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

arXiv:cond-mat/9903197 · Physical review. B, Condensed matter

Bound-state instability of the chiral Luttinger liquid in one dimension

A. F. Ho, Piers Coleman

Published Jul 15, 2000• 0 citations
arXivHTMLPDF

Abstract

We have developed a ``bootstrap'' method for solving a class of interacting one-dimensional chiral fermions. The conventional model for interacting right-moving electrons with spin has an SO(4) symmetry, and can be written as four interacting Majorana fermions, each with the same velocity. We have found a method for solving some cases when the velocities of these Majorana fermions are no longer equal. We demonstrate in some detail the remarkable result that corrections to the skeleton self-energy identically vanish for these models, and this enables us to solve them exactly. For the cases where the model can be solved by bosonization, our method can be explicitly checked. However, we are also able to solve some cases where the excitation spectrum differs qualitatively from a Luttinger liquid. Of particular interest is the so-called SO(3) model, where a triplet of Majorana fermions, moving at one velocity, interact with a single Majorana fermion moving at another velocity. Using our method we show, that a sharp bound (or antibound) state splits off from the original Luttinger-liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.

DOI
10.1103/physrevb.62.1688

Related papers

Breakdown of the Chiral Luttinger Liquid in One Dimension

A. F. Ho, Piers Coleman

We have developed a fermionic boot-strap method to solve a class of chiral one-dimensional fermion models. Using this scheme, we show that Luttinger liquid behavior in a gas of four interacting chiral Majorana fermions is highly sensitive to the velocity degeneracy. Upon changing the velocity of one chiral fermion, a sharp bound (or antibound) state splits off from the original Luttinger liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.

arXiv:cond-mat/9812441•Physical Review Letters•Aug 16, 1999•5 citations

- Non-chiral Bosonization of Fermions in One Dimension

Girish S. Setlur

In this chapter, an alternative to the conventional approach to bosonization in one dimension that invokes the Dirac equation in 1+1 dimension with chiral ‘rightmovers’ and ‘left-movers’ is proposed (the conventional approach is discussed, e.g., in the text book by Giamarchi, Quantum Physics in One Dimension, Oxford). This technique allows us to use a basis different from the plane wave basis that makes this non-chiral approach ideally suited to study Luttinger liquids that have a boundary(s) or impurities that break translational symmetry. We provide a simple solution to the electron Green function for the problem of a Luttinger liquid (LL) with a boundary and also for a LL with a single impurity. The present method is significantly easier than the g-ology based standard bosonization and other methods that require a combination of RG (renormalization group) along with bosonization/refermionization techniques. This method is superior because it reproduces, as a limiting case, the exact closed formula for the (random phase approximation) RPA Green function of free fermions with the single impurity without mutual interactions obtainable trivially using Fermi algebra, whereas the conventional bosonization/RG approaches are not able to do so.

CRC Press eBooks•Dec 5, 2013•0 citations

Luttinger liquid in a solvable two-dimensional model

R. Hlubina

We consider spinless electrons in two dimensions with the bare spectrum ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{p}}$=${\mathit{v}}_{\mathit{F}}$(\ensuremath{\Vert}${\mathit{p}}_{\mathit{x}}$\ensuremath{\Vert}+\ensuremath{\Vert}${\mathit{p}}_{\mathit{y}}$\ensuremath{\Vert}). In momentum space, the interactions among electrons have a finite range ${\mathit{q}}_{0}$, which is small compared to the Fermi momentum. A golden-rule calculation of the electron lifetime indictes a breakdown of Landau's Fermi-liquid theory in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.

arXiv:cond-mat/9312083•Physical review. B, Condensed matter•Sep 15, 1994•17 citations

Interacting fermions in one dimension: The Tomonaga-Luttinger model

Kurt Schönhammer

The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.

arXiv:cond-mat/9710330•Oct 30, 1997•12 citations

Generalized hard-core fermions in one dimension: An exactly solvable Luttinger liquid

G. Gómez-Santos

A model for interacting spinless fermions in one dimension is presented and solved exactly. The solution is shown to be a Luttinger liquid with charge density wave instabilities for well characterized filling factors. Under very plausible assumptions, the exact solution is extended to the spin case with the usual charge-spin separation into two different Luttinger liquids. Required technicalities are almost trivial but the solution is very rich: Periodicity of instabilities and correlation exponents can be chosen at will.

Physical Review Letters•Jun 14, 1993•30 citations

Consistent bosonization treatment of interacting fermion systems in dimensions higher than one: Requirement for a Luttinger-liquid state

Y. M. Li

We present a consistent bosonization treatment for interacting fermion systems in dimensions higher than one. In our analysis, all physically relevant excitations corresponding to forward scattering are included and physical results do not depend on the cutoff parameter introduced in the bosonization process. We calculate the fermion momentum distribution function, from which a condition for the appearance of a Luttinger-liquid state is derived. We find that, for nonsingular interactions, the zero-frequency particle-hole pair excitations directly destroy the orthogonality catastrophe so that there is no Luttinger-liquid fixed point. We further consider the unscreened Coulomb interaction and find that the system remains in a Fermi-liquid state. We observe that the class of singular interactions which are expected to generate a Luttinger-liquid state is very tightly constrained by these zero-frequency excitations.

Physical review. B, Condensed matter•May 15, 1995•2 citations
We have developed a ``bootstrap'' method for solving a class of interacting one-dimensional chiral fermions. The conventional model for interacting right-moving electrons with spin has an SO(4) symmetry, and can be written as four interacting Majorana fermions, each with the same velocity. We have found a method for solving some cases when the velocities of these Majorana fermions are no longer equal. We demonstrate in some detail the remarkable result that corrections to the skeleton self-energy identically vanish for these models, and this enables us to solve them exactly. For the cases where the model can be solved by bosonization, our method can be explicitly checked. However, we are also able to solve some cases where the excitation spectrum differs qualitatively from a Luttinger liquid. Of particular interest is the so-called SO(3) model, where a triplet of Majorana fermions, moving at one velocity, interact with a single Majorana fermion moving at another velocity. Using our method we show, that a sharp bound (or antibound) state splits off from the original Luttinger-liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.
We have developed a fermionic boot-strap method to solve a class of chiral one-dimensional fermion models. Using this scheme, we show that Luttinger liquid behavior in a gas of four interacting chiral Majorana fermions is highly sensitive to the velocity degeneracy. Upon changing the velocity of one chiral fermion, a sharp bound (or antibound) state splits off from the original Luttinger liquid continuum, cutting off the x-ray singularity to form a broad incoherent excitation with a lifetime that grows linearly with frequency.
In this chapter, an alternative to the conventional approach to bosonization in one dimension that invokes the Dirac equation in 1+1 dimension with chiral ‘rightmovers’ and ‘left-movers’ is proposed (the conventional approach is discussed, e.g., in the text book by Giamarchi, Quantum Physics in One Dimension, Oxford). This technique allows us to use a basis different from the plane wave basis that makes this non-chiral approach ideally suited to study Luttinger liquids that have a boundary(s) or impurities that break translational symmetry. We provide a simple solution to the electron Green function for the problem of a Luttinger liquid (LL) with a boundary and also for a LL with a single impurity. The present method is significantly easier than the g-ology based standard bosonization and other methods that require a combination of RG (renormalization group) along with bosonization/refermionization techniques. This method is superior because it reproduces, as a limiting case, the exact closed formula for the (random phase approximation) RPA Green function of free fermions with the single impurity without mutual interactions obtainable trivially using Fermi algebra, whereas the conventional bosonization/RG approaches are not able to do so.
We consider spinless electrons in two dimensions with the bare spectrum ${\mathrm{\ensuremath{\epsilon}}}_{\mathbf{p}}$=vF{\mathit{v}}_{\mathit{F}}vF​(\ensuremath{\Vert}px{\mathit{p}}_{\mathit{x}}px​\ensuremath{\Vert}+\ensuremath{\Vert}py{\mathit{p}}_{\mathit{y}}py​\ensuremath{\Vert}). In momentum space, the interactions among electrons have a finite range q0{\mathit{q}}_{0}q0​, which is small compared to the Fermi momentum. A golden-rule calculation of the electron lifetime indictes a breakdown of Landau's Fermi-liquid theory in the model. At the one-loop level of perturbation theory, we show that the density wave and the superconducting instabilities cancel each other and there is no symmetry breaking. We solve the model via bosonization; the excitation spectrum is found to consist of gapless bosonic modes as in a one-dimensional Luttinger liquid.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.
A model for interacting spinless fermions in one dimension is presented and solved exactly. The solution is shown to be a Luttinger liquid with charge density wave instabilities for well characterized filling factors. Under very plausible assumptions, the exact solution is extended to the spin case with the usual charge-spin separation into two different Luttinger liquids. Required technicalities are almost trivial but the solution is very rich: Periodicity of instabilities and correlation exponents can be chosen at will.
We present a consistent bosonization treatment for interacting fermion systems in dimensions higher than one. In our analysis, all physically relevant excitations corresponding to forward scattering are included and physical results do not depend on the cutoff parameter introduced in the bosonization process. We calculate the fermion momentum distribution function, from which a condition for the appearance of a Luttinger-liquid state is derived. We find that, for nonsingular interactions, the zero-frequency particle-hole pair excitations directly destroy the orthogonality catastrophe so that there is no Luttinger-liquid fixed point. We further consider the unscreened Coulomb interaction and find that the system remains in a Fermi-liquid state. We observe that the class of singular interactions which are expected to generate a Luttinger-liquid state is very tightly constrained by these zero-frequency excitations.