Physical review. B, Solid state

Two-cutoff scaling for the one-dimensional electron gas

Gary S. Grest, Elihu Abrahams, S. T. Chui, Patrick A. Lee, A. Zawadowski

Published Aug 1, 1976• 37 citations

Abstract

We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency ${\ensuremath{\omega}}_{D}$, while the electron bandwidth is of order ${E}_{F}$, with ${\ensuremath{\omega}}_{D}<{E}_{F}$. Our results, consistent only in the region of weak coupling, give the scaling trajectories for the coupling constants and cutoffs. The relationship to the conventional one-cutoff renormalization-group approach is discussed.

DOI: 10.1103/physrevb.14.1225

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One dimensional acoustic phonon linearly coupled to electron density

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Boundary Conditions for Renormalization-Group Equations in One-Dimensional Fermi Gas

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Physical review. B, Solid state

Two-cutoff scaling for the one-dimensional electron gas

Gary S. Grest, Elihu Abrahams, S. T. Chui, Patrick A. Lee, A. Zawadowski

Published Aug 1, 1976• 37 citations

Publisher

Abstract

We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency

{\ensuremath{\omega}}_{D}

, while the electron bandwidth is of order

{E}_{F}

, with

{\ensuremath{\omega}}_{D}<{E}_{F}

. Our results, consistent only in the region of weak coupling, give the scaling trajectories for the coupling constants and cutoffs. The relationship to the conventional one-cutoff renormalization-group approach is discussed.

DOI: 10.1103/physrevb.14.1225

Related papers

One dimensional acoustic phonon linearly coupled to electron density

Sanjeev Kumar, Gautam Kumar Sinha

We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.We have studied one dimensional acoustic phonon linearly coupled to the electron density. The phonon spectrum is linear with cutoff at the Debye frequency, it is straightforward modification of three dimensional optical phonons. Phonon field results in substituting the dynamical coupling for the screened coulomb interaction. We have shown that the electron-phonon coupling quantitatively changed the phase diagram of Luttinger liquid with a single impurity. The existence of slow and fast polaron modes with different weights in different regimes resulted in different renormalization group flow for a weak scatterer and a weak tunneling link. The resulting phase diagram depending on the parameters of the problem, regimes corresponding to purely metallic or purely insulating behavior and an intermediate regime with two stable fixed points and one unstable, finite conductance fixed point. we have found that the direction of renormalization group flow change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. The obtained results were found in good agreement with previously obtained results.

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arXiv:0708.0012Physical Review BNov 16, 200728 citations

Renormalization group for one-dimensional Fermi systems: Three-loop scaling equations

Edward H. Rezayi, Josef Sak, S. Talukdar

The one-dimensional Fermi system with Tomonaga-type ( ${g}_{2}$ ) and back-scattering ( ${g}_{1}$ ) 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 ${g}_{1}$ depends on ${g}_{1}$ 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 one-dimensional Fermi system with Tomonaga-type (${g}_{2}$) and back-scattering (${g}_{1}$) 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 ${g}_{1}$ depends on ${g}_{1}$ 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.

Physical review. B, Condensed matterMay 1, 197912 citations

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A. I. Larkin, Josef Sak

Physical Review LettersOct 17, 197729 citations

Renormalization group for one-dimensional Fermi systems with long-range potential, exchange and umklapp interactions

Edward H. Rezayi, Josef Sak, Jenő Sólyom

Physical review. B, Condensed matterAug 1, 19797 citations

We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency

{\ensuremath{\omega}}_{D}

, while the electron bandwidth is of order

{E}_{F}

, with

{\ensuremath{\omega}}_{D}<{E}_{F}

\frac{1}{2}

fermions interacting with either intra- or intermolecular phonons, the quantum corrections to the Peierls gap at half-filling are determined at arbitrary phonon frequency. The nature of quantum-classical transitions is clarified in weak coupling.

The one-dimensional Fermi system with Tomonaga-type (

{g}_{2}

) and back-scattering (

{g}_{1}

) 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

{g}_{1}

depends on

{g}_{1}

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.