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arXiv:cond-mat/0609754 · Physical Review B

Bosonization approach to charge and spin dynamics of one-dimensional spin- 12 fermions with band curvature in a clean quantum wire

S. Teber

Published Jul 9, 2007• 33 citations
arXivHTMLPDF

Abstract

We consider one-dimensional spin-$\frac{1}{2}$ fermions in a clean quantum wire, with forward scattering interactions and a nonlinear single-particle spectrum ${\ensuremath{\xi}}_{k}=v\ensuremath{\mid}k\ensuremath{\mid}+{k}^{2}∕2m$, where $v$ is the Fermi velocity and $1∕m$ is the band curvature. We calculate the dynamical structure factor (DSF) of the model at small wave vector $q$ with the help of the bosonization technique. For spinless fermions, we show that, starting from the single-parametric spectrum $\ensuremath{\omega}=u\ensuremath{\mid}q\ensuremath{\mid}$, bosonization emulates the two-parametric excitation spectrum $\ensuremath{\omega}=u\ensuremath{\mid}q\ensuremath{\mid}\ifmmode\pm\else\textpm\fi{}{q}^{2}∕2{m}^{*}$, where ${m}^{*}$ decreases with increasing repulsive interactions. Moreover, away from the excitation cone, i.e., $\ensuremath{\omega}⪢u\ensuremath{\mid}q\ensuremath{\mid}$, bosonization yields the two-pair excitation continuum of the DSF. For spinful fermions, we show that the spin-charge coupling (SCC) due to band curvature affects the charge and spin DSFs in an asymmetric way. For the charge DSF, SCC manifests as a two-peak structure: a charge peak at $\ensuremath{\omega}={u}_{\ensuremath{\rho}}\ensuremath{\mid}q\ensuremath{\mid}$ but also a spin peak at $\ensuremath{\omega}={u}_{\ensuremath{\sigma}}\ensuremath{\mid}q\ensuremath{\mid}$, as charge fluctuations may decay via chargeless spin-singlet excitations. For the magnetic DSF, SCC manifests as a continuous transfer of magnetic spectral weight to frequencies $\ensuremath{\omega}>{u}_{\ensuremath{\sigma}}\ensuremath{\mid}q\ensuremath{\mid}$, as spin fluctuations decay via pairs of chargeless spin and spinless charge-neutral excitations.

DOI
10.1103/physrevb.76.045309

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The low energy properties of different one-dimensional fermionic lattice models are investigated using the bosonization technique. We attach much importance to a proper consideration of the Klein factors which are neglected or inaccurately treated in the majority of the literature. In order to deal with the nonlinear terms, arising from the dimerization or similar scattering processes, we use the self-consistent harmonic approximation (SCHA), a variational method where the nonlinearities are replaced by a quadratic potential. We construct an extension of the SCHA in order to account for the existence of Klein factors in bosonized Hamiltonians. As prototypical models we consider one-dimensional dimerized spinless fermions and Hubbard models with periodic modulation of the hopping and of the chemical potential, respectively. In an attempt to describe more realistic systems we also consider a model where one-dimensional electrons are coupled to a three-dimensional conduction band via a local interaction. In order to see how the one-dimensional electron system is modified in the presence of the three-dimensional environment, we calculate the charge and the spin susceptibility, and analyze the collective excitations within RPA. We find that the spin excitations remain unaffected while in the charge channel, the collective modes of the one-dimensional and three-dimensional subsystems are coupled.
We revisit the dynamic spin susceptibility $\ensuremath{\chi}(q,\ensuremath{\omega})$ of one-dimensional interacting fermions. To second order in the interaction, backscattering results in a logarithmic correction to $\ensuremath{\chi}(q,\ensuremath{\omega})$ at $q\ensuremath{\ll}{k}_{F}$, even if the single-particle spectrum is linearized near the Fermi points. Consequently, the dynamic spin structure factor $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ is nonzero at frequencies above the single-particle continuum. In the boson language, this effect results from the marginally irrelevant backscattering operator of the sine-Gordon model. Away from the threshold, the high-frequency tail of $\mathrm{Im}\ensuremath{\chi}(q,\ensuremath{\omega})$ due to backscattering is larger than that due to finite mass by a factor of kF/q{k}_{F}/qkF​/q. We derive the renormalization group equations for the coupling constants of the $g\ensuremath{-}\text{ology}$ model at finite $\ensuremath{\omega}$ and qqq and find the corresponding expression for $\ensuremath{\chi}(q,\ensuremath{\omega})$, valid to all orders in the interaction but not in the immediate vicinity of the continuum boundary, where the finite-mass effects become dominant.
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.
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We study the influence of spin on the quantum interference of interacting electrons in a single-channel disordered quantum wire within the framework of the Luttinger liquid (LL) model. The nature of the electron interference in a spinful LL is particularly nontrivial because the elementary bosonic excitations that carry charge and spin propagate with different velocities. We extend the functional-bosonization approach to treat the fermionic and bosonic degrees of freedom in a disordered spinful LL on an equal footing. We analyze the effect of spin-charge separation at finite temperature both on the spectral properties of single-particle fermionic excitations and on the conductivity of a disordered quantum wire. We demonstrate that the notion of weak localization, related to the interference of multiple-scattered electron waves and their decoherence due to electron-electron scattering, remains applicable to the spin-charge separated system. The relevant dephasing length, governed by the interplay of inelastic electron-electron interactions and spin-charge separation, is found to be parametrically shorter than in a spinless LL. We calculate both the quantum (weak localization) and classical (memory effect) corrections to the conductivity of a disordered spinful LL. The classical correction is shown to dominate in the limit of high temperature.