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Annals of the Israel Physical Society

Classical calculation of correlation functions in a one-dimensional Fermi gas

J. N. Fields, Richard A. Klemm

Published Jan 1, 1978• 0 citations
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Abstract

The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.

Related papers

Classical calculation of correlation functions in a one-dimensional Fermi gas

J. N. Fields, Richard A. Klemm

The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.

Physical review. B, Solid state•Dec 1, 1977•3 citations

Equivalence of a One-Dimensional Fermion Model and the Two-Dimensional Coulomb Gas

S. T. Chui, Patrick A. Lee

We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for ${U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|$ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.

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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.

Physical review. B, Solid state•Nov 1, 1974•53 citations

Coupling between charge and spin degrees of freedom in the one-dimensional Fermi gas with backscattering

F. D. M. Haldane

A slightly corrected version of the 'boson representation' of fermions in the Luttinger model (that avoids the alpha to 0 limit) is used to obtain explicitly the lowest-order interactions between charge- and spin-density waves in the generalised model that includes backscattering. This is in contrast to the decoupling of these degrees of freedom reported by Luther and Emery. Instead, the results obtained by these authors are identified as applying to a model with long-range coupling between spin densities at different points.

Journal of Physics C Solid State Physics•Nov 28, 1979•53 citations

Properties of an interacting one-dimensional fermion system

W. I. Friesen

For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.

•Jan 1, 2010•0 citations

Correlation functions of one-dimensional quantum systems

Thierry Giamarchi, Heinz J. Schulz

A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.

Physical review. B, Condensed matter•Mar 1, 1989•213 citations
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.
We show that the one-dimensional Luttinger model generalized to include spin and backward scattering is equivalent to a two-dimensional Coulomb gas. Scaling equations are derived and correlation functions are given simple physical interpretation in terms of the Coulomb gas; e.g., existence of an energy gap can be understood in terms of Debye screening. We conclude that an energy gap exists for ${U}_{\ensuremath{\parallel}}<|{U}_{\ensuremath{\perp}}|$ so that triplet excitations are nondivergent, and we provide physical arguments to support the exponents proposed by Luther and Emery for singlet excitations for general coupling constant.
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, ${g}_{1}\ensuremath{-}2{g}_{2}>0$ 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 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.
A slightly corrected version of the 'boson representation' of fermions in the Luttinger model (that avoids the alpha to 0 limit) is used to obtain explicitly the lowest-order interactions between charge- and spin-density waves in the generalised model that includes backscattering. This is in contrast to the decoupling of these degrees of freedom reported by Luther and Emery. Instead, the results obtained by these authors are identified as applying to a model with long-range coupling between spin densities at different points.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.
A number of one-dimensional quantum systems like interacting fermions or spin chains can be described in terms of a generalized quantum sine-Gordon Hamiltonian. The transfer-matrix formulation of critical two-dimensional models also frequently leads to sine-Gordon Hamiltonians. We compute here the different correlation functions of such a model using a real-space renormalization technique. Our renormalization calculation gives us the exponents of the correlation functions as well as the corrections to the usual power-law behavior due to marginally irrelevant operators. Except on the critical line, where logarithmic corrections exist at all length scales, the correlation functions decay like power laws, with amplitude corrections from the marginal operators. The logarithmic corrections always appear at short length scale. The complete crossover between the two behaviors is given by our equations. The implication of our calculation for physical systems like the one-dimensional electron gas, quantum spin chains, and two-dimensional statistical systems is discussed.