Introduction to bosonization

Brazilian Journal of Physics, vol. 33, no. 1, March, 2003 3 Introduction to Bosonization E. Miranda Instituto de Fı́sica Gleb Wataghin, Unicamp, Caixa Postal 6165, 13083-970, Campinas, SP, Brazil Received on 27 August, 2002 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-12 Luttinger models, working out both single and two particle correlation functions. The implications of these results for the specific cases of the (anisotropic) Heisenberg and the Hubbard models are discussed. Although everything in these notes can be found in the published literature, detailed and explicit calculations of most of the results are given, which may prove useful to beginning graduate students or researchers in this area. I Introduction These notes formed the basis of a series of lectures given at the Brazilian Statistical Mechanics School, which took place from February 18 to 29, 2002, at the Universidade de São Paulo in São Carlos. While writing them, I had in mind a beginning graduate student in physics, already familiar with basic Quantum Mechanics, including the formalism of second quantization, but not with very much more. I follow through the mathematical details necessary to establish the bosonization technique of one-dimensional systems, which is by now a rigorous and mature method that underlies much of our understanding of these systems. It has found many applications in real quasi-one-dimensional systems such as quantum wires [1], carbon nanotubes [2] and edge states of the quantum Hall effect [3]. For the sake of motivation, I focus on two models: the Hubbard model of spin- 21 fermions and the anisotropic (XXZ) Heisenberg spin- 21 model. I should stress that all the material covered in these lectures can be found in one way or another in the published literature, so there is no claim of originality. However, the detail and care with which some calculations are done may be useful for the uninitiated, who are the main targets of these notes. The topic of bosonization is covered in many review articles. Some of then are [4, 5, 6, 7, 8]. Some of the original articles are [9, 10, 11, 12, 13, 14, 15, 16]. I have drawn extensively from Haldane [15], von Delft and Schoeller [8], Voit[7] and Affleck [6]. These notes are organized as follows. Section II introduces the two basic models. The fundamental tools of bosonization are developed in Sections III to XIII. Section XIV focuses on the basic interacting model solved by bosonization, the Luttinger model. This is then applied to the XXZ model in Section XV. Section XVI is devoted to the important Luttinger liquid conjecture by Haldane. The case of spin- 21 fermions is studied in Section XVII. We end with a brief discussion of gaps and the sine-Gordon theory in Section XVIII. II The Hubbard and the Heisenberg models Our aim will be to study strongly correlated systems in one spatial dimension. These are typically systems of interacting electrons but we will be interested in spin systems as well. The prototypical interacting electron system is the Hubbard model. This is a lattice model whose Hamiltonian in one dimension is HHub = t X j  cyj cj+1 + h:c: + U X j cyj" cj" cyj# cj# : (1) The first term describes the hopping process, in which an electron can move from one site to the next with amplitude t while preserving its spin projection  (taken arbitrarily along the z -axis). The second term describes the local Coulomb repulsion (U > ) between opposite spin electrons residing on the same site. This is the so-called Hubbard U interaction term, named after one of the first people to work on this model in a series of classic papers [17, 18, 19]. This is one of the simplest interacting fermionic models one can write and has been extensively studied. 0 The cj operators are the usual annihilation operators with anti-commutation relations E. Miranda 4 Another important model is the spin -1/2 XXZ model, fcj ; cj0 0 g = n o cj ; cyj0 0 n cy ; cy0 0 o j j  = 0; HXXZ = J (2) = Æj;j0 Æ;0 : (3) X j
Sjx Sjx+1 + Sjy Sjy+1 +
Sjz Sjz+1 : (4) Here, Sja are spin- 21 operators with commutation relations c and a
2 P a Sj  a b S ;S j
j0 = iÆj;j0 "abcSjc a = x; y; z orequivalently 1; 2; 3; (5) = 34 = 12 12 + 1 . The symbol "abc is the totally anti-symmetric Levi-Civita tensor 8 if therearerepeated indicesamong (a; b; c) < 0 1 if (a; b; c) isan evenpermutation of (1; 2; 3) : "abc = : 1 if (a; b; c) is anodd permutation of (1; 2; 3) (6) d J is the exchange coupling and
the anisotropy parameter. A special important case of (4) is at isotropic Heisenberg model HHeis = J X j
= 1, the so-called Sj
Sj+1 : (7) Both models (1) and (4) can be solved exactly in one dimension (and only in one dimension) by means of the celebrated Bethe Ansatz [20, 21]. However, though the Bethe Ansatz can give the spectrum of eigenvalues and eigenvectors (plus a bit more), there is still a lot of important information that it cannot give, such as correlation functions. The technique of bosonization, specially suited for one spatial dimension, is a powerful field-theoretical tool that enables one to calculate correlation functions. In fact, it gives us a very great deal of insight into the physics of onedimensional systems by classifying them into “universality classes” and by characterizing their spectrum of low-lying excitations. Getting ahead of ourselves, it consists of a systematic mapping of a fermionic system (states, operators, Hamiltonians, etc.) into an auxiliary bosonic one. It turns out that the bosonic language is often more suited for the understanding of the physics of the system, sometimes even allowing for its exact solution, as we will see. We will embark on this construction taking the Hubbard model as a guide and it will become clear how it can be generalized to other models. ). Let us first look at the non-interacting limit (U In this case, the Hamiltonian can be easily diagonalized by means of Fourier transformation. Define (we work with the ) lattice spacing a =0 =1 L ikj X e = = eikj p ck : k2BZ L X (9) Note that we have “put the system in a box (ring)”, which is short for working on a finite lattice of L sites, with periodic boundary conditions cyj+L; = e ikjpe ikL y ck = cyj : L k2BZ X The last equality follows if e ikL = 1 ) k = where n = 0; 1; 2; : : : ;   2 n; (11) L L (10)  L 2 1 ; 2: (12) We take even values of L even for simplicity. Higher values 2 L
, then of n are redundant since, if k L 2 ( ) = i 2L L2 +1 j eikj = e ( = +1 ) = e i 2L ( L2 1)j = eik0 j ; 2 L ei L L 2 +1 j (13) 2  L
 and k is completely equivalent where k 0 L 2 to k 0 . The set (12) is called the first (...truncated)