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arXiv:1606.03465 · Physical review. B./Physical review. B

Tunneling into quantum wires: Regularization of the tunneling Hamiltonian and consistency between free and bosonized fermions

Michele Filippone, Piet W. Brouwer

Published Dec 21, 2016• 9 citations
arXivHTMLPDF

Abstract

Tunneling between a point contact and a one-dimensional wire is usually described with the help of a tunneling Hamiltonian that contains a $\ensuremath{\delta}$ function in position space. Whereas the leading-order contribution to the tunneling current is independent of the way this $\ensuremath{\delta}$ function is regularized, higher-order corrections with respect to the tunneling amplitude are known to depend on the regularization. Instead of regularizing the $\ensuremath{\delta}$ function in the tunneling Hamiltonian, one may also obtain a finite tunneling current by invoking the ultraviolet cutoffs in a field-theoretic description of the electrons in the one-dimensional conductor, a procedure that is often used in the literature. For the latter case, we show that standard ultraviolet cutoffs lead to different results for the tunneling current in fermionic and bosonized formulations of the theory, when going beyond leading order in the tunneling amplitude. We show how to recover the standard fermionic result using the formalism of functional bosonization and revisit the tunneling current to leading order in the interacting case.

DOI
10.1103/physrevb.94.235426

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Tunneling between a point contact and a one-dimensional wire is usually described with the help of a tunneling Hamiltonian that contains a \ensuremathδ\ensuremath{\delta}\ensuremathδ function in position space. Whereas the leading-order contribution to the tunneling current is independent of the way this \ensuremathδ\ensuremath{\delta}\ensuremathδ function is regularized, higher-order corrections with respect to the tunneling amplitude are known to depend on the regularization. Instead of regularizing the \ensuremathδ\ensuremath{\delta}\ensuremathδ function in the tunneling Hamiltonian, one may also obtain a finite tunneling current by invoking the ultraviolet cutoffs in a field-theoretic description of the electrons in the one-dimensional conductor, a procedure that is often used in the literature. For the latter case, we show that standard ultraviolet cutoffs lead to different results for the tunneling current in fermionic and bosonized formulations of the theory, when going beyond leading order in the tunneling amplitude. We show how to recover the standard fermionic result using the formalism of functional bosonization and revisit the tunneling current to leading order in the interacting case.
Bosonization technique for one-dimensional fermions out of equilibrium is developed in the framework of the Keldysh action formalism. We first demonstrate how this approach is implemented for free fermions and for the problem of nonequilibrium Fermi edge singularity. We then employ the technique to study an interacting quantum wire attached to two electrodes with arbitrary energy distributions. The nonequilibrium electron Green's functions, which can be measured via tunneling spectroscopy technique and carry the information about energy distribution, zero-bias anomaly, and dephasing, are expressed in terms of functional determinants of single-particle ``counting'' operators. The corresponding time-dependent scattering phase is found to be intrinsically related to ``fractionalization'' of electron-hole excitations in the tunneling process and at boundaries with leads. Results are generalized to the case of spinful particles as well to Green's functions at different spatial points (relevant to the problem of dephasing in Luttinger liquid interferometers). For double-step distributions, the dephasing rates are oscillatory functions of the interaction strength.
Using the self-consistent Hartree-Fock approximation for spinless electrons at zero temperature, we study tunneling of the interacting electron gas through a single δ barrier in a finite one-dimensional wire connected to contacts.Our results exhibit features known from correlated many-body models.In particular, the conductance decays with the wire length as ∝ L -2α , where the power α is universal.We also show that a similar result for a wire conductance can be extracted from the persistent current (I) through the δ barrier in a one-dimensional ring, where it is known that I ∝ L -1-α .
We use our recently developed functional bosonization approach to bosonize interacting fermions in arbitrary dimension d beyond the Gaussian approximation. Even in d=1 the finite curvature of the energy dispersion at the Fermi surface gives rise to interactions between the bosons. In higher dimensions scattering processes describing momentum transfer between different patches on the Fermi surface (around-the-corner processes) are an additional source for corrections to the Gaussian approximation. We derive an explicit expression for the leading correction to the bosonized Hamiltonian and the irreducible self-energy of the bosonic propagator that takes the finite curvature as well as around-the-corner proceses into account. In the special case that around-the-corner scattering is negligible, we show that the self-energy correction to the Gaussian propagator is negligible if the dimensionless quantities (qc{\mathit{q}}_{\mathit{c}}qc​/kF{\mathit{k}}_{\mathit{F}}kF​)d{)}^{\mathit{d}})dF0{\mathit{F}}_{0}F0​ [1+F0{\mathit{F}}_{0}F0​${]}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\mu}/${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$\ensuremath{\Vert}\ensuremath{\partial}${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$/\ensuremath{\partial}\ensuremath{\mu}\ensuremath{\Vert} are small compared with unity for all patches \ensuremath{\alpha}. Here qc{\mathit{q}}_{\mathit{c}}qc​ is the cutoff of the interaction in wave-vector space, kF{\mathit{k}}_{\mathit{F}}kF​ is the Fermi wave vector, \ensuremath{\mu} is the chemical potential, F0{\mathit{F}}_{0}F0​ is the usual dimensionless Landau interaction parameter, and ${\ensuremath{\nu}}^{\mathrm{\ensuremath{\alpha}}}$ is the local density of states associated with patch \ensuremath{\alpha}. We also show that the well-known cancellation between vertex and self-energy corrections in one-dimensional systems, which is responsible for the fact that the random-phase approximation (RPA) for the density-density correlation function is exact in d=1, exists also in d>1, provided (1) the interaction cutoff qc{\mathit{q}}_{\mathit{c}}qc​ is small compared with kF{\mathit{k}}_{\mathit{F}}kF​, and (2) the energy dispersion is locally linearized at the Fermi surface. Finally, we suggest a systematic method to calculate corrections to the RPA, which is based on the perturbative calculation of the irreducible bosonic self-energy arising from the non-Gaussian terms of the bosonized Hamiltonian.
Non-chiral bosonization technique adapted to study chiral quantum wires with non-interacting fermions coupled through a point-contact with a constant bias between the wires is introduced and is shown to reproduce the exact Green functions of this system which was previously derived by the present authors analytically using standard methods. The tunneling I-V characteristics are obtained using the bosonized Green functions. The proposed unconventional bosonization scheme is also shown to be internally consistent as the four-point functions evaluated using NCBT are shown to be related to the two-point functions through Wick's theorem as it should be. In equilibrium, the equal space-time NCBT Green functions for an interacting Luttinger liquid with impurities obtained in a previous work shows universal scaling behaviour in accordance with Bethe ansatz and functional renormalization group predictions. We expect to obtain a similar scaling form of the tunneling properties out of equilibrium in presence of interparticle interactions.
The calculation of the full counting statistics of the charge within a finite interval of an interacting one-dimensional system of electrons is a fundamental, yet as of now, unresolved problem. Even in the noninteracting case, charge counting turns out to be more difficult than anticipated because it necessitates the calculation of a nontrivial determinant and requires regularization. Moreover, interactions in a one-dimensional system are best described using bosonization. However, this technique rests on a long-wavelength approximation and is a priori inapplicable for charge counting due to the sharp boundaries of the counting interval. To mitigate these problems, we investigate the counting statistics using several complementary approaches. To treat interactions, we develop a diagrammatic approach in the fermionic basis, which makes it possible to obtain the cumulant generating function up to arbitrary order in the interaction strength. Importantly, our formalism preserves charge quantization in every perturbative order. We derive an exact expression for the noise and analyze its interaction-dependent logarithmic cutoff. We compare our fermionic formalism with the results obtained by other methods, such as the Wigner crystal approach and numerical calculations using the density-matrix renormalization group. Surprisingly, we show good qualitative agreement with the Wigner crystal for weak interactions, where the latter is in principle not expected to apply.
The calculation of the full counting statistics of the charge within a finite interval of an interacting one-dimensional system of electrons is a fundamental, yet as of now unresolved problem. Even in the non-interacting case, charge counting turns out to be more difficult than anticipated because it necessitates the calculation of a nontrivial determinant and requires regularization. Moreover, interactions in a one-dimensional system are best described using bosonization. However, this technique rests on a long-wavelength approximation and is a priori inapplicable for charge counting due to the sharp boundaries of the counting interval. To mitigate these problems, we investigate the counting statistics using several complementary approaches. To treat interactions, we develop a diagrammatic approach in the fermionic basis, which makes it possible to obtain the cumulant generating function up to arbitrary order in the interaction strength. Importantly, our formalism preserves charge quantization in every perturbative order. We derive an exact expression for the noise and analyze its interaction-dependent logarithmic cutoff. We compare our fermionic formalism with the results obtained by other methods, such as the Wigner crystal approach and numerical calculations using the density-matrix renormalization group. Surprisingly, we show good qualitative agreement with the Wigner crystal for weak interactions, where the latter is in principle not expected to apply.