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arXiv:cond-mat/9805275 · Annalen der Physik

Bosonization for beginners - refermionization for experts

Jan von Delft, Herbert Schoeller

Published Nov 1, 1998• 508 citations
arXivHTMLPDF

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, $ρ_{dos} (ω)$, 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 $ρ_{dos}(ω) \sim ω$. 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 $e^{-i Φ(x)}$ and the boson field $Φ(x)$ itself.

DOI
10.1002/(sici)1521-3889(199811)7:4<225::aid-andp225>3.0.co;2-l

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This tutorial gives an elementary and self-contained review of abelian bosonization in 1 dimension in a system of finite size LLL, 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, ρdos(ω)ρ_{dos} (ω)ρdos​(ω), 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 ρdos(ω)∼ωρ_{dos}(ω) \sim ωρdos​(ω)∼ω. 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−iΦ(x)e^{-i Φ(x)}e−iΦ(x) and the boson field Φ(x)Φ(x)Φ(x) itself.
This volume provides a detailed account of bosonization. The first part of the book examines the technical aspects of bosonization including one-dimensional fermions, the Gaussian model, the structure of Hilbert space in conformal theories, Bose-Einstein condensation in two dimensions, non-Abelian bosonization, and the Ising and WZNW models. The second part presents applications of the bosonization technique to realistic models including the Tomonaga-Luttinger liquid, spin liquids in one dimension and the spin-1/2 Heisenberg chain with alternative exchange. The third part addresses the problems of quantum impurities. Chapters cover potential scattering, the X-ray edge problem, impurities in Tomonaga-Luttinger liquids and the multi-channel Kondo problem.
Abstract This tutorial review gives an elementary and self‐contained derivation of the standard identities (ψη(x) ∼ Fη e , etc.) for 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, ρdos(ω), 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 ρdos(ω) ∼ ω. 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 with little or no prior knowledge of bosonization, who are interested in seeing “all the details” explicitly; it is written at the level of beginning graduate students, requiring only knowledge of second quantization, but not of field theory (which is not needed here). At the same time, we hope that experts too might find useful our explicit treatment of certain subtleties that can often be swept under the rug, but are crucial for some applications, such as the calculation of ρdos(ω) — 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 F e—iΦ(x) and the boson field Φ(x) itself.
This is a pedagogical introduction to the general technique of bosonization of one-dimensional systems starting from scratch and assuming very little besides basic quantum mechanics and second quantization. The formalism is developed in a self-contained fashion and applied to the spinless and spin- Luttinger models, working out both single and two particle correlation functions. The implications of these results for the specific cases ofthe (anisotropic) Heisenberg and the Hubbard models are discussed. Although everything in these notes can befound in the published literature, detailed and explicit calculations of most of the results are given, which mayprove useful to beginning graduate students or researchers in this area.
In their comment Fabrizio and Gogolin dispute our result of the enhancement of the tunneling density of states in a Tomonaga-Luttinger liquid at the location of a backward scattering defect [Phys. Rev. Lett. 76, 4230(1996); cond-mat/9601020]. They state that the anticommutativity of the fermion operators of the left and right moving electrons was not considered properly in the Letter. We show in the Reply that the result of the Letter can be reproduced following the Comment when its calculations are performed correctly. This clearly indicates that the question about the anticommutation relations was raised by Fabrizio and Gogolin without serious grounds.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension, and transport in clean quantum wires and in the presence of isolated impurities.
We discuss the technique of bosonization for studying systems of interacting fermions in one dimension. After briefly reviewing the low-energy properties of Fermi and Luttinger liquids, we present some of the relations between bosonic and fermionic operators in one dimension. We use these relations to calculate the correlation functions and the renormalization group properties of various operators for a system of spinless fermions. We then apply the methods of bosonization to study the Heisenberg antiferromagnetic spin 1/2 chain, the Hubbard model in one dimension and transport in clean quantum wires and in the presence of isolated impurities.