repulsion, and superconductivity for arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and a Landau function
F, with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.