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arXiv:cond-mat/9612173 · Physical review. B, Condensed matter

Interacting electrons with spin in a one-dimensional dirty wire connected to leads

Inès Safi A, Heinz J. Schulz

Published Jan 15, 1999• 76 citations
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Abstract

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 ${2k}_{F}$ charge-density wave at $T<{L}^{(3/2)(K\ensuremath{-}1)}$ for short-range interactions with parameter $K<1/3$. 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, $K<1/3$. This leads to various regimes, depending on temperature and on $K$. 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 $\mathrm{Var}(\mathcal{R})/{\mathcal{R}}^{2}$ stays always of the same order: it is equal to ${L}_{T}/L\ensuremath{\ll}1$ in the high-temperature limit, then saturates at $1/2$ in the low-temperature limit, indicating that the relative fluctuations of $\mathcal{R}$ are universal.

DOI
10.1103/physrevb.59.3040

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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≫\ensuremathωT\ensuremath{\gg}\ensuremath{\omega}T\ensuremath≫\ensuremathω. 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)T<{L}^{(3/2)(K\ensuremath{-}1)}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 R=g\ensuremath−2e2/h\mathcal{R}=g\ensuremath{-}{2e}^{2}/hR=g\ensuremath−2e2/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 LT/L\ensuremath≪1{L}_{T}/L\ensuremath{\ll}1LT​/L\ensuremath≪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.
We investigate transport through a finite interacting wire connected to noninteracting leads. The conductance of the pure wire is not renormalized by the interactions for any spatial variation of the interaction parameters u,Ku,Ku,K, and not even for Coulomb interactions restricted to the wire. We rigorously relate the conductance to the transmission, that turns out to be perfect. If KKK varies abruptly at the contacts, an electron incident on the wire is reflected into a series of partial spatially separated charges which sum up to unity. For attractive interactions, the reflection at the contact is similar to Andreev reflection on a gapless superconductor. This process affects the density-density or pairing correlation functions: they are enhanced on the bulk of the wire as in an infinite Luttinger liquid, then extend to the external noninteracting leads in a way reminiscent of the proximity effect. The effect of impurities is governed by the wire parameter but is affected by the leads close to the contacts. Our results give a possible explanation to recent experiments on quantum wires by Tarucha et al.
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 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 $\ensuremath{\Elzxh}\ensuremath{\Omega}\ensuremath{\ll}{\ensuremath{\omega}}_{\ensuremath{\Gamma}},$ 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)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 $\ensuremath{\Gamma}(\ensuremath{\Omega}t)$ as it traverses a closed path C\mathcal{C}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 C\mathcal{C}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 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.
Transport through a one-dimensional wire of interacting electrons connected to semi infinite leads is investigated using a bosonization approach. The dynamic nonlocal conductivity is rigorously expressed in terms of the transmission. For abrupt variations of the interaction parameters at the junctions, an incident electron is transmitted as a sequence of partial charges: the central wire acts as a Fabry-Perot resonator. The dc conductance is shown to be given by the total transmission which turns out to be perfect. When the wire has a tendency towards superconducting order, partial Andreev reflection of an incident electron occurs. Finally, we study the role of a weak barrier at one contact or inside the wire by a renormalization group method at finite temperature. We compute the conductance in the presence of localized or extended disorder, and compare our results to recent experiments on quantum wires.
We study the transport of electrons through a long quantum wire connecting two bulk leads. As the electron density in the wire is lowered, the Coulomb interactions lead to short-range crystalline ordering of electrons. In this Wigner crystal state the spins of electrons form an antiferromagnetic Heisenberg spin chain with exponentially small exchange coupling JJJ. Inhomogeneity of the electron density due to the coupling of the wire to the leads results in violation of spin-charge separation in the device. As a result the spins affect the conductance of the wire. At zero temperature the low-energy spin excitations propagate freely through the wire, and its conductance remains 2e2∕h2{e}^{2}∕h2e2∕h. Since the energy of the elementary excitations in the spin chain (spinons) cannot exceed $\ensuremath{\pi}J∕2$, the conductance of the wire acquires an exponentially small negative correction $\ensuremath{\delta}\phantom{\rule{0.2em}{0ex}}G\ensuremath{\propto}\ensuremath{-}\mathrm{exp}(\ensuremath{-}\ensuremath{\pi}J∕2T)$ at low temperatures T⪡JT⪡JT⪡J. At higher temperatures, T⪢JT⪢JT⪢J, most of the spin excitations in the leads are reflected by the wire, and the conductance levels off at a new universal value e2∕h{e}^{2}∕he2∕h.
The physics of interacting quantum wires has attracted a lot of attention recently. When the density of electrons in the wire is very low, the strong repulsion between electrons leads to the formation of a Wigner crystal. We review the rich spin and orbital properties of the Wigner crystal, in both the one-dimensional and the quasi-one-dimensional regimes. In the one-dimensional Wigner crystal the electron spins form an antiferromagnetic Heisenberg chain with exponentially small exchange coupling. In the presence of leads, the resulting inhomogeneity of the electron density causes a violation of spin-charge separation. As a consequence the spin degrees of freedom affect the conductance of the wire. Upon increasing the electron density, the Wigner crystal starts deviating from the strictly one-dimensional geometry, forming a zigzag structure instead. Spin interactions in this regime are dominated by ring exchanges, and the phase diagram of the resulting zigzag spin chain has a number of unpolarized phases as well as regions of complete and partial spin polarization. Finally we address the orbital properties in the vicinity of the transition from a one-dimensional to a quasi-one-dimensional state. Due to the locking between chains in the zigzag Wigner crystal, only one gapless mode exists. Manifestations of Wigner crystal physics at weak interactions are explored by studying the fate of the additional gapped low-energy mode as a function of interaction strength.
We study the renormalization of a single impurity potential in one-dimensional interacting electron systems in the presence of magnetic field. Using the bosonization technique and Bethe ansatz solutions, we determine the renormalization group flow diagram for the amplitudes of scattering of up- and down-spin electrons by the impurity in a quantum wire at low electron density and in the Hubbard model at less than half filling. In the absence of magnetic field the repulsive interactions are known to enhance backscattering and make the impurity potential impenetrable in the low-energy limit. On the contrary, we show that in a strong magnetic field the interaction may suppress the backscattering of majority-spin electrons by the impurity potential in the vicinity of the weak-potential fixed point. This implies that in a certain temperature range the impurity becomes almost transparent for the majority-spin electrons while it is impenetrable for the minority-spin ones. The impurity potential can thus have a strong spin-filtering effect.