Bosonization for Beginners --- Refermionization for Experts

TL;DR

This tutorial provides an accessible, detailed introduction to abelian bosonization in one dimension, and applies finite-size refermionization to resolve a controversy about the tunneling density of states in a Tomonaga-Luttinger liquid.

Contribution

It offers a self-contained, explicit treatment of bosonization and refermionization, including subtleties like Klein factors and finite-size effects, with a rigorous application to impurity problems.

Findings

Confirmed the low-energy behavior of the tunneling density of states at g=1/2 as proportional to ω

Resolved a controversy by showing the asymptotic behavior matches Furusaki's results

Captured effects beyond mean-field treatments of Coulomb gas models

Abstract

This tutorial gives an elementary and self-contained review of abelian bosonization in 1 dimension in a system of finite size $L$, following and simplifying Haldane's constructive approach. As a non-trivial application, we rigorously resolve (following Furusaki) a recent controversy regarding the tunneling density of states, $\rho_{dos} (\omega)$, at the site of an impurity in a Tomonaga-Luttinger liquid: we use finite-size refermionization to show exactly that for g=1/2 its asymptotic low-energy behavior is $\rho_{dos}(\omega) \sim \omega$. This agrees with the results of Fabrizio & Gogolin and of Furusaki, but not with those of Oreg and Finkel'stein (probably because we capture effects not included in their mean-field treatment of the Coulomb gas that they obtained by an exact mapping; their treatment of anti-commutation relations in this mapping is correct, however, contrary to recent suggestions in the literature). --- The tutorial is addressed to readers unfamiliar with bosonization, or for those interested in seeing ``all the details'' explicitly; it requires knowledge of second quantization only, not of field theory. At the same time, we hope that experts too might find useful our explicit treatment of certain subtleties -- these include the proper treatment of the so-called Klein factors that act as fermion-number ladder operators (and also ensure the anti-commutation of different species of fermion fields), the retention of terms of order 1/L, and a novel, rigorous formulation of finite-size refermionization of both $e^{-i \Phi(x)}$ and the boson field $\Phi (x)$ itself.