Schulz and Shastry recently introduced a new class of exactly solvable one-dimensional models with non-Fermi-liquid physics. First, we show that this behavior is caused by the existence of anyonic excitations ( i.e. states with non-trivial conformal spin) with charge [(δ)/(2π)], where δ is the phase shift of the potential. Since δ is an arbitrary real number, these anyons may therefore carry an irrational charge. Next, in view of this result we revisit the general classification of non-Fermi-liquid two-species gapless 1D models: while it is usually thought that c = 1 × c = 1 theories form a four-parameter space, we establish that the most general non-relativistic conformal field theory has ten parameters. This defines a single larger universality class with sub-manifolds associated with the standard Luttinger liquid (with spin-charge separation), and the Schulz-Shastry class of models.
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains. We focus on itinerant Ising (Majorana) and Fibonacci anyons, which are, respectively, related to SU
(2)2 and SU
(2)3 anyons and also, respectively, describe quasiparticles of the Moore-Read and
Z3-Read-Rezayi fractional quantum Hall states. Following the derivation of the electronic large-U effective Hubbard model, we derive effective anyonic
t-
J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (neutral) anyonic degrees of freedom---a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum can be performed by combining spectra for charge and anyonic sectors. The anyonic sector is that of a squeezed chain of localized interacting anyons and, hence, is described by the same conformal field theory (CFT), with central charge
c=1/2 for Ising anyons and
c=7/10 or
c=4/5 for Fibonacci anyons with antiferromagnetic or ferromagnetic coupling, respectively. The charge sector is the spectrum of a chain of hard-core bosons subject to phase shifts which coincide with the momenta of the combined anyonic eigenstates, revealing a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization. The combined central charge extracted from the entanglement entropy between segments of the chain is shown to be
1+c, where
c is the central charge of the underlying CFT of the localized anyon (squeezed) chain.
We review the progress in the theory of one-dimensional (ID) Fermi liquids which has occurred over the past decade. The usual Fermi liquid theory, based on a quasi-particle picture, breaks down in one dimension because of the Peierls divergence in the particle-hole bubble, producing anomalous dimensions of operators, and because of charge-spin separation. Both are related to the importance of scattering processes transferring finite momentum. A description of the low-energy properties of gapless 1D quantum systems can be based on the exactly solvable Luttinger model which incorporates these features, and whose correlation functions can be calculated. Special properties of the eigenvalue spectrum, parameterized by one renormalized velocity and one effective coupling constant per degree of freedom, fully describe the physics of this model. Other gapless 1D models share these properties in a low-energy subspace. The concept of a "Luttinger liquid" implies that their low-energy properties are described by an effective Luttinger model, and constitutes the universality class of these quantum systems. Once the mapping on the Luttinger model is achieved, one has an asymptotically exact solution of the 1D many-body problem. Lattice models identified as Luttinger liquids include the 1D Hubbard model off half-filling, and variants such as the t-J- or the extended Hubbard model. In addition, 1D electron-phonon systems or metals with impurities can be Luttinger liquids, as well as the edge states in the quantum Hall effect.
We study finite-size properties of Schulz-Shastry-Luttinger liquids to reveal anyonic signatures, realized as low-energy excitations on top of the helical ground state in saturated spin-1/2 zigzag chains. The model features asymmetric and marginal couplings of density and phase gradients and belongs to the Schulz-Shastry class. We investigate periodic and Dirichlet boundary conditions and discuss its diagonalization as well as its stability. Although Dirichlet boundary conditions require a fine-tuning of coupling constants and universal parameters, only their magnitude is restricted for cyclic systems. We derive boundary characteristic quantities like Friedel oscillations and persistent currents. Finally, we discuss the bulk and boundary behavior of the longitudinal spin correlations including subleading corrections.
We construct models of interacting itinerant non-Abelian anyons moving along one-dimensional chains, focusing, in particular, on itinerant Ising anyon chains, and derive effective anyonic t-J models for the low-energy sectors. Solving these models by exact diagonalization, we find a fractionalization of the anyons into charge and (non-Abelian) anyonic degrees of freedom--a generalization of spin-charge separation of electrons which occurs in Luttinger liquids. A detailed description of the excitation spectrum by combining spectra for charge and anyonic sectors requires a subtle coupling between charge and anyonic excitations at the microscopic level (which we also find to be present in Luttinger liquids), despite the macroscopic fractionalization.
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.