[Paper Review] Bosonization and Strongly Correlated Systems

Alexander O. Gogolin, A. A. Nersesyan+1 more|ArXiv.org|Sep 8, 1999

Cold Atom Physics and Bose-Einstein Condensates7 references1,109 citations

TL;DR

This book provides a comprehensive treatment of bosonization, a nonperturbative technique for analyzing strongly correlated one-dimensional quantum systems. It establishes connections between fermionic models, conformal field theory, and integrable systems, demonstrating how bosonic representations—via Ward identities and correlation functions—enable exact solutions of critical and non-critical models like the Tomonaga-Luttinger liquid and spin chains.

ABSTRACT

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.

Motivation & Objective

• To bridge the gap between abstract mathematical formalism and physical applications in strongly correlated systems.
• To present bosonization as a nonperturbative method for solving interacting quantum many-body problems in (1+1) dimensions.
• To unify concepts from high-energy physics and condensed matter physics by highlighting deep parallels in their theoretical frameworks.
• To provide a systematic treatment of both Abelian and non-Abelian bosonization, including applications to spin chains and impurity problems.
• To demonstrate how conformal field theory and Ward identities enable exact computation of correlation functions without relying on a Hamiltonian formalism.

Proposed method

• Utilizes the duality between fermionic and bosonic fields in (1+1)D to map interacting fermion systems onto non-interacting bosonic theories.
• Applies the Gaussian model and conformal field theory (CFT) to describe critical systems with gapless linear spectra.
• Employs Ward identities derived from conformal symmetry to determine multi-point correlation functions as solutions to differential equations.
• Uses the Dotsenko-Fateev representation to express CFT correlation functions in terms of bosonic exponentials.
• Applies non-Abelian bosonization to systems with internal symmetries, such as the WZNW and Ising models.
• Analyzes quantum impurities via the X-ray edge problem and multi-channel Kondo effect, using bosonized field-theoretic techniques.

Experimental results

Research questions

• RQ1 How can bosonization be systematically applied to one-dimensional fermionic systems with strong correlations?
• RQ2 What is the role of conformal symmetry in determining the structure of correlation functions in critical (1+1)-dimensional systems?
• RQ3 How do Ward identities derived from conformal invariance replace the need for a Hamiltonian in solving quantum many-body problems?
• RQ4 In what ways do non-Abelian bosonization and CFT extend the reach of the original Abelian bosonization technique?
• RQ5 How can bosonization techniques be used to describe quantum impurities and their effects in Tomonaga-Luttinger liquids?

Key findings

• Bosonization provides a nonperturbative framework to exactly solve strongly correlated systems in (1+1) dimensions, including the Tomonaga-Luttinger liquid and spin-1/2 Heisenberg chain.
• Conformal field theory establishes that gapless (1+1)-dimensional systems possess infinite-dimensional conformal symmetry, leading to an infinite set of Ward identities.
• Correlation functions in critical systems are uniquely determined by solving differential equations derived from Ward identities, replacing the need for explicit Hamiltonian diagonalization.
• The Hilbert space of interacting theories is not equivalent to that of free bosons; certain states must be projected out, which can be handled via operator constraints.
• Non-Abelian bosonization extends the method to systems with non-Abelian symmetries, such as the WZNW model and the SU(2) spin chain.
• The X-ray edge problem and multi-channel Kondo effect are successfully analyzed using bosonized field-theoretic techniques, revealing universal scaling behavior in impurity systems.