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Europhysics Letters (EPL)

Spin transport through a 1D Mott-Hubbard insulator of finite length

Vadim V. Ponomarenko

Published Jan 1, 2014• 2 citations
Publisher

Abstract

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.

DOI
10.1209/0295-5075/105/17008

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Journal of Physics A Mathematical and Theoretical•Jun 9, 2014•1 citation
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.
Transport between two Fermi liquid reservoirs through a one channel wire of length LLL is examined when the 1D electron system has an energy gap $2M\ensuremath{\gg}{v}_{c}/L$ ( vc{v}_{c}vc​ is the charge velocity in the wire). In the spinless case transformation of reservoir electrons into solitons of fractional charge entails a crossover from a Fermi liquid regime of integer charge transfer below the crossover energy ${T}_{x}\ensuremath{\ll}{T}_{L}$ to an insulating regime of the fractional charge transfer in the current vs voltage, conductance vs temperature, and the shot noise. Similar behavior is predicted for a spin Mott insulator of filling factor $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}n/({2m}^{\ensuremath{'}})$ ( ${n,m}^{\ensuremath{'}}$ are positive integers).
Transport through a 1D Mott-Hubbard insulator of a finite length LLL is studied beyond the perturbative approach. At a special value of the low-energy constant of the interaction we have mapped the problem onto exactly solvable models and found current vs voltage at a high temperature T>max(m,TL)T>\mathrm{max}({m,T}_{L})T>max(m,TL​) and at a low energy T,V<TLT,V<{T}_{L}T,V<TL​ ( TL=vc/L{T}_{L}{\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}v}_{c}/LTL​=vc​/L; vc{v}_{c}vc​: charge velocity). The result shows that for the strong interaction creating a large Mott-Hubbard gap $2m\ensuremath{\gg}{T}_{L}$ inside the wire, the transport is suppressed near half filling everywhere inside the gap except for a small region where $V,T<{T}_{L}\mathrm{exp}(\ensuremath{-}2m/{T}_{L})$.
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 one-dimensional (1D) Hubbard model at zero magnetic field, h=0, and zero chemical potential, μ=0, and, thus, at the h=μ=0[SO(4)×U(1)]/Z2-symmetry point, is the paradigmatic quantum system for low-dimensional strongly correlated electron 1D Mott-Hubbard insulators. Studies relying on hydrodynamic theory and Kardar-Parisi-Zhang (KPZ) scaling have found that for the 1D Hubbard model at the h=μ=0 point, both spin and charge transports are for all temperatures T>0 that are anomalous superdiffusive. In this paper, we review recent results that correct the hydrodynamic theory and KPZ scaling prediction concerning the finite-temperature charge transport, as it is normal diffusive for all finite temperatures T>0. This failure of the hydrodynamic theory and KPZ scaling stems from the misleading assumption that the h=μ=0[SO(4)×U(1)]/Z2-symmetry point rather is a h=μ=0SO(4)-symmetry point. Such an assumption has ignored the quantum effects associated with the τ-translational U(1) symmetry beyond SO(4).
Finite-temperature T > 0 transport properties of integrable and nonintegrable one-dimensional (1D) many-particle quantum systems are rather different, showing ballistic and diffusive behavior, respectively. The repulsive 1D Hubbard model is a prominent example of an integrable correlated system. For electronic densities n ≠ 1 (and spin densities m ≠ 0) it is an ideal charge (and spin) conductor, with ballistic charge (and spin) transport for T ⩾ 0. In spite of the fact that it is solvable by the Bethe ansatz, at n = 1 (and m = 0) its T > 0 charge (and spin) transport properties are an issue that remains poorly understood. Here we combine this solution with symmetry and the explicit calculation of current-operator matrix elements between energy eigenstates to show that for on-site repulsion U > 0 and at n = 1 the charge stiffness D η ( T ) vanishes for T > 0 in the thermodynamic limit. A similar behavior is found by such methods for the spin stiffness D s ( T ) for U > 0 and T > 0, which vanishes at m = 0. This absence of finite temperature n = 1 ballistic charge transport and m = 0 ballistic spin transport are exact results that clarify long-standing open problems.