The description of solids at a microscopic level is complex, involving the interaction of a huge number of its constituents, such as ions or electrons. It is impossible to solve the corresponding many-body problems analytically or numerically, although much insight can be gained from the analysis of simplified models. An important example is the Hubbard model, which describes interacting electrons in narrow energy bands, and which has been applied to problems as diverse as high-Tc superconductivity, band magnetism, and the metal-insulator transition. This 2005 book presents a coherent, self-contained account of the exact solution of the Hubbard model in one dimension. The early chapters will be accessible to beginning graduate students with a basic knowledge of quantum mechanics and statistical mechanics. The later chapters address more advanced topics, and are intended as a guide for researchers to some of the more topical results in the field of integrable models.

Using results on the scaling of energies with the size of the system and the principles of conformal quantum field theory, we calculate the asymptotics of correlation functions for the one-dimensional Hubbard model in the repulsive regime in the presence of an external magnetic field. The critical exponents are given in terms of a dressed charge matrix that is defined in terms of a set of integral equations obtained from the Bethe-Ansatz solution for the Hubbard model. An interpretation of this matrix in terms of thermodynamical coefficients is given, and several limiting cases are considered.

We present a general method for the calculation of correlation functions in the repulsive one-dimensional Hubbard model at less than half-filling in a magnetic field k. We describe the dependence of the critical exponents that drive their long-distance asymptotics on the Coulomb coupling, the density, and h. This dependence can be described in terms of a set of coupled Bethe-Ansatz integral equations. It simplifies significantly in the strong-coupling limit, where we give explicit formulas for the dependence of the critical exponents on the magnetic field. In particular, we find that at small field the functional dependence of the critical exponents on 6 can be algebraic or logarithmicdepending on the operators involved, In addition, we evaluate the singularities of the Fourier images of the correlation functions. It turns out that switching on a magnetic field gives rise to singularities in the dynamic field-field correlation functions that are absent at k=o.

By a combination of analytical and numerical techniques, we analyze the continuum limit of the integrable 3 ⊗3 ⊗ 3 ⊗3 . . . sl(2/1) superspin chain. We discover profoundly new features, including a continuous spectrum of conformal weights, whose numerical evidence is infinite degeneracies of the scaled gaps in the thermodynamic limit. This indicates that the corresponding conformal field theory has a non compact target space (even though our lattice model involves only finite dimensional representations). We argue that our results are compatible with this theory being the level k = 1, 'SU (2/1) WZW model' (whose precise definition requires some care). In doing so, we establish several new results for this model. With regard to potential applications to the spin quantum Hall effect, we conclude that the continuum limit of the 3 ⊗3 ⊗ 3 ⊗3 . . . sl(2/1) integrable superspin chain is not the same as (and is in fact very different from) the continuum limit of the corresponding chain with two-superspin interactions only, which is known to be a model for the spin quantum Hall effect. The study of possible RG flows between the two theories is left for further study.

Using a model of an open spin chain that is exactly soluble by means of a Bethe ansatz, we study the effects of a boundary magnetic field and an impurity spin coupled to the chain. An impurity spin only scatters forward, while the boundary is purely a back-scatterer. Two parameters for the impurity and one for the boundary permit us to mimic the effect of real magnetic impurity, with both forward and backward scattering.