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arXiv:cond-mat/9812441 · Physical Review Letters

Breakdown of the Chiral Luttinger Liquid in One Dimension

A. F. Ho, Piers Coleman

Published Aug 16, 1999• 5 citations
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Abstract

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.

DOI
10.1103/physrevlett.83.1383

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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.
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 revisit the low-energy physics of one-dimensional spinless fermion liquids, showing that with sufficiently strong interactions the conventional Luttinger liquid can give way to a strong-pairing phase. While the density fluctuations in both phases are described by a gapless Luttinger liquid, single fermion excitations are gapped only in the strong-pairing phase. Smooth spatial Interfaces between the two phases lead to topological degeneracies in the ground-state and low-energy spectrum of the density fluctuations. Using a concrete microscopic model, with both single-particle and pair hopping, we show that the strong-pairing state is established through emergence of a new low-energy fermionic mode. We characterize the two phases with numerical calculations using the density matrix renormalization group. In particular we find enhancement of the central charge from c=1c=1c=1 in the two Luttinger liquid phases to c=3/2c=3/2c=3/2 at the critical point, which gives direct evidence for an emergent critical Majorana mode. Finally, we confirm the existence of topological degeneracies in the low-energy phonon spectrum, associated with spatial interfaces between the two phases.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
The explicitly soluble Luttinger model is used as a basis for the description of the general interacting Fermi gas in one dimension, which will be called 'Luttinger liquid theory', by analogy with Fermi liquid theory. The excitation spectrum of the Luttinger model is described by density-wave, charge and current excitations; its spectral properties determine a characteristic parameter that controls the correlation function exponents. These relations are shown to survive in non-soluble generalisations of the model with a non-linear fermion dispersion. It is proposed that this low-energy structure is universal to a wide class of 1D systems with conducting or fluid properties, including spin chains.
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.
When a local potential changes abruptly in time, an electron gas responds by shifting to a new state which at long times is orthogonal to the one in the absence of the local potential. This is known as Anderson's orthogonality catastrophe and it is relevant for the so-called x-ray edge or Fermi-edge singularity, and for tunneling into an interacting one-dimensional system of fermions. It often happens that the finite frequency response of the photon absorption or the tunneling density of states exhibits a singular behavior as a function of frequency: [Formula: see text], where ω(th) is a threshold frequency and α is an exponent characterizing the singular response. In this review singular responses of spin-incoherent Luttinger liquids are reviewed. Such responses most often do not fall into the familiar form above, but instead typically exhibit logarithmic corrections and display a much higher universality in terms of the microscopic interactions in the theory. Specific predictions are made, the current experimental situation is summarized and key outstanding theoretical issues related to spin-incoherent Luttinger liquids are highlighted.