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arXiv:1210.1398 · Physical Review B

Charge and spin fractionalization beyond the Luttinger-liquid paradigm

A. Moreno, Alejandro Muramatsu, J. M. P. Carmelo

Published Feb 4, 2013• 18 citations
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Abstract

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 $t$-$J$ 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 ($J=2t$) 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.

DOI
10.1103/physrevb.87.075101

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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 investigate the phenomenon of spin and charge fractionalization of an electron injected into a strongly correlated chain, namely the 1D t-J model, by means of the time-dependent DMRG method. In order to compare and interpret our results we use the Luttinger Liquid (LL) theory and the exact solution of the model at the supersymmetric point as analytical framework. We show that fractionalization of charge and spin beyond the forms described by LL theory takes place at high energies, when a spin-1/2 fermion is injected into the t-J model.
We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows us to make contact with the Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wave function is also given for that spinon state. Our construction of the spectroscopy based on fractional states leads to several unique physical insights: in the low-energy limit, we find that the Sz=0{S}_{z}=0Sz​=0 continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the one-dimensional counterpart of the Laughlin fractional Quantum Hall effect quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states that we are able to describe.
Using the time-dependent density-matrix renormalization group, we study the time evolution of electronic wave packets in the one-dimensional extended Hubbard model with on-site and nearest-neighbor repulsion, UUU and VVV, respectively. As expected, the wave packets separate into spin-only and charge-only excitations (spin-charge separation). Charge and spin velocities exhibit nonmonotonic dependence on VVV. For small and intermediate values of VVV, both velocities increase with VVV. However, the charge velocity exhibits a stronger dependence than that of the spin, leading to a more pronounced spin-charge separation. Charge fractionalization, on the other hand, is weakly affected by VVV. The results are explained in terms of Luttinger liquid theory in the weak-coupling limit and an effective model in the strong-coupling regime.
At low temperature, collective excitations of one-dimensional (1D) interacting fermions exhibit spin-charge separation, a unique feature predicted by the Tomonaga-Luttinger liquid (TLL) theory, but a rigorous understanding remains challenging. Using the thermodynamic Bethe ansatz (TBA) formalism, we analytically derive universal properties of a 1D repulsive spin-1/2 Fermi gas with arbitrary interaction strength. We show how spin-charge separation emerges from the exact TBA formalism, and how it is disrupted by the interplay between the two degrees of freedom that brings us beyond the TLL paradigm. Based on the exact low-lying excitation spectra, we further evaluate the spin and charge dynamical structure factors (DSFs). The peaks of the DSFs exhibit distinguishable propagating velocities of spin and charge as functions of interaction strength, which can be observed by Bragg spectroscopy with ultracold atoms.
The authors introduce a new representation of the Bethe ansatz solutions for 1D Luttinger liquids which describes the spectral properties and asymptotic behaviour of the correlation functions of the 1D Hubbard model in terms of the renormalized interaction of charge and spin pseudoparticles. The study of the low-lying eigenstates is reduced to the familiar languages of band theory and of the Fermi liquid. The authors results provide a better understanding of the decoupling of charge and spin degrees of freedom in 1D interacting systems, generalize the concept of a Landau liquid and may be relevant to the physics of higher dimensional systems.
The authors introduce a new representation of the Bethe ansatz solutions for 1D Luttinger liquids which describes the spectral properties and asymptotic behaviour of the correlation functions of the 1D Hubbard model in terms of the renormalized interaction of charge and spin pseudoparticles. The study of the low-lying eigenstates is reduced to the familiar languages of band theory and of the Fermi liquid. The authors results provide a better understanding of the decoupling of charge and spin degrees of freedom in 1D interacting systems, generalize the concept of a Landau liquid and may be relevant to the physics of higher dimensional systems.