Introduction and guide to the chapter

Linear response functions contain a wealth of information about the physical properties of a many-body system. In the case of the electron liquid, for example, the density–density response function provides a unified framework for the understanding of different phenomena such as static screening, effective interaction, collective modes, electron energy loss spectra (inelastic scattering of electrons), and Raman spectra (inelastic scattering of photons). The spin–spin response function provides the corresponding information for the spin density fluctuations as probed, for example, by cross-polarized Raman scattering experiments, in which the incident and scattered photon have perpendicular polarizations, or by spin-flip electron energy loss spectroscopy, in which the incoming and outgoing electrons have opposite spin orientations.

As we have seen, the Hartree–Fock approximation describes each electron as an independent particle moving in a self-consistent field generated by all the other electrons. There is no correlation between this self-consistent field and the instantaneous position of the electron. Reality is however quite different: whenever an electron moves, it acts on the surrounding electrons, causing a collective disturbance which eventually feeds back on its own motion. In order to study these effects we need, first of all, to learn how the electron liquid as a whole responds to disturbances caused by the charge, the spin, and the current of a single electron. The linear response theory of Chapter 3 provides the natural framework for this description. Strictly speaking, the linear response functions describe the readjustments of the electronic density, spin, or current, in response to externally controlled fields.

Introduction

The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory, which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7.

In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.

Introduction to the quantum inverse scattering method

The quantum inverse scattering method is the modern algebraic theory of exactly solvable quantum systems. It arose [404, 410, 411] as an attempt to carry over the concepts of the inverse scattering method for classical non-linear evolution equations [2, 134] into quantum mechanics. As a result, our understanding of both the theory of integrable partial differential equations and the theory of exactly solvable quantum systems changed, and the algebraic roots of the exact solvability became apparent. These roots originate from the Yang-Baxter equation and its classical counterpart.

Before turning to our actual subject, which is the application of the quantum inverse scattering method to the Hubbard model, we give a brief general introduction. We shall limit our exposition basically to the material which is needed later for the understanding of the algebraic structure of the Hubbard model. The reader who is interested in the general scope of the method and in the history of its development is referred to the excellent books and review articles [131, 270, 276, 277, 407].

Integrability

As a motivation for the definition of the Yang-Baxter algebra in the following subsection we shall first recall the concept of integrability in classical mechanics. Then, by considering the elementary example of the harmonic oscillator, we shall see that this concept does not directly apply to quantum mechanical systems and needs to be extended.

In this chapter we will reveal another piece of the algebraic structure of the Hubbard model. As was first observed by Uglov and Korepin [462], the Hubbard Hamiltonian on the infinite line is invariant under the action of the direct sum of two so-called Yangian quantum groups, extending the rotational and the η-pairing su(2) symmetries we encountered earlier. Following [172] we shall address the issue in a more general context. We present two pairs of fermionic representations of the Y(su(2)) Yangian quantum group which commute with the trigonometric [162] and hyperbolic [40, 41] versions of a Hubbard Hamiltonian with non-nearest-neighbour hopping. In both cases the two representations are also mutually commuting, hence can be combined into a representation of Y(su(2))⊕Y(su(2)). The generators of the Yangian symmetry of the ordinary Hubbard model (with nearest-neighbour hopping) and of a number of other interesting models like the Haldane-Shastry spin-chain [194, 394] are obtained as special cases of our general result.

Introduction

Quantum groups were introduced by Drinfeld [107, 109]. His original intention was to put what we called the Yang-Baxter algebra into the mathematically more conventional context of Hopf algebras. The Yangians are special quantum groups. Their representation theory [80, 81] is intimately related to the classification of integrable quantum systems with rational R-matrices.

On account of Lieb and Wu's 1968 Bethe ansatz solution, the one-dimensional Hubbard model has become a laboratory for theoretical studies of non-perturbative effects in strongly correlated electron systems. Many of the tools available for the analysis of such systems have been applied to this model, both to provide complementary insights to what is known from the exact solution or as an ultimate test of their quality. In parallel, due to the synthesis of new quasi one-dimensional materials and the refinement of experimental techniques, the one-dimensional Hubbard model has evolved from a toy model to a paradigm of experimental relevance for strongly correlated electron systems.

Due to the ongoing efforts to improve our understanding of one-dimensional correlated electron systems, there exists a large number of review articles and books covering various aspects of the general theory, as well as the Bethe ansatz and field theoretical methods. A collection of these works is listed in the General Bibliography below.

Still we felt – and many of our colleagues shared this view – that there would be a need for a coherent account of all of these aspects in a unified framework and from the perspective of the one-dimensional Hubbard model, which, moreover, would be accessible to beginners in the field. This motivated us to write this volume. It is intended to serve both as a textbook and as a monograph.

A very curious situation arises in the context of the calculation of the partition function from the spectrum of an integrable Hamiltonian. Despite the validity of the Bethe ansatz equations for all energy eigenvalues of the model the direct evaluation of the partition function is rather difficult. In contrast to ideal quantum gases the eigenstates are not explicitly known: the Bethe ansatz equations provide just implicit descriptions that pose problems of their own kind. Yet, knowing the behaviour of quantum chains at finite temperatures is important for many reasons. As a matter of fact, the groundstate is strictly inaccessible due to the very fundamentals of thermodynamics. Therefore the study of finite temperatures is relevant for theoretical as well as experimental reasons. At high temperatures, quantum systems show only trivial static properties without correlations. Lowering the temperature, the systems enter a large regime with non-universal correlations and finally approach the quantum critical point at exactly zero temperature showing universal, non-trivial properties with divergent correlation lengths governed by conformal field theory [51].

In Chapter 5 of this book we have discussed the traditional Thermodynamical Bethe Ansatz (TBA) as developed for the Heisenberg model and the Hubbard model [155, 433–435] on the basis of a method [496] invented for the Bose gas.

The purpose of this opening chapter is threefold: to introduce the Hubbard model, to discuss its origin and significance and to give a brief summary of its history. Rather than beginning with more general and historical considerations we will start with a concrete albeit somewhat technical discussion of how the Hubbard model arises as an effective description of electronic degrees of freedom in solids.

On the origin of the Hubbard model

The Hubbard model is named after John Hubbard, who in a series of influential articles [201–206] introduced the Hamiltonian in order to model electronic correlations in narrow energy bands and proposed a number of approximate treatments of the associated many-body problem. Our following discussion of how the Hubbard Hamiltonian arises in an approximate description of interacting electrons in a solid loosely parallels Hubbard's original work. We will assume that the reader is familiar with the basic concepts of solid state theory (see e.g. [25, 509]) and with the formalism of second quantization (e.g. [283]). For further reading we refer to the original literature [188, 201, 233] and to the monographs [27, 158, 498].

A solid consists of ions and electrons condensed in a three-dimensional crystalline structure. Since the ions are much heavier than the electrons, it is often a good phenomenological starting point for the exploration of the electronic properties of solids to think of the ions as forming a static lattice.

The bare existence of this book is due to the amazing fact that the solution of the stationary Schrödinger equation for the one-dimensional Hubbard model can be reduced to a set of algebraic equations, which is tractable in the thermodynamic limit. These equations will be derived in this chapter. We will call them the Lieb-Wu equations to honour E. H. Lieb and F. Y. Wu, who first obtained them [298]. The derivation is based on a method, called the nested (coordinate) Bethe ansatz, which goes back to the seminal articles of C. N. Yang [493] and M. Gaudin [154], who generalized earlier work [60, 296] on exactly solvable models to models with internal degrees of freedom.

The roots of the Lieb-Wu equations parameterize the eigenvalues and the eigenstates of the Hamiltonian of the one-dimensional Hubbard model. They encode the complete information about the model. These roots are not explicitly known in the general N-particle case. In the thermodynamic limit (N →∞), however, only the distributions of the roots in the complex plane matter, and many physical quantities can be calculated as solutions of linear or non-linear Fredholm type integral equations. Moreover, it is sometimes possible to use the Lieb-Wu equations in an implicit way even for finite N, e.g., in the proof of the SO(4) highest weight properties of the eigenstates in Sections 3.D and 3.F of the appendix or in the calculation of their norm in Section 3.5.

In this chapter we carry out the thermodynamic limit on the level of the monodromy matrix introduced in Chapter 12. This means to change the strategy as compared to the Bethe ansatz solutions put forward in Chapters 3 and 12 which depended crucially on the use of periodic boundary conditions. The discreteness of the quasi momenta kj in the Bethe ansatz wave function was due to the finite length L of the system. For infinite L there will be no Lieb-Wu equations which were our main tool for studying the Hubbard model in this book. Instead the commutation relations for the elements of the infinite interval monodromy matrix will utilized in the calculations shown below.

We basically follow the articles [335, 336], where the quantum inverse scattering method, in the way as originally designed in [131, 404, 454], was applied to the Hubbard model. Our account will be restricted to the case of zero electron density. Quite generally, the quantum inverse scattering method, as originally conceived in the spirit of the ‘inverse scattering theory’ for classical non-linear evolution equations, is restricted to uncorrelated vacua (ground states) which limits the applicability of the method. Nevertheless, applying it to the empty band ground state of the Hubbard model we shall obtain valuable additional insights into its structure. We shall reveal the connection between Shastry's R-matrix and the Yangian symmetry discussed in Chapter 14.