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Physical review. B, Condensed matter

Reexamination of the third-order renormalization-group calculation for a one-dimensional interacting Fermi system

C. S. Ting, Xian-xi Dai, Bambi Hu

Published Apr 1, 1983• 3 citations
Publisher

Abstract

A one-dimensional interacting Fermi system with weak-coupling parameters ${g}_{1} (<0)$ and ${g}_{2}$ 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.

DOI
10.1103/physrevb.27.4384

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A one-dimensional interacting Fermi system with weak-coupling parameters g1(<0){g}_{1} (<0)g1​(<0) and g2{g}_{2}g2​ is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling g1\ensuremath′{g}_{1}^{\ensuremath{'}}g1\ensuremath′​ varies, depending on whether the limit \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→0, q=0q=0q=0 or \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0, q\ensuremath→0q\ensuremath{\rightarrow}0q\ensuremath→0 is taken. For \ensuremathω\ensuremath→0\ensuremath{\omega}\ensuremath{\rightarrow}0\ensuremathω\ensuremath→0, q=0q=0q=0, the invariant coupling g1\ensuremath′{g}_{1}^{\ensuremath{'}}g1\ensuremath′​ scales to g1\ensuremath′\ensuremathπv\ensuremath≃\ensuremath−1.24\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}1.24\ensuremathπvg1\ensuremath′​​\ensuremath≃\ensuremath−1.24, and for \ensuremathω=0\ensuremath{\omega}=0\ensuremathω=0, q\ensuremath→0q\ensuremath{\rightarrow}0q\ensuremath→0, it scales to g1\ensuremath′\ensuremathπv\ensuremath≃\ensuremath−0.716\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}0.716\ensuremathπvg1\ensuremath′​​\ensuremath≃\ensuremath−0.716. We also argue that the invariant coupling g1\ensuremath′{g}_{1}^{\ensuremath{'}}g1\ensuremath′​ obtained from neither limit is appropriate for calculating the critical exponents of various response functions.
A one-dimensional system of electrons with interaction parameters g1{g}_{1}g1​ and g2{g}_{2}g2​ is studied by applying the third-order renormalization-group technique. For g1>0{g}_{1}>0g1​>0 our result is essentially the same as that obtained previously in the first-order scaling. For g1<0{g}_{1}<0g1​<0 the invariant coupling $\frac{{{g}^{\ensuremath{'}}}_{1}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}0.87$ is considerably weaker than that $\frac{{{g}^{\ensuremath{'}}}_{1}}{\ensuremath{\pi}v}=\ensuremath{-}2$ in the second-order renormalization obtained by Menyh\'ard and S\'olyom in the low-frequency limit. The phases of the ground state in the present approximation are the same as those predicted in the second-order renormalization. The critical exponents for response functions which describe various critical fluctuations have also been calculated.
The one-dimensional Fermi system with Tomonaga-type (g2{g}_{2}g2​) and back-scattering (g1{g}_{1}g1​) interactions is studied by a weak-coupling renormalization-group method at the three-loop level. When the transfer cutoff procedure is used, the Gell-Mann-Low function for g1{g}_{1}g1​ depends on g1{g}_{1}g1​ alone, $\frac{d{g}_{1}}{d\ensuremath{\xi}}={{g}_{1}}^{2}+\frac{1}{2}{{g}_{1}}^{3}+\frac{1}{8}{{g}_{1}}^{4}$. Further, in the absence of umklapp processes the combination ${g}_{1}\ensuremath{-}2{g}_{2}$ is a scale invariant. These conclusions agree with the results of Luther and Emery derived by the boson representation method. Some novel calculational techniques for the derivation of the scaling equations are described and compared to the conventional scheme.
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=1d=1d=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=2and3d=2 \mathrm{and} 3d=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 FFF, 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 1N\frac{1}{N}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=2d=2d=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.
The one-dimensional Ising model with ferromagnetic interactions which decay as 1/${\mathit{r}}^{\mathrm{\ensuremath{\alpha}}}$ is considered. Using a real-space renormalization group scheme (RG) we calculate the critical temperature and the correlation-length critical exponent as a function of \ensuremath{\alpha}. General asymptotic properties are obtained for arbitrary values of the rescaling length b of the RG transformation. Several rigorous results are recovered exactly in the limit b\ensuremath{\rightarrow}\ensuremath{\infty}. We obtain a b=\ensuremath{\infty} extrapolation of the critical temperature for arbitrary values of \ensuremath{\alpha}>1, which we conjecture aproximates with high precision the exact one. In particular, we obtain the value Tc{\mathit{T}}_{\mathit{c}}Tc​/J=${\mathrm{\ensuremath{\pi}}}^{2}$/12 for the 1/r2{\mathit{r}}^{2}r2 model.
We revisit the issue of the temperature dependence of the specific heat C(T)C(T)C(T) for interacting fermions in one dimension. The charge component Cc(T){C}_{c}(T)Cc​(T) scales linearly with TTT, but the spin component Cs(T){C}_{s}(T)Cs​(T) displays a more complex behavior with TTT as it depends on the backscattering amplitude, g1{g}_{1}g1​, which scales down under renormalization group transformation and eventually behaves as ${g}_{1}(T)\ensuremath{\sim}1∕\mathrm{log}\phantom{\rule{0.2em}{0ex}}T$. We show, however, by direct perturbative calculations that Cs(T){C}_{s}(T)Cs​(T) is strictly linear in TTT to order g12{g}_{1}^{2}g12​ as it contains the renormalized backscattering amplitude not on the scale of TTT, but at the cutoff scale set by the momentum dependence of the interaction around 2kF2{k}_{F}2kF​. The running amplitude g1(T){g}_{1}(T)g1​(T) appears only at third order and gives rise to an extra T∕log3TT∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}TT∕log3T term in Cs(T){C}_{s}(T)Cs​(T). This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in g1{g}_{1}g1​ within the sine-Gordon model.
We consider a Callan-Symanzik and Wilson renormalization-group (RG) approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third-order (two-loop) approximation of the $\ensuremath{\beta}$ function using both methods, and compare it with the well-known multiplicative Gell--Mann Low approach. We point out a previously unnoticed strong instability of the third-order fixed point with respect to an arbitrary dimensionless parameter, which suggests a RG flow toward a strong-coupling phase.