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arXiv:2204.13517

Unconventional bosonization of chiral quantum wires coupled through a point-contact driven out of equilibrium

Nikhil Danny Babu, Girish S. Setlur

Published Apr 28, 2022• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2204.13517

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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.
Abstract We study a system of two non-interacting quantum wires with fermions of opposite chirality with a point contact junction at the origin across which tunneling can take place when an arbitrary time-dependent bias between the wires is applied. We obtain the exact dynamical non-equilibrium Green function by solving Dyson’s equation analytically. Both the space–time dependent two and four-point functions are written down in a closed form in terms of simple functions of position and time. This allows us to obtain, among other things, the I–V characteristics for an arbitrary time-dependent bias. Our method is a superior alternative to competing approaches to non-equilibrium as we are able to account for transient phenomena as well as the steady state. We study the approach to steady state by computing the time evolution of the equal-time one-particle Green function. Our method can be easily applied to the problem of a double barrier contact whose internal properties can be adjusted to induce resonant tunneling leading to a conductance maximum. We then consider the case of a finite bandwidth in the point contact and calculate the non-equilibrium transport properties which exhibit non-Markovian behaviour. When a subsequently constant bias is suddenly switched on, the current shows a transient build up before approaching its steady state value in contrast to the infinite bandwidth case. This transient property is consistent with numerical simulations of lattice systems using time-dependent density matrix renormalization group suggesting thereby that this transient build up is merely due to the presence of a short distance cutoff in the problem description and not on the other details.
We study a system of two non-interacting quantum wires with fermions of opposite chirality with a point-contact junction at the origin across which tunneling can take place when an arbitrary time-dependent bias between the wires is applied. We obtain the exact dynamical non-equilibrium Green function by solving Dyson's equation analytically. Both the space-time dependent two and four-point functions are written down in a closed form in term of simple functions of position and time. This allows us to obtain among other things, the I-V characteristics for an arbitrary time-dependent bias. Our method is a superior alternative to competing approaches to non-equilibrium as we are able to account for transient phenomena as well as the steady state. We study the approach to steady state by computing the time-evolution of the equal-time one-particle Green function. We also show how the presence of the point-contact affects the density profile of the electrons by inducing Friedel oscillations in the system and how the presence of a bias influences these oscillations.
The non-chiral bosonization technique (NCBT) enables the most singular part of the asymptotic two-point and four-point functions of a strongly inhomogenous Luttinger liquid (i.e. in potentials with sharp discontinuities such as barriers, wells, step, etc.) to be written in terms of elementary functions of positions and times. On the other hand, the exact two-point function obeys the Schwinger-Dyson equation which relates the two-point function to a certain four-point function. Here we show that the formulas for the Green functions derived using NCBT obey the Schwinger-Dyson equation. This exercise is a non-perturbative confirmation of the NCBT. We also use the functional bosonization method of Lee and Chen and show that the resulting Green functions are inconsistent with both conventional fermionic perturbation theory and the NCBT. The NCBT has been previously shown to be consistent with fermionic perturbation theory to leading order.
Non-chiral bosonization (NCBT) is a non-trivial modification of the standard Fermi-Bose correspondence in one spatial dimensions made in order to facilitate the study of strongly inhomogeneous Luttinger liquids (LL) where the properties of free fermions plus the source of inhomogeneities are reproduced exactly. The formalism of NCBT is introduced and limiting case checks, fermion commutation rules, point splitting constraints, etc. are discussed. The Green functions obtained from NCBT are expanded in powers of the fermion-fermion interaction strength (forward scattering short-range only) and compared with the corresponding terms obtained using standard fermionic perturbation theory. Lastly, the Green functions obtained from NCBT are inserted into the Schwinger-Dyson equation which is the equation of motion of the Green functions and serves as a non-perturbative confirmation of the method. Some other analytical approaches like functional bosonization and numerical techniques like DMRG, which can be used to obtain the correlation functions in 1D, are briefly discussed.
Nonchiral bosonization (NCBT) is a nontrivial modification of the standard Fermi—Bose correspondence in one spatial dimension done to facilitate studying strongly inhomogeneous Luttinger liquids where the properties of free fermions plus the source of inhomogeneities are reproduced exactly. We introduce the NCBT formalism and discuss limit case checks, fermion commutation rules, point-splitting constraints, etc. We expand the Green’s functions obtained from NCBT in powers of the fermion—fermion interaction strength (only short-range forward scattering) and compare them with the corresponding terms obtained using standard fermionic perturbation theory. Finally, we substitute the Green’s functions obtained from NCBT in the Schwinger—Dyson equation, which is the equation of motion of the Green’s functions and serves as a nonperturbative confirmation of the method. We briefly discuss some other analytic approaches such as functional bosonization and numerical techniques like the density-matrix renormalization group, which can be used to obtain the correlation functions in one dimension.
In this chapter, an alternative to the conventional approach to bosonization in one dimension that invokes the Dirac equation in 1+1 dimension with chiral ‘rightmovers’ and ‘left-movers’ is proposed (the conventional approach is discussed, e.g., in the text book by Giamarchi, Quantum Physics in One Dimension, Oxford). This technique allows us to use a basis different from the plane wave basis that makes this non-chiral approach ideally suited to study Luttinger liquids that have a boundary(s) or impurities that break translational symmetry. We provide a simple solution to the electron Green function for the problem of a Luttinger liquid (LL) with a boundary and also for a LL with a single impurity. The present method is significantly easier than the g-ology based standard bosonization and other methods that require a combination of RG (renormalization group) along with bosonization/refermionization techniques. This method is superior because it reproduces, as a limiting case, the exact closed formula for the (random phase approximation) RPA Green function of free fermions with the single impurity without mutual interactions obtainable trivially using Fermi algebra, whereas the conventional bosonization/RG approaches are not able to do so.