Physical review. B, Condensed matter

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

Edward H. Rezayi, Josef Sak, Jenő Sólyom

Published Aug 1, 1979• 7 citations

Abstract

A model of a one-dimensional Fermi gas with momentum-transfer cutoff is studied by renormalization group within the framework of perturbation theory at the two-loop level. The interactions are allowed to carry spin and charge. It is shown that for this particular model the spin- and charge-density degrees of freedom are governed by two decoupled sets of scaling equations. This agrees with the results obtained by the boson-representation method.

DOI: 10.1103/physrevb.20.1129

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Physical review. B, Condensed matter

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

Edward H. Rezayi, Josef Sak, Jenő Sólyom

Published Aug 1, 1979• 7 citations

Publisher

Abstract

DOI: 10.1103/physrevb.20.1129

Related papers

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

Symmetry properties of one-dimensional Fermi systems: Some nonperturbative results

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We investigate the Fermi gas model for one-dimensional (1D) conductors as a function of chemical potential and of the applied magnetic field. The bosonization method allows to prove the equivalence between these parameters, the first acting on charge degrees of freedom, the second on spin degrees of freedom. A number of results available in the literature can thus be connected. We also use the mapping to the 2D classical Coulomb gas and a two-cutoff renormalization-group method, the two energy scales being the bandwidth and the Zeeman energy (or the chemical potential). We obtain a coherent description of the Fermi gas phase diagram as a function of these two ``fields.''We investigate the Fermi gas model for one-dimensional (1D) conductors as a function of chemical potential and of the applied magnetic field. The bosonization method allows to prove the equivalence between these parameters, the first acting on charge degrees of freedom, the second on spin degrees of freedom. A number of results available in the literature can thus be connected. We also use the mapping to the 2D classical Coulomb gas and a two-cutoff renormalization-group method, the two energy scales being the bandwidth and the Zeeman energy (or the chemical potential). We obtain a coherent description of the Fermi gas phase diagram as a function of these two ``fields.''

Physical review. B, Condensed matterJun 1, 198611 citations

Solution of the one-dimensional electron gas on a lattice

Victor J. Emery, Alan H. Luther, Ingo Peschel

An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.An exact solution of the one-dimensional electron gas on a lattice is given for the special case in which two of the four coupling constants have particular values. It is shown that umklapp scattering has the same effect on the charge-density waves as backward scattering across the Fermi "surface" has on the spin-density waves and that the method of Luther and Emery can be used to solve this more general problem. For repulsive electron-electron interactions the umklapp scattering produces a gap in the charge-density wave spectrum and this appears for low-lying excitations when there is a half-filled band. The use of renormalization-group scaling to solve for general values of the coupling constants is discussed.

Physical review. B, Solid stateFeb 1, 197691 citations

Some properties of the one-dimensional Fermi model

Hiroyuki Fukuyama, Thomas Maurice Rice, Chandra Mohan Varma +1 more

A model of one-dimensional fermions characterized by a large momentum-transfer interaction ${g}_{1}$ and a small momentum-transfer interaction ${g}_{2}$ is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of ${g}_{1}$ dominates the behavior of the system. For ${g}_{1}>0$ , ${g}_{1}\ensuremath{-}2{g}_{2}>0$ the triplet Cooper response diverges as $T\ensuremath{\rightarrow}0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for ${g}_{1}>0$ and diverges to zero for ${g}_{1}<0$ . The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.A model of one-dimensional fermions characterized by a large momentum-transfer interaction ${g}_{1}$ and a small momentum-transfer interaction ${g}_{2}$ is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of ${g}_{1}$ dominates the behavior of the system. For ${g}_{1}>0$, ${g}_{1}\ensuremath{-}2{g}_{2}>0$ the triplet Cooper response diverges as $T\ensuremath{\rightarrow}0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for ${g}_{1}>0$ and diverges to zero for ${g}_{1}<0$. The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.

Physical review. B, Solid stateNov 1, 197453 citations

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.

A model of one-dimensional fermions characterized by a large momentum-transfer interaction

{g}_{1}

and a small momentum-transfer interaction

{g}_{2}

is considered. Results obtained previously by the renormalization-group method are extended by studying the triplet Cooper-pair response function and uniform charge and spin response functions. The sign of

{g}_{1}

dominates the behavior of the system. For

{g}_{1}>0

, ${g}_{1}\ensuremath{-}2{g}_{2}>0$ the triplet Cooper response diverges as $T\ensuremath{\rightarrow}0$ and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for

{g}_{1}>0

and diverges to zero for

{g}_{1}<0

. The specific heat is found to be linear in temperature in all regions. These results, obtained by the second-order renormalization group, yield the correct qualitative results in regions where exact solutions are available.