Nonchiral Bosonization of Strongly Inhomogeneous Luttinger Liquids

Abstract

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 shortrange 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.

Friedel oscillations and dynamical density of states of an inhomogeneous Luttinger liquid

“…. This can be done using a non standard harmonic analysis suited to study inhomogeneous Luttinger liquids like the one under study [12].…”

“…More recently, a new technique has been developed which is capable of extracting the most singular part of the asymptotic correlation functions of a Luttinger liquid with an impurity of arbitrary strength in terms of elementary functions of positions and times, taking into account arbitrary strength of mutual interactions between the particles [11,12]. This method, which goes by the name 'Non-chiral bosonization technique' has been successfully applied to study a Luttinger liquid with a cluster of impurities around an origin [11], the one step fermionic ladder (two 1D wires placed parallel to each other with hopping between a pair of opposing points) [13] and slowly moving heavy impurities in a Luttinger liquid [14].…”

“…This method, which goes by the name 'Non-chiral bosonization technique' has been successfully applied to study a Luttinger liquid with a cluster of impurities around an origin [11], the one step fermionic ladder (two 1D wires placed parallel to each other with hopping between a pair of opposing points) [13] and slowly moving heavy impurities in a Luttinger liquid [14]. The correlation functions obtained using this method are validated using perturbative comparison as well as by inserting them into the Schwinger Dyson equation to obtain an identity [12]. Once the Green function of a given system is obtained, it may be used to predict different physical phenomena occurring in the system.…”

“…As discussed in our earlier works [11], for doing so one just needs to double the holon exponent and allow the spinon exponents to vanish. In equation (12), the term -g 2 comes from the holon part which doubles while the −1/2 comes from the spinon part which vanishes for the spinless case. Thus the Friedel oscillation decays asx g which is another well established result [15].…”

Density-density correlation function of strongly inhomogeneous Luttinger liquids

“…A recently developed alternative to this, which goes by the name 'Non chiral bosonization technique (NCBT)', does a better job of avoiding RG methods and tackling impurities of arbitrary strengths [21,22]. But it can yield only the most singular part of the Green functions.…”