Non-chiral bosonization of strongly inhomogeneous Luttinger liquids

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.