In this book, we have discussed quantum phase transitions in transverse field Ising and XY models in one and higher dimensions, particularly in the context of quantum information processing and non-equilibrium dynamics. We have illustrated how these models relate to actual quantum annealing protocols in physical systems and their connection to efficient quantum computation. Quantum phase transitions for pure models as well as for models with random interactions or random fields have also been considered. These models were introduced in the early 1960's in the context of order-disorder ferroelectric systems. This book focuses on the salient issues for which these models continue to be important and interesting even fifty years after their first appearance.

Our starting point involved generic theories of behavior of information theoretic measures, which were then illustrated using the transverse field models. Related models, e.g. those which are exactly soluble using Jordan-Wigner transformations and those exhibiting topological quantum phase transitions, were presented in the book at some length. These models are relevant to low-dimensional condensed matter physics as well as to quantum information and quantum dynamical studies.

On the theoretical side, the integrability of the transverse field Ising/XY models in one dimension has provided an ideal testing ground for field theoretical and information theoretical studies. Numerous predictions of theoretical studies have been verified experimentally in recent years. [180]. The one-dimensional version of these models have been extremely useful in studying information theoretic measures like concurrence, entanglement entropy, fidelity and fidelity susceptibility, and also the scaling of the defect density generated by quantum critical and multicritical quenches, namely, the Kibble–Zurek scaling. These models have played a crucial role in the development of quantum annealing techniques and adiabatic quantum algorithms. The equilibration or thermalization following quantum quenches have been studied using variants of transverse field models.

An advantageous feature of transverse field Ising models is the quantum-classical mapping which renders these models ideally suited for quantum Monte Carlo studies for d > 1.

In the previous chapters we have been dealing mainly with effects involving a single electron, such as the solution of the Schrödinger equation for a hydrogen-like atom. However, the majority of interesting problems in spectroscopy deal with systems that contain many electrons. In this chapter we shall see how to construct many-body wave functions from single-particle wave functions, and how to build a many-body Hamiltonian in matrix form and apply this to the Coulomb interaction for many-electron atoms.

Many-body wave functions

A many-body wave function needs to satisfy several characteristics. First, we need to ensure that the particles in the wave function are identical. This is a clear difference from classical physics, where we can distinguish one object from another. Second, it is with many-body wave functions that the distinction between fermions and bosons comes to the forefront. The construction of many-body basisfunctions starts by choosing a basis of one-particle basisfunctions. Let us denote these by ϕk(r), where k is a generic quantum number describing the quantum states (for example, momentum and spin k → kσ with spin projection σ = ↑, ↓ or k → nlmσ for hydrogen-like atomic orbitals). The basis can consist of eigenfunctions of the one-particle problem. In this case, the one-particle interactions H1 (r1) are already solved and we are only dealing with the two-particle interactions H2(r1, r2). However, this is not essential and we often choose a basis that is convenient to work with. For example, for atoms and ions, convenient basisfunctions are the hydrogen-like atomic orbitals. In the presence of spin-orbit coupling, these states are not eigenfunctions. In this case, both one- and two-particle interactions couple different many-body basisfunctions. In principle, the basis of many-body wave functions has to be complete. In practice, this is impossible since it requires an infinite number of basisfunctions.