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Non-chiral bosonization and the Schwinger-Dyson Equation

Joy Prakash Das, Girish S. Setlur

Published Dec 25, 2017• 0 citations
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Abstract

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.

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Quantum steeplechase is the study of a Luttinger liquid (LL) in one dimension in the presence of a finite number of barriers and wells clustered around an origin. The powerful nonchiral bosonization technique (NCBT) is introduced to write down closed formulas for the two-point functions in the sense of the random phase approximation (RPA). Unlike g-ology-based methods that are tied to the translationally invariant, free particle basis, the NCBT explicitly makes use of the translationally noninvariant single particle wave functions. The present method, that provides the most singular part of the asymptotically exact Green function in a closed form, is in contrast to competing methods that require a combination of renormalization group and/or numerical methods in addition to the bosonization techniques.

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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.
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 In this work, we show in pedagogical detail that the most singular contributions to the slow part of the asymptotic density-density correlation function of Luttinger liquids with fermions interacting mutually with only short-range forward scattering and also with localised scalar static impurities (where backward scattering takes place) has a compact analytical expression in terms of simple functions that have second order poles and involve only the scale-independent bare transmission and reflection coefficients. This proof uses conventional fermionic perturbation theory resummed to all orders, together with the idea that for such systems, the (connected) moments of the density operator all vanish beyond the second order—the odd ones vanish identically and the higher order even moments are less singular than the second order moment which is the only one included. This important result is the crucial input to the recently introduced ‘Non-Chiral Bosonization Technique’ (NCBT) to study such systems. The results of NCBT cannot be easily compared with the results obtained using conventional bosonization as the former only extracts the most singular parts of the correlation functions albeit for arbitrary impurity strengths and mutual interactions. The latter, ambitiously attempts to study all the parts of the asymptotic correlation functions and is thereby unable to find simple analytical expressions and is forced to operate in the vicinity of the homogeneous system or the half line (the opposite extreme). For a fully homogeneous system or its antithesis viz. the half-line, all the higher order connected moments of the density vanish identically which means the results of chiral bosonization and NCBT ought to be the same and indeed they are.
In this work, analytical expressions for the Green function of a Luttinger liquid are derived with one and two mobile impurities (heavy particles) using a combination of bosonization and perturbative approaches. The calculations are done in the random phase approximation (RPA) limit using the powerful non-chiral bosonization technique (NCBT) which is nothing but the resummation of the most singular parts of the RPA terms of the Green function expanded out in powers of the forward scattering between fermions with the source of inhomogeneities treated exactly . The force acting on the heavy particle(s) is studied as a function of its terminal velocity, both in the linear and non-linear regime. Linear mobility (which is valid for impurities moving much slower than a certain crossover speed) has a power law temperature dependence whose exponent has a closed algebraic expression in terms of the various parameters in the problem. This expression interpolates between the ballistic regime of no-coupling with the fermions and the no-tunneling regime. When the speed of the impurity is much larger than this crossover speed, the applied force depends non-linearly on the speed and this too is a power law with a closely related exponent. The case of two mobile impurities is also studied whose mobility exhibits peculiar resonances when their mutual separation is appropriately chosen.
Quantum steeplechase is the study of a Luttinger liquid (LL) in one dimension in the presence of a finite number of barriers and wells clustered around an origin. The powerful nonchiral bosonization technique (NCBT) is introduced to write down closed formulas for the two-point functions in the sense of the random phase approximation (RPA). Unlike g-ology-based methods that are tied to the translationally invariant, free particle basis, the NCBT explicitly makes use of the translationally noninvariant single particle wave functions. The present method, that provides the most singular part of the asymptotically exact Green function in a closed form, is in contrast to competing methods that require a combination of renormalization group and/or numerical methods in addition to the bosonization techniques.