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arXiv:0906.0811 · Physical Review B

Sharp fractional charges in Luttinger liquids

Jon Magne Leinaas, Mats Horsdal, T. H. Hansson

Published Sep 29, 2009• 15 citations
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Abstract

We examine charge fractionalization by chiral separation in a one-dimensional fermion system described by Luttinger liquid theory. The focus is on the question of whether the fractional charges are quantum-mechanically sharp, and in the analysis we make a distinction between the global charge, which is restricted by boundary conditions, and the local charge where a background contribution is subtracted. We show, by way of examples, that fractional charges of arbitrary values, all which are quantum-mechanically sharp, can be introduced by different initial conditions. Since the system is gapless, excitations of arbitrary low frequency contribute to the fluctuations, it is important to make a precise definition of sharp charges, and this we we do by subtraction of the ground-state contribution. We very briefly comment on the relevance of our analysis for proposed experiments.

DOI
10.1103/physrevb.80.115327

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We examine charge fractionalization by chiral separation in a one-dimensional fermion system described by Luttinger liquid theory. The focus is on the question of whether the fractional charges are quantum-mechanically sharp, and in the analysis we make a distinction between the global charge, which is restricted by boundary conditions, and the local charge where a background contribution is subtracted. We show, by way of examples, that fractional charges of arbitrary values, all which are quantum-mechanically sharp, can be introduced by different initial conditions. Since the system is gapless, excitations of arbitrary low frequency contribute to the fluctuations, it is important to make a precise definition of sharp charges, and this we we do by subtraction of the ground-state contribution. We very briefly comment on the relevance of our analysis for proposed experiments.
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It is well established that at low energies one-dimensional (1D) fermionic systems are described by the Luttinger-liquid (LL) theory, which predicts phenomena such as spin-charge separation and charge fractionalization into chiral modes. Here we show through the time evolution of an electron injected into a 1D ttt-JJJ model, obtained with time-dependent density matrix renormalization group, that a further fractionalization of both charge and spin takes place beyond the hydrodynamic limit. Its dynamics can be understood at the supersymmetric point (J=2tJ=2tJ=2t) in terms of the excitations of the Bethe-ansatz solution. Furthermore we show that fractionalization with similar characteristics extends to the whole region corresponding to a repulsive LL.
We discuss how one-dimensional interacting fermion systems, which in the low-energy approximation are described by Luttinger liquid theory, can be reformulated as systems of weakly interacting particles with fractional charge and statistics. Our approach is to use Landau's phenomenological approach to Fermi liquid theory, where the quasiparticles are interpreted as adiabatically dressed fermions. In an earlier publication, the local charge carried by these excitations has been shown to be fractional. We focus here on the statistics of the quasiparticles and show that by a change of momentum variables the Landau parameters of the generalized Fermi fluid can be transformed to zero. This change in interaction is compensated by a change of the entropy function, which is consistent with the interpretation of the quasiparticles as satisfying generalized exclusion statistics.
In this paper, we present a description of Haldane's Luttinger liquid which parallels Laughlin's theory of the Fractional Quantum Hall (FQH) incompressible fluid, both exhibiting similar ground states as well as fractional excitations. These two non-Fermi liquids are instances of a generic structure for low-dimensional quantum liquids which we propose to dub Jastrow-Luttinger Fractional Liquids. An important feature of such liquids is the complete fractionalization of the parent particle. In particular, in both one and two dimensions spin-charge separation can be achieved and is indeed suggested to occur for unpolarized quantum Hall systems both at the edge and in the bulk.
We investigate the generic transport in a one-dimensional strongly correlated fermionic chain beyond linear response. Starting from a Gaussian wave packet with positive momentum on top of the ground state, we find that the numerical time evolution splits the signal into at least three distinct fractional charges moving with different velocities. A fractional left-moving charge is expected from conventional Luttinger liquid theory, but for the prediction of the two separate right-moving packets the nonlinearity of the dispersion must also be taken into account. This out-of-equilibrium protocol therefore allows a direct measurement of nonlinear interaction parameters, which also govern threshold singularities of dynamic response functions. The nonlinear Luttinger liquid theory also predicts the correct dynamics at low energies, where it agrees with the conventional Luttinger liquid. Moreover, at high energies, the wave packet dynamics reveals signatures of composite excitations containing two-particle bound states. Our results uncover a simple strategy to probe the nonlinear regime in time-resolved experiments in quantum wires and ultracold-atom platforms.
We show that one-dimensional quantum systems with gapless degrees of freedom and open boundary conditions form a universality class of quantum critical behavior, which we propose to call ``bounded Luttinger liquids.'' They share the following properties with ordinary (periodic) Luttinger liquids: the absence of fermionic quasiparticle excitations, charge-spin separation, and anomalous power-law correlations with exponents whose scaling relations are parametrized by a single coupling constant per degree of freedom, ${K}_{\ensuremath{\nu}}.$ The values of ${K}_{\ensuremath{\nu}}$ are independent of boundary conditions, but the representation of the critical exponents in terms of these ${K}_{\ensuremath{\nu}}'\mathrm{s}$ depends on boundary conditions. We illustrate these scaling relations by exploring general rules for boundary critical exponents derived earlier using the Bethe ansatz solution of the one-dimensional Hubbard model together with boundary conformal field theory, and the theory of Luttinger liquids in finite-size systems. We apply this theory to the photoemission properties of the organic conductors (TMTSF)2X,(\mathrm{TMTSF}{)}_{2}X,(TMTSF)2​X, where TMTSF is tetramethyltetraselenafulvalene, and X=ClO9,X={\mathrm{ClO}}_{9},X=ClO9​, PF6,{\mathrm{PF}}_{6},PF6​, ReO4,\mathrm{Re}{\mathrm{O}}_{4},ReO4​, and discuss to what extent the assumption of finite strands with open boundaries at the sample surface can reconcile the experimental results with independent information on the Luttinger-liquid state in these materials.