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arXiv:cond-mat/0212201 · Physical review. B, Condensed matter

Adiabatic charge and spin transport in interacting quantum wires

Prashant Sharma, Claudio Chamon

Published Jul 22, 2003• 42 citations
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Abstract

We study charge and spin transport through an interacting quantum wire, caused by backscattering of electrons off an effective impurity potential with a periodic time dependence. The adiabatic regime of this pump for charge and spin is shown to depend on the presence of interactions in the wire. For the noninteracting case, the transported quantities in a period are calculated and found to be adiabatic (independent of the pumping frequency) for all frequencies $\ensuremath{\Omega}\ensuremath{\ll}{v}_{F}/\mathcal{l},$ where $\mathcal{l}$ is the range of the scattering potential and ${v}_{F}$ is the Fermi velocity; this result is along the lines of adiabatic pumping in mesoscopic systems. On the other hand, we show that for a wire with repulsive electron-electron interactions, the adiabatic regime is confined to $\ensuremath{\Elzxh}\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\Gamma}},$ where ${\ensuremath{\omega}}_{\ensuremath{\Gamma}}$ is an energy scale set by the backscattering potential. Using symmetry and scaling properties of the quantum wire Hamiltonian, we relate the charge Q and spin S transported through the wire in a period $2\ensuremath{\pi}/\ensuremath{\Omega}$ to an integral involving quasistatic backscattering conductances. We also show that the pumped charge Q (or the spin $S)$ is quantized in the adiabatic limit if the conductance of the system is zero at the stable fixed point of the renormalization-group (RG) transformations. The quanta transported are given by the winding number of a complex coupling $\ensuremath{\Gamma}(\ensuremath{\Omega}t)$ as it traverses a closed path $\mathcal{C}$ enclosing the origin $\ensuremath{\Gamma}=0.$ No adiabatic charge or spin is transported when the RG fixed point corresponds to a perfectly conducting wire, or if the path $\mathcal{C}$ does not enclose the point $\ensuremath{\Gamma}=0.$ By contrast, for a RG marginal conductance---which is the case for noninteracting electrons---the charge transported in a cycle is not quantized, rather it is proportional to the area enclosed by the path or circuit $\mathcal{C},$ in agreement with previous studies of noninteracting mesoscopic systems. Finite size, temperature, and frequency effects on the transported quantities follow from the relation between adiabatic transport coefficients (backscattering conductances) and the quasistatic conductance of the system.

DOI
10.1103/physrevb.68.035321

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We study charge and spin transport through an interacting quantum wire, caused by backscattering of electrons off an effective impurity potential with a periodic time dependence. The adiabatic regime of this pump for charge and spin is shown to depend on the presence of interactions in the wire. For the noninteracting case, the transported quantities in a period are calculated and found to be adiabatic (independent of the pumping frequency) for all frequencies \ensuremathΩ\ensuremath≪vF/l,\ensuremath{\Omega}\ensuremath{\ll}{v}_{F}/\mathcal{l},\ensuremathΩ\ensuremath≪vF​/l, where l\mathcal{l}l is the range of the scattering potential and vF{v}_{F}vF​ is the Fermi velocity; this result is along the lines of adiabatic pumping in mesoscopic systems. On the other hand, we show that for a wire with repulsive electron-electron interactions, the adiabatic regime is confined to \ensuremath\Elzxh\ensuremathΩ\ensuremath≪\ensuremathω\ensuremathΓ,\ensuremath{\Elzxh}\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\Gamma}},\ensuremath\Elzxh\ensuremathΩ\ensuremath≪\ensuremathω\ensuremathΓ​, where \ensuremathω\ensuremathΓ{\ensuremath{\omega}}_{\ensuremath{\Gamma}}\ensuremathω\ensuremathΓ​ is an energy scale set by the backscattering potential. Using symmetry and scaling properties of the quantum wire Hamiltonian, we relate the charge Q and spin S transported through the wire in a period 2\ensuremathπ/\ensuremathΩ2\ensuremath{\pi}/\ensuremath{\Omega}2\ensuremathπ/\ensuremathΩ to an integral involving quasistatic backscattering conductances. We also show that the pumped charge Q (or the spin S)S)S) is quantized in the adiabatic limit if the conductance of the system is zero at the stable fixed point of the renormalization-group (RG) transformations. The quanta transported are given by the winding number of a complex coupling \ensuremathΓ(\ensuremathΩt)\ensuremath{\Gamma}(\ensuremath{\Omega}t)\ensuremathΓ(\ensuremathΩt) as it traverses a closed path C\mathcal{C}C enclosing the origin \ensuremathΓ=0.\ensuremath{\Gamma}=0.\ensuremathΓ=0. No adiabatic charge or spin is transported when the RG fixed point corresponds to a perfectly conducting wire, or if the path C\mathcal{C}C does not enclose the point \ensuremathΓ=0.\ensuremath{\Gamma}=0.\ensuremathΓ=0. By contrast, for a RG marginal conductance---which is the case for noninteracting electrons---the charge transported in a cycle is not quantized, rather it is proportional to the area enclosed by the path or circuit C,\mathcal{C},C, in agreement with previous studies of noninteracting mesoscopic systems. Finite size, temperature, and frequency effects on the transported quantities follow from the relation between adiabatic transport coefficients (backscattering conductances) and the quasistatic conductance of the system.
We investigate a one-dimensional wire of interacting electrons connected to one-dimensional noninteracting leads in the absence and in the presence of a backscattering potential. The ballistic wire separates the charge and spin parts of an incident electron even in the noninteracting leads. The Fourier transform of nonlocal correlation functions is computed for $T\ensuremath{\gg}\ensuremath{\omega}$. In particular, this allows to study the proximity effect, related to the Andreev reflection. In addition, a new type of proximity effect emerges when the wire has normally a tendency towards Wigner crystal formation. The latter is suppressed by the leads below a space-dependent crossover temperature; it gets dominated everywhere by the 2kF{2k}_{F}2kF​ charge-density wave at $T<{L}^{(3/2)(K\ensuremath{-}1)}$ for short-range interactions with parameter K<1/3K<1/3K<1/3. The lowest-order renormalization equations of a weak backscattering potential are derived explicitly at finite temperature. A perturbative expression for the conductance in the presence of a potential with arbitrary spatial extension is given. It depends on the interactions, but is also affected by the noninteracting leads, especially for very repulsive interactions, K<1/3K<1/3K<1/3. This leads to various regimes, depending on temperature and on KKK. For randomly distributed weak impurities, the conductance fluctuations, equal to that of $\mathcal{R}=g\ensuremath{-}{2e}^{2}/h$, are computed. They depend on the interaction parameters, and are different for electrons with or without spin. But the ratio Var(R)/R2\mathrm{Var}(\mathcal{R})/{\mathcal{R}}^{2}Var(R)/R2 stays always of the same order: it is equal to ${L}_{T}/L\ensuremath{\ll}1$ in the high-temperature limit, then saturates at 1/21/21/2 in the low-temperature limit, indicating that the relative fluctuations of R\mathcal{R}R are universal.
Using the equation of motion, we investigate theoretically the dynamical ac conductance of a clean Luttinger-liquid quantum wire adiabatically coupled to Fermi liquid electron reservoirs in the presence of short-ranged electron-electron interactions. For a perfect single mode quantum wire, in the limit of zero-ranged interaction we conclude that the static dc conductance of ω→0 is e 2 / h , which is independent of the electron interactions. While in the dynamical case of ω≠0, the ac conductance oscillates with the amplitude e 2 / h and the period which depends on the interaction strength and the driving frequency as well as the position in the wire.
We study the effects of interelectron interactions on the charge pumped through an adiabatic quantum electron pump. The pumping is through a system of barriers, whose heights are deformed adiabatically. (Weak) interaction effects are introduced through a renormalisation group flow of the scattering matrices and the pumped charge is shown to always approach a quantized value at low temperatures or long length scales. The maximum value of the pumped charge is set by the number of barriers and is given by ${Q}_{\mathrm{max}}={n}_{b}\ensuremath{-}1$. The correlation between the transmission and the charge pumped is studied by seeing how much of the transmission is enclosed by the pumping contour. The (integer) value of the pumped charge at low temperatures is determined by the number of transmission maxima enclosed by the pumping contour. The extra dissipation at finite temperatures leading to the nonquantized values of the pumped charge scales as a power law with the temperature $(Q\ensuremath{-}{Q}_{\mathrm{int}}\ensuremath{\propto}{T}^{2\ensuremath{\alpha}})$, or with the system size $(Q\ensuremath{-}{Q}_{\mathrm{int}}\ensuremath{\propto}{L}_{W}^{\ensuremath{-}2\ensuremath{\alpha}})$, where $\ensuremath{\alpha}$ is a measure of the interactions, and vanishes at $T=0({L}_{W}=\ensuremath{\infty})$. For a double barrier system, our result agrees with the quantisation of pumped charge seen in Luttinger liquids.
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We study two different parametric pumps--one for pumping spin currents, the other for charge currents--in interacting quantum wires. We find that, as a function of pumping frequency, the spin or charge pumped per cycle has a nonuniversal crossover--depending on pumping details--between two universal fixed point values of 0 and twice the electronic spin or charge quantum number. The direction of flow between these two fixed points depends on whether the interactions are repulsive or attractive, while the quantization itself is a signature of interactions.
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