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Physical review. B, Solid state

Properties of a one-dimensional metallic system in the third-order renormalization-group approximation

C. S. Ting

Published May 1, 1976• 9 citations
Publisher

Abstract

A one-dimensional system of electrons with interaction parameters ${g}_{1}$ and ${g}_{2}$ is studied by applying the third-order renormalization-group technique. For ${g}_{1}>0$ our result is essentially the same as that obtained previously in the first-order scaling. For ${g}_{1}<0$ 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.

DOI
10.1103/physrevb.13.4029

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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 ${g}_{1}^{\ensuremath{'}}$ varies, depending on whether the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$, q=0q=0q=0 or $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$ is taken. For $\ensuremath{\omega}\ensuremath{\rightarrow}0$, q=0q=0q=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.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
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We apply the functional renormalization group method to the calculation of dynamical properties of zero-dimensional interacting quantum systems. As case studies we discuss the anharmonic oscillator and the single impurity Anderson model. We truncate the hierarchy of flow equations such that the results are at least correct up to second order perturbation theory in the coupling. For the anharmonic oscillator energies and spectra obtained within two different functional renormalization group schemes are compared to numerically exact results, perturbation theory, and the mean field approximation. Even at large coupling the results obtained using the functional renormalization group agree quite well with the numerical exact solution. The better of the two schemes is used to calculate spectra of the single impurity Anderson model, which then are compared to the results of perturbation theory and the numerical renormalization group. For small to intermediate couplings the functional renormalization group gives results which are close to the ones obtained using the very accurate numerical renormalization group method. In particulare the low-energy scale (Kondo temperature) extracted from the functional renormalization group results shows the expected behavior.
Scaling behavior for first-order phase transitions can be derived alternatively but consistently from renormalization-group, phenomenological, or finite-size considerations. A general analysis of densities at a renormalization-group fixed point demonstrates that if the coexistence of ppp distinct phases is possible, then ppp distinct eigenvalue exponents must equal the spatial dimensionality. This basic eigenvalue (or scaling) exponent condition can also be derived phenomenologically by various arguments not depending on detailed renormalization-group considerations. A scaling description of first-order phase transitions is presented and extended to finite systems with linear dimensions LLL, leading to a rounding proportional to ${L}^{\ensuremath{-}d}$, response-function maxima varying as ${L}^{d}\ensuremath{\propto}N$, and boundary-condition-dependent shifts which may be as large as $\ensuremath{\sim}{L}^{\ensuremath{-}1}$.