One-dimensional chains arranged into a many-dimensional mesh are analyzed. The model is used to study the relevance of strong forward- and backward-scattering processes in systems that look isotropic over large scales. The model combines Luttinger- and Fermi-liquid features, depending on the position in the Fermi surface. Over most of it, Luttinger behavior prevails, although, at isolated points in the Brillouin zone, finite quasiparticle peaks exist.

Physical review. B, Condensed matterJan 1, 199326 citations

arXiv:cond-mat/0004451 · Philosophical Magazine B

On Fermi systems with strong forward scattering in d spatial dimensions

Behnam Farid

Published Sep 11, 2000• 0 citations

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Abstract

DOI: 10.1080/014186300419517

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On Fermi systems with strong forward scattering indspatial dimensions

Behnam Farid

We consider a specific generalization to spatial dimensions d greater than one of a formalism, based upon conservation laws and the associated ward identities, that exactly solves the one-dimensional Luttinger model, and expose its inadequacies for d > 1. We conclude that findings arrived at through application of this generalized formalism concerning systems of fermions with strong forward scattering, in particular that d = 1 would be the cross-over dimension from Luttinger- to Fermi-liquid state, are open to question.We consider a specific generalization to spatial dimensions d greater than one of a formalism, based upon conservation laws and the associated ward identities, that exactly solves the one-dimensional Luttinger model, and expose its inadequacies for d > 1. We conclude that findings arrived at through application of this generalized formalism concerning systems of fermions with strong forward scattering, in particular that d = 1 would be the cross-over dimension from Luttinger- to Fermi-liquid state, are open to question.

Philosophical Magazine BSep 1, 20003 citations

Fermi systems with strong forward scattering

Walter Metzner, Claudio Castellani, C. Di Castro

We review the theory of interacting Fermi systems whose low-energy physics is dominated by forward scattering, that is scattering processes generated by effective interactions with small momentum transfers. These systems include Fermi liquids as well as several important non-Fermi-liquid phases: one-dimensional Luttinger liquids, systems with long-range interactions, and fermions coupled to a gauge field. We report results for the critical dimensions separating different 'universality classes' and discuss the behaviour of physical quantities such as the momentum distribution function, the single-particle propagator and low-energy response functions in each class. The renormalization group for Fermi systems will be reviewed and applied as a link between microscopic models and effective lowenergy theories. Particular attention is paid to conservation laws, which constrain any effective low-energy theory of interacting Fermi systems. In scattering processes with small momentum transfers the velocity of each scattering particle is (almost) conserved. This asymptotic conservation law leads to non-trivial cancellations of Feynman diagrams and other simplifications, making thus possible a non-perturbative treatment of forward scattering via Ward identities or bosonization techniques.We review the theory of interacting Fermi systems whose low-energy physics is dominated by forward scattering, that is scattering processes generated by effective interactions with small momentum transfers. These systems include Fermi liquids as well as several important non-Fermi-liquid phases: one-dimensional Luttinger liquids, systems with long-range interactions, and fermions coupled to a gauge field. We report results for the critical dimensions separating different 'universality classes' and discuss the behaviour of physical quantities such as the momentum distribution function, the single-particle propagator and low-energy response functions in each class. The renormalization group for Fermi systems will be reviewed and applied as a link between microscopic models and effective lowenergy theories. Particular attention is paid to conservation laws, which constrain any effective low-energy theory of interacting Fermi systems. In scattering processes with small momentum transfers the velocity of each scattering particle is (almost) conserved. This asymptotic conservation law leads to non-trivial cancellations of Feynman diagrams and other simplifications, making thus possible a non-perturbative treatment of forward scattering via Ward identities or bosonization techniques.

arXiv:cond-mat/9701012Advances In PhysicsMay 1, 1998154 citations

Bosonization and fermion liquids in dimensions greater than one

A. Houghton, John Bradley Marston

We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the ${\mathit{T}}^{3}$ ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.We develop and describe new approaches to the problem of interacting fermions in spatial dimensions greater than 1. These approaches are based on generalizations of powerful tools previously applied to problems in one spatial dimension. We begin with a review of one-dimensional interacting fermions. We then introduce a simplified model in two spatial dimensions to study the role that spin and perfect nesting play in destabilizing fermion liquids. The complicated functional renormalization-group equations of the full problem are made tractable in our model by replacing the continuum of points that make up the closed Fermi line with four Fermi points. Despite this drastic approximation, the model exhibits physically reasonable behavior both at half-filling (where instabilities occur) and away from half-filling (where a Luttinger liquid arises). Next we implement the bosonization of higher-dimensional Fermi surfaces introduced by Luther and advocated most recently by Haldane. Bosonization incorporates the phase-space and small-angle scattering processes neglected in our model (but does not, as yet, address questions of stability). The charge sector is equivalent to an exactly solvable Gaussian quantum field theory; the spin sector, however, must be solved semiclassically. Using the Luther-Haldane approach we recover the collective mode equation of Fermi-liquid theory and in three dimensions reproduce the ${\mathit{T}}^{3}$ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.

arXiv:cond-mat/9210007Physical review. B, Condensed matterSep 15, 1993186 citations

Conservation laws and correlation functions in the Luttinger liquid

Walter Metzner, C. Di Castro

The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.The low-energy properties of interacting Fermi systems are highly constrained by conservation laws. They generally simplify the structure of the underlying renormalization group by reducing the number of independent renormalization constants. In one dimension, all properties of normal metallic fixed points are uniquely determined by separate charge and spin conservation for states near the left and right Fermi points, respectively. We construct the general Luttinger-liquid theory of one-dimensional (1D) metals directly from these conservation laws. Luttinger-liquid parameters emerge naturally from the velocities associated with the conserved currents at the Luttinger-liquid fixed point. Instead of bosonization, one may thus use techniques familiar from Fermi-liquid theory, i.e., Feynman diagrams, equations of motion, and Ward identities. The choice of a technique comprising both Fermi- and Luttinger-liquid theory makes the similarities and differences of both theories particularly transparent, and sets the stage for constructing non-Fermi-liquid metallic fixed points in d>1. Several generic properties and asymptotic conservation laws of 2D non-Fermi-liquid metals are discussed.

Physical review. B, Condensed matterJun 15, 199381 citations

Ward identities and the β function in the Luttinger liquid

C. Di Castro, Walter Metzner

One-dimensional metals have a particular symmetry associated with the discrete structure of the Fermi surface: separate charge conservation in low-energy-scattering processes for particles near the left and right Fermi points, respectively. The field-theoretical renormalization group allows for an efficient exploitation of the Ward identities following from this symmetry. As a first application we prove that the \ensuremath{\beta} function of the Luttinger model vanishes identically. The same symmetry ensures the finiteness of the compressibility, thus making possible the existence of stable metallic phases with anomalous dimensions in d=1.One-dimensional metals have a particular symmetry associated with the discrete structure of the Fermi surface: separate charge conservation in low-energy-scattering processes for particles near the left and right Fermi points, respectively. The field-theoretical renormalization group allows for an efficient exploitation of the Ward identities following from this symmetry. As a first application we prove that the \ensuremath{\beta} function of the Luttinger model vanishes identically. The same symmetry ensures the finiteness of the compressibility, thus making possible the existence of stable metallic phases with anomalous dimensions in d=1.

Physical Review LettersDec 30, 199144 citations

Luttinger liquids in higher dimensions

F. Guinea, Gergely T. Zimányi

Physical review. B, Condensed matterJan 1, 199326 citations

{\mathit{T}}^{3}

ln(T) contribution to the specific heat due to small-angle scattering processes. We conclude with a discussion of our results and some speculation about future possibilities.