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Physical Review Letters

Tunneling between Two Luttinger Liquids Induced by a Driving Field

D. Schmeltzer

Published Nov 6, 2000• 7 citations
Publisher

Abstract

Tunneling between two Luttinger liquids driven by a time-dependent field with a frequency $f$ is investigated using the zero-mode bosonization. We show that inclusion of the zero modes is essential in order to obtain correct results in the limit ${L}_{T}/L\ensuremath{\gg}1$ ( $L$ is the channel length and ${L}_{T}$ is the thermal length). We find that the tunneling current is quantized in units of `` $\mathrm{ef}$'' and takes the form $I\ensuremath{\sim}\mathrm{ef}\ensuremath{\Sigma}{n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}\ensuremath{\delta}[{\ensuremath{\mu}}_{F}\ensuremath{-}\frac{2\ensuremath{\pi}}{L}v\ensuremath{\Elzxh}(n\ensuremath{-}\frac{1}{2})]$.

DOI
10.1103/physrevlett.85.4132

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Tunneling between two Luttinger liquids driven by a time-dependent field with a frequency fff is investigated using the zero-mode bosonization. We show that inclusion of the zero modes is essential in order to obtain correct results in the limit LT/L\ensuremath≫1{L}_{T}/L\ensuremath{\gg}1LT​/L\ensuremath≫1 ( LLL is the channel length and LT{L}_{T}LT​ is the thermal length). We find that the tunneling current is quantized in units of `` ef\mathrm{ef}ef'' and takes the form I\ensuremath∼ef\ensuremathΣn=1\ensuremath∞\ensuremathδ[\ensuremathμF\ensuremath−2\ensuremathπLv\ensuremath\Elzxh(n\ensuremath−12)]I\ensuremath{\sim}\mathrm{ef}\ensuremath{\Sigma}{n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{\ensuremath{\infty}}\ensuremath{\delta}[{\ensuremath{\mu}}_{F}\ensuremath{-}\frac{2\ensuremath{\pi}}{L}v\ensuremath{\Elzxh}(n\ensuremath{-}\frac{1}{2})]I\ensuremath∼ef\ensuremathΣn=1\ensuremath∞\ensuremathδ[\ensuremathμF​\ensuremath−L2\ensuremathπ​v\ensuremath\Elzxh(n\ensuremath−21​)].
We apply the bosonization technique to the problem of tunneling in a Fermi liquid, and present a semiclassical theory of tunneling rates. To test the method, we derive and evaluate an expression for the tunneling current in the problem of two parallel Luttinger liquids. Next we study tunneling between parallel two dimensional Fermi liquids. In the absence of a magnetic field, the conductance I/VI/VI/V has a resonance peak at zero bias. Our estimate for the peak broadening agrees with the many-body theory results. In a magnetic field perpendicular to the plane, we find a tunneling gap linear in the field, in agreement with the experiment. In this regime the tunneling is spatially coherent over a distance of the order of the cyclotron radius. The coherence can be probed by a magnetic field parallel to the plane, which gives an Aharonov-Bohm phase to the tunneling amplitude. As a result, the current is an oscillatory function of the field.
We show that the macroscopic current and charge can be formulated as a quantum-mechanical zero-mode problem. We find that the current is given by the velocity operator of a particle restricted to move around a circle. As an explicit example we investigated a Luttinger liquid of length L that is perturbed by a time-dependent impurity. Using the statistical mechanics of zero modes we computed the nonequilibrium current. In particular we show that in the low-temperature limit, LT/L>1,{L}_{T}/L>1,LT​/L>1, the zero-mode method introduced here becomes essential for computing the current.
The space-time tunneling bosonization has been introduced to study N-coupled Luttinger liquids. The interaction is separated into a forward exchange and a cooperon part. At T=0 the cooperon part is described by tunneling events which form a Coulomb gas in the plasma phase. At T\ensuremath{\ne}0 we ignore the cooperon part and show that the correlation functions show a Luttinger-liquid behavior. In the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, we extract the two-dimensional behavior and show that for a certain range of parameters the Luttinger liquid survives.
We consider the problem of tunneling between spinless one-dimensional (1D) Luttinger-liquid chains. We show how to map the problem onto the four-state chiral clock model together with a free boson for the total charge mode. We use a variety of results, some of them exact, from the integrable chiral Potts model and the study of commensurate/incommensurate transitions to deduce the physics of coupled Luttinger liquids. For those intrachain interaction strengths for which interchain tunneling is relevant, we find that it can lead to the formation of symmetric and antisymmetric bands with split Fermi surfaces, depending on the relative strengths of interchain tunneling and interchain interactions. With split Fermi surfaces, interchain transport is coherent. It is not possible to have two gapless Fermi surfaces with the same Fermi momentum when interchain tunneling is relevant. However, interchain interactions can drive the formation of a gap, in which case interchain transport will be incoherent. We comment on the possible relevance of our results to c-axis transport in high-Tc{T}_{c}Tc​ superconductors.
We have calculated the two-probe Landauer conductance of a one-channel quantum wire containing a Luttinger liquid and connected to two noninteracting leads through tunnel barriers. The tunneling conductance shows broad resonances as a function of the bias voltage, this being a manifestation of the spin-charge separation. In the limit of zero bias and zero barrier, the tunneling conductance reduces to the ballistic contact value e2h\frac{{e}^{2}}{h}he2​ per channel.
Entanglement dynamics depend not only on how a quantum system is partitioned, but critically on how interactions across that partition are structured. For a spatial bipartition of a locally interacting system, entanglement is generated near the boundary and then propagates into the bulk. By contrast, when two extended quantum fields are coupled locally along their entire length, the interaction crosses the field-space partition everywhere, and this generates correlations throughout the system. Here, we study the entanglement dynamics between two gapless one-dimensional quantum many-body systems described by Luttinger liquid theory. The systems are initially decoupled and prepared at zero or finite temperature, after which a time-dependent tunneling interaction is activated uniformly along their length. Within a Gaussian approximation, we derive general analytical expressions for the logarithmic negativity, mutual information, and Rényi entropies under arbitrary coupling protocols. At zero temperature, entanglement displays an early-time power-law growth whose exponent is fixed solely by the first non-null derivative of the tunneling protocol. Once the coupling saturates, we obtain exact long-time averages of the information-theoretic quantities and characterise how the correlations scale with temperature and the final coupling strength. We also analyse how mutual information and logarithmic negativity approach the adiabatic limit for a very slow protocol with respect to intrinsic system timescale. This work extends the study of entanglement dynamics in nonequilibrium field theory to field-space partitions and mixed initial states.