Bosonization for Beginners --- Refermionization for Experts
Abstract
This tutorial gives an elementary and self-contained review of abelian bosonization in 1 dimension in a system of finite size , 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, , 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 . 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 and the boson field itself.
Keywords
Cite
@article{arxiv.cond-mat/9805275,
title = {Bosonization for Beginners --- Refermionization for Experts},
author = {Jan von Delft and Herbert Schoeller},
journal= {arXiv preprint arXiv:cond-mat/9805275},
year = {2010}
}
Comments
Revtex, 70 pages. Changes: Regarding the controversial tunneling density of states at an impurity in a g=1/2 Luttinger liquid, we (1) give a new, more explicit calculation, (2) show that contrary to recent suggestions (including our own), Oreg and Finkel'stein treat fermionic anticommutation relations CORRECTLY (see Appendix K), but (3) suggest that their MEAN-FIELD treatment of their Coulomb gas may not be sufficiently accurate