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Journal of Physics Condensed Matter

Charge and spin transport in the one-dimensional Hubbard model

Nuno M. R. Peres, Pedro D Sacramento, J. M. P. Carmelo

Published May 18, 2001• 4 citations
Publisher

Abstract

We study the charge and spin currents transported by the elementary excitations of the one-dimensional (1D) Hubbard model and derive the corresponding current spectra. We present results both for finite-size systems and in the thermodynamic limit. This includes finding the couplings of both the low-energy and finite-energy (string) excitations to external charge and spin probes. At zero magnetic field the general structure of the charge-spin separation survives at all energy scales and the effective charges of both the low-energy and finite-energy charge excitations are studied as functions of the on-site Coulomb interaction U , electronic density n , and applied magnetic field H . In some limits the effective charge of the low-energy excitations equals that of the electrons, whereas that of the finite-energy charge-string excitations of rapidity length γ is found to be 2γ times the electronic charge. At U = ∞ the spin excitations do not contribute to spin transport, whereas the low-energy charge excitations feel an effective flux given by (ϕ ↑ N ↑ -ϕ ↓ N ↓ )/( N ↑ + N ↓ ), where N σ is the number of electrons of spin σ and ϕ σ is a spin-dependent flux. This reveals that at zero magnetic field and U = ∞ there is no spin transport, while at finite magnetic field the low-energy charge excitations also carry spin. In the U >> t limit the spin is carried both by holons and spinons. Finally, we find that the charge- and spin-current spectra can be derived from a semi-classical approach.

DOI
10.1088/0953-8984/13/22/310

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We study the charge and spin currents transported by the elementary excitations of the one-dimensional (1D) Hubbard model and derive the corresponding current spectra. We present results both for finite-size systems and in the thermodynamic limit. This includes finding the couplings of both the low-energy and finite-energy (string) excitations to external charge and spin probes. At zero magnetic field the general structure of the charge-spin separation survives at all energy scales and the effective charges of both the low-energy and finite-energy charge excitations are studied as functions of the on-site Coulomb interaction U , electronic density n , and applied magnetic field H . In some limits the effective charge of the low-energy excitations equals that of the electrons, whereas that of the finite-energy charge-string excitations of rapidity length γ is found to be 2γ times the electronic charge. At U = ∞ the spin excitations do not contribute to spin transport, whereas the low-energy charge excitations feel an effective flux given by (ϕ ↑ N ↑ -ϕ ↓ N ↓ )/( N ↑ + N ↓ ), where N σ is the number of electrons of spin σ and ϕ σ is a spin-dependent flux. This reveals that at zero magnetic field and U = ∞ there is no spin transport, while at finite magnetic field the low-energy charge excitations also carry spin. In the U >> t limit the spin is carried both by holons and spinons. Finally, we find that the charge- and spin-current spectra can be derived from a semi-classical approach.
The transport of charge and spin at finite energies is studied for the Hubbard chain in a magnetic field by means of the pseudoparticle perturbation theory. In the general case, this involves the solution of an infinite set of Bethe-ansatz equations with a flux. Our results refer to all densities and magnetizations. We express the charge and the spin-diffusion currents in terms of elementary currents associated with the charge and spin carriers. We show that these are the α,γpseudoparticles (with α=c,s and γ=0,1,2,3,...) and we find their couplings to charge and spin. We also study the ratios of the pseudoparticle charge and spin transport masses over the corresponding static mass. These ratios provide valuable information on the effects of electronic correlations in the transport properties of the quantum system. We show that the transport of charge and spin in the Hubbard chain can, alternatively, be described by means of pseudoparticle kinetic equations. This follows from the occurrence of only forward-scattering pseudoparticle interactions at all energies.
Most studies on the one-dimensional (1D) Hubbard model with transfer integral ttt and on-site repulsion UUU have assumed that for u=U/4t>0u=U/4t>0u=U/4t>0 and $h=\ensuremath{\mu}=0$, where hhh denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the U>0U>0U>0 Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond SO(4)\mathrm{SO}(4)SO(4) is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the h>0h>0h>0 spin and $\ensuremath{\mu}>0$ charge stiffnesses and the h=0h=0h=0 spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.
We study low-energy spin and charge transport through a 1D Mott-Hubbard insulator of finite length L attached to Fermi liquid reservoirs, which, in the presence of spin accumulation, are characterized by different electrochemical potentials for electrons of opposite spin polarizations. At temperatures less than ( v c : charge velocity in the wire) and under the assumption that the Hubbard gap 2 M is large enough, M > T L , we calculate the average currents (charge and spin) and their zero-frequency correlators. The average spin (charge) current depends only on the difference (sum) of the spin-dependent voltages and even a weak electron backscattering of low rate leads to the spin current suppression at smaller than . The spin current recovers its free mode behavior at spin voltage or temperature larger than Γ. Suppression of the spin-charge correlator suggesting the appearance of spin-charge separation needs both to be larger than Γ. In the absence of the average charge current at its shot noise is proportional to the average spin backscattered current defined by V s and can be used to measure the spin accumulation in the reservoirs. The relation of these results to Kondo dot transport in the Toulouse limit is also clarified.
We study low energy spin and charge transport through a 1D Mott-Hubbard insulator of finite length LLL attached to Fermi liquid reservoirs characterized by different chemical potentials for electrons of opposite spin polarizations as it happens in quantum spin Hall insulators. We calculate the average currents (charge and spin) and their correlators and demonstrate how a transition induced by the reservoirs to the low energy Fermi liquid regime results in breakdown of the spin-charge separation, which is visible in the presence of the spin dependent voltages and a weak one electron scattering in the system. These calculations are carried out under assumption that the Hubbard gap 2M is large enough: M>TL≡vc/LM > T_L \equiv v_c/LM>TL​≡vc​/L (vcv_cvc​: charge velocity in the wire) and the scattering rate Γs≪TLΓ_s \ll T_LΓs​≪TL​. Relation of these results to Kondo dot transport in the Toulouse limit is also clarified.
The problem of spin-charge separation is addressed by considering current carrying states in a one-dimensional system of interacting electrons. Exact results for the Hubbard model and the bosonization method are exploited. We find that away from half filling both spin and charge excitations carry charge currents which are proportional to their momenta for most cases. Being in a qualitative agreement with the noninteracting ld Fermi gas picture, this result contradicts the spin-charge separation concept as it is usually derived e.g. from the bosonization approach or in a strong repulsion picture. For weak coupling the paradox is resolved by taking into account the Fermi velocity dispersion and by reexamining the structure of the current operator within the bosonization approach. For strong coupling we recover an effect of hale drag by a spinon as reflected in their momentum shifts.
In this paper we study the charge and spin currents transported by the elementary excitations of the one-dimensional Hubbard model. The corresponding current spectra are obtained by both analytic methods and numerical solution of the Bethe-ansatz equations. For the case of half-filling, we find that the spin-triplet excitations carry spin but no charge, while charge ηηη-spin triplet excitations carry charge but no spin, and both spin-singlet and charge ηηη-spin-singlet excitations carry neither spin nor charge currents.