Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical review. B, Solid state

Some properties of the one-dimensional Fermi model

Hiroyuki Fukuyama, Thomas Maurice Rice, Chandra Mohan Varma, B. I. Halperin

Published Nov 1, 1974• 53 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.10.3775

Related papers

Specific heat of a one-dimensional interacting Fermi system: Role of anomalies

Andrey V. Chubukov, Dmitrii L. Maslov, Ronojoy Saha

We revisit the issue of the temperature dependence of the specific heat $C(T)$ for interacting fermions in one dimension. The charge component ${C}_{c}(T)$ scales linearly with $T$, but the spin component ${C}_{s}(T)$ displays a more complex behavior with $T$ as it depends on the backscattering amplitude, ${g}_{1}$, 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 ${C}_{s}(T)$ is strictly linear in $T$ to order ${g}_{1}^{2}$ as it contains the renormalized backscattering amplitude not on the scale of $T$, but at the cutoff scale set by the momentum dependence of the interaction around $2{k}_{F}$. The running amplitude ${g}_{1}(T)$ appears only at third order and gives rise to an extra $T∕{\mathrm{log}}^{3}\phantom{\rule{0.2em}{0ex}}T$ term in ${C}_{s}(T)$. This agrees with the results obtained by a variety of bosonization techniques. We also show how to obtain the same expansion in ${g}_{1}$ within the sine-Gordon model.

arXiv:0707.4422•Physical Review B•Feb 12, 2008•10 citations

Order in metallic chains. I. The single chain

Hanoch Gutfreund, Richard A. Klemm

The nature of the order as $T\ensuremath{\rightarrow}0$ in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer (${g}_{2}$ and ${g}_{4}$) and large-momentum-transfer (${g}_{1}$ and ${g}_{3}$) electron-electron interactions, focusing upon the particular value of ${g}_{1}$ for which Luther and Emery have recently found an exact solution. For this value of ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, we calculate the low-frequency behavior of the zero-temperature response functions explicitly. For the singlet superconducting and charge-density-wave responses, our results are consistent with those of Luther and Emery, implying that these response functions are divergent as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for certain values of ${g}_{2}$. For the triplet superconducting and spin-density-wave responses, however, our results differ from those of Luther and Emery, as we show explicitly that these responses have a gap for low frequencies, as was predicted by Lee. Furthermore, they do not diverge at the gap edge for any value of ${g}_{2}$. We also consider the effect of interactions between electrons on the same side of the Fermi "surface," and find that the low-momentum-transfer process of that type (${g}_{4}$) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" (${g}_{3}$), we may solve the problem exactly for ${g}_{2}=0$ and ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, and find that the ground state of the system exhibits only long-range charge-density-wave order. By a mapping onto the classical two-dimensional Coulomb gas problem, we may extend those results for ${g}_{3}\ensuremath{\ne}0$ to the region ${g}_{1}<0$ and ${g}_{1}\ensuremath{-}2{g}_{2}<|{g}_{3}|$.

Physical review. B, Solid state•Aug 1, 1976•56 citations

Comments on a Solution of a One-Dimensional Fermi-Gas Model

Patrick A. Lee

A recent solution by Luther and Emery of a one-dimensional interacting Fermi-gas model contains a flaw in its final step which when corrected yields qualitatively different behavior. In particular spin-density wave and triplet superconducting pairing excitations are not divergent. The result is consistent with a gap in the uniform spin susceptibility and the gap can now be given a simple physical interpretation. The bulk of the Luther-Emery solution including exponents for the charge-density wave and singlet pairing remains correct.

Physical Review Letters•May 12, 1975•75 citations

Interacting Fermions in One Dimension. I. Repulsive Potential

James McGuire

The exact energies and wavefunctions for the ground state and low-lying excited states of a system of N − 1 one-dimensional fermions all of the same spin and one fermion of the opposite spin are calculated in the large-volume, finite-density limit when the particles interact via a repulsive delta function potential. A number of properties of the system such as pair correlation functions and the effective mass of a certain class of excitations are also discussed.

Journal of Mathematical Physics•Mar 1, 1965•220 citations

Response behavior of the two-chain Fermi gas

Richard A. Klemm

We consider a model of TTF-TCNQ (tetrahiafulvalenium-tetracyanoquinodimethanide) and similar charge-transfer linear conductors in which there are two chains. The electrons on the acceptor (TCNQ) chains have a usual metallic band, but the electrons on the donor (TTF) chains have an inverted or "hole" band. The electrons on each chain interact via the forward and backward scattering ${g}_{2}$ and ${g}_{1}$ processes, respectively. Electrons on different chains interact via the analogous processes ${W}_{2}$ and ${W}_{1}$. The model is also applicable to two metallic chains, with the signs of ${W}_{1}$ and ${W}_{2}$ reversed. Using the techniques of Luther and Emery and of Chui and Lee we show that the problem may be mapped onto a similar problem of three coupled classical two-dimensional Coulomb gases. In addition to gaps in the spin degrees of freedom that were present in the single-chain problem, we find that the antisymmetric form of the resultant charge degrees of freedom may have a gap. We then investigate the predominant two-, four-, six-, and eight-particle response functions and calculate their temperature dependence. In addition to the four divergent two-particle response functions present in the single chain, there is a region for weak interaction strengths in which the $2{k}_{F}$ interchain Cooper pairing response is the most divergent as $T\ensuremath{\rightarrow}0$. The excitonic insulator response may also be important at high temperatures. For the single chain, either the $4{k}_{F}$ response or a four-particle $q=0$ response is the most divergent for strong coupling and repulsive ${g}_{1}$, depending upon the sign of ${g}_{2}\ensuremath{-}\frac{1}{2}{g}_{1}$. For the two-chain system, the most divergent response as $T\ensuremath{\rightarrow}0$ for strongly attractive interactions (${g}_{1}$ and ${g}_{2}$) is the response to the formation of excitonic "molecules" or exciton pairs. For strong coupling and repulsive ${g}_{1}$ either $8{k}_{F}$ or octet responses may be observable. We discuss these results in conjunction with our previous renormalization-group treatment, which indicates that for certain values of ${g}_{2}$, ${W}_{2}$, and ${W}_{1}$ the line ${g}_{1}=0$ may be crossed. We use these results to interpret the recent x-ray data on TTF-TCNQ showing a crossover from $4{k}_{F}$ to $2{k}_{F}$ behavior above the crossover from one-dimensional to three-dimensional behavior.

Physical review. B, Condensed matter•Aug 1, 1979•6 citations

Conductivity of One-Dimensional Interacting Fermions

Daniel Charles Mattis

Using an exactly soluble model, the decay rate of a current-carrying state of one-dimensional fermions is calculated in the presence of random scatterers at finite temperature and the dc conductivity thereby inferred. For interacting fermions it is modified by a factor ${(\frac{T}{T}+\ensuremath{\theta})}^{g}$, where $g$ is a positive (negative) coupling constant for repulsive (attractive) two-body forces. While the conductivity could be greatly enhanced for $g<0$ and $T\ensuremath{\ll}\ensuremath{\theta}$, one-dimensional superconductivity appears ruled out at any finite temperature.

Physical Review Letters•Apr 1, 1974•53 citations
A model of one-dimensional fermions characterized by a large momentum-transfer interaction g1{g}_{1}g1​ and a small momentum-transfer interaction g2{g}_{2}g2​ 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 g1{g}_{1}g1​ dominates the behavior of the system. For g1>0{g}_{1}>0g1​>0, g1\ensuremath−2g2>0{g}_{1}\ensuremath{-}2{g}_{2}>0g1​\ensuremath−2g2​>0 the triplet Cooper response diverges as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0 and is seen to dominate the singlet Cooper response which is also divergent. The uniform static spin susceptibility is nondivergent for g1>0{g}_{1}>0g1​>0 and diverges to zero for g1<0{g}_{1}<0g1​<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.
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.
The nature of the order as $T\ensuremath{\rightarrow}0$ in the one-dimensional interacting electron gas is investigated. We consider both low-momentum-transfer (g2{g}_{2}g2​ and g4{g}_{4}g4​) and large-momentum-transfer (g1{g}_{1}g1​ and g3{g}_{3}g3​) electron-electron interactions, focusing upon the particular value of g1{g}_{1}g1​ for which Luther and Emery have recently found an exact solution. For this value of ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, we calculate the low-frequency behavior of the zero-temperature response functions explicitly. For the singlet superconducting and charge-density-wave responses, our results are consistent with those of Luther and Emery, implying that these response functions are divergent as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for certain values of g2{g}_{2}g2​. For the triplet superconducting and spin-density-wave responses, however, our results differ from those of Luther and Emery, as we show explicitly that these responses have a gap for low frequencies, as was predicted by Lee. Furthermore, they do not diverge at the gap edge for any value of g2{g}_{2}g2​. We also consider the effect of interactions between electrons on the same side of the Fermi "surface," and find that the low-momentum-transfer process of that type (g4{g}_{4}g4​) does not change the response behavior qualitatively. For the large-momentum-transfer process between electrons on the same side of the Fermi "surface" (g3{g}_{3}g3​), we may solve the problem exactly for g2=0{g}_{2}=0g2​=0 and ${g}_{1}=\ensuremath{-}\frac{6}{5\ensuremath{\pi}{\ensuremath{\nu}}_{F}}$, and find that the ground state of the system exhibits only long-range charge-density-wave order. By a mapping onto the classical two-dimensional Coulomb gas problem, we may extend those results for ${g}_{3}\ensuremath{\ne}0$ to the region g1<0{g}_{1}<0g1​<0 and ${g}_{1}\ensuremath{-}2{g}_{2}<|{g}_{3}|$.
A recent solution by Luther and Emery of a one-dimensional interacting Fermi-gas model contains a flaw in its final step which when corrected yields qualitatively different behavior. In particular spin-density wave and triplet superconducting pairing excitations are not divergent. The result is consistent with a gap in the uniform spin susceptibility and the gap can now be given a simple physical interpretation. The bulk of the Luther-Emery solution including exponents for the charge-density wave and singlet pairing remains correct.
We consider a model of TTF-TCNQ (tetrahiafulvalenium-tetracyanoquinodimethanide) and similar charge-transfer linear conductors in which there are two chains. The electrons on the acceptor (TCNQ) chains have a usual metallic band, but the electrons on the donor (TTF) chains have an inverted or "hole" band. The electrons on each chain interact via the forward and backward scattering g2{g}_{2}g2​ and g1{g}_{1}g1​ processes, respectively. Electrons on different chains interact via the analogous processes W2{W}_{2}W2​ and W1{W}_{1}W1​. The model is also applicable to two metallic chains, with the signs of W1{W}_{1}W1​ and W2{W}_{2}W2​ reversed. Using the techniques of Luther and Emery and of Chui and Lee we show that the problem may be mapped onto a similar problem of three coupled classical two-dimensional Coulomb gases. In addition to gaps in the spin degrees of freedom that were present in the single-chain problem, we find that the antisymmetric form of the resultant charge degrees of freedom may have a gap. We then investigate the predominant two-, four-, six-, and eight-particle response functions and calculate their temperature dependence. In addition to the four divergent two-particle response functions present in the single chain, there is a region for weak interaction strengths in which the 2kF2{k}_{F}2kF​ interchain Cooper pairing response is the most divergent as $T\ensuremath{\rightarrow}0$. The excitonic insulator response may also be important at high temperatures. For the single chain, either the 4kF4{k}_{F}4kF​ response or a four-particle q=0q=0q=0 response is the most divergent for strong coupling and repulsive g1{g}_{1}g1​, depending upon the sign of ${g}_{2}\ensuremath{-}\frac{1}{2}{g}_{1}$. For the two-chain system, the most divergent response as $T\ensuremath{\rightarrow}0$ for strongly attractive interactions (g1{g}_{1}g1​ and g2{g}_{2}g2​) is the response to the formation of excitonic "molecules" or exciton pairs. For strong coupling and repulsive g1{g}_{1}g1​ either 8kF8{k}_{F}8kF​ or octet responses may be observable. We discuss these results in conjunction with our previous renormalization-group treatment, which indicates that for certain values of g2{g}_{2}g2​, W2{W}_{2}W2​, and W1{W}_{1}W1​ the line g1=0{g}_{1}=0g1​=0 may be crossed. We use these results to interpret the recent x-ray data on TTF-TCNQ showing a crossover from 4kF4{k}_{F}4kF​ to 2kF2{k}_{F}2kF​ behavior above the crossover from one-dimensional to three-dimensional behavior.
Using an exactly soluble model, the decay rate of a current-carrying state of one-dimensional fermions is calculated in the presence of random scatterers at finite temperature and the dc conductivity thereby inferred. For interacting fermions it is modified by a factor ${(\frac{T}{T}+\ensuremath{\theta})}^{g}$, where ggg is a positive (negative) coupling constant for repulsive (attractive) two-body forces. While the conductivity could be greatly enhanced for g<0g<0g<0 and $T\ensuremath{\ll}\ensuremath{\theta}$, one-dimensional superconductivity appears ruled out at any finite temperature.