Multivariable optimization problems belong to an interdisciplinary field of research connecting diverse fields like computer science, information theory and statistical physics. It involves finding an optimal solution out of the many available states or configurations, possibly satisfying a set of constraints. For physicists, the problem can be visualized as finding the global minima of the energy landscape of a given complex Hamiltonian, which is quite difficult to obtain by the obvious method of examining the energy of each state. This is due to the huge number of available states which grows exponentially with the system size (for N Ising spins, the configuration space grows as 2N). The search for such a global minimum becomes all the more complex when the landscape consists of several local minima separated by macroscopic barriers (O(N)) arising from frustrating constraints in the systems. To solve such an optimization problem in a stochastic case, the concept of simulated annealing (SA) was put forward by Kirkpatrick et al. [437]. In this method, a tunable noise is artificially induced (thermal-like fluctuations equivalent to that of a metallurgical annealing) which helps the system to escape from the local minima by hopping above the energy barriers, and staying longer in minimum energy valley(s) as the noise variable is slowly reduced to zero starting from a high value. At the same time, it is also known that in complex systems, due to many closely spaced low lying states, the annealing needs to be performed infinitely slowly to reach the true ground state; this is indeed a major drawback. We shall elaborate on this point later when we discuss the effect of system size on the minimum gap between the ground state and the first excited state of the quantum many body system under consideration. The numerical methods used in such studies are variants of Monte Carlo methods.

In recent years, there has been an upsurge of studies interconnecting the phenomena of quantum phase transitions, non-equilibrium dynamics, and quantum information and computation. These studies are important from the viewpoint of fundamental physics as well as for developing new quantum technologies. This book is the first attempt to connect these different fields, mentioning both the promises and the problems and incorporating discussions of the most recent technological developments. While there are several books on quantum phase transitions, for example, those by S. Sachdev (Cambridge University Press, 2011) and S. Suzuki et al., (Springer, 2013), the present book emphasizes several different aspects not discussed in earlier books or reviews. We build up from preliminary discussions of the basic phenomenology in the introductory chapter to full exegeses of important models, with further details presented in the appendices. We hope that this structure will enable the beginner to navigate smoothly through the more involved discussions. Concise summaries at the end of each chapter should permit the reader to easily get a sense of the scope of the book.

The book describes generic theories of the scaling of quantum information theoretic measures close to a quantum critical point (QCP) and of the residual energy in the final state reached following a passage through a QCP. This non-adiabatic passage in turn generates non-trivial quantum correlations in the final state which, in some cases, are found to satisfy some intriguing scaling relations. All these theories are illustrated employing the transverse Ising and other transverse field models and their variants. The advantage of using the transverse field Ising model is two-fold: (i) the one-dimensional version with a nearest-neighbor interaction is exactly soluble (and the QCP is conformally invariant), and (ii) the model can be mapped to a classical Ising model with one added dimension using the Suzuki–Trotter or the path integral formalism. These two remarkable properties of the these models have been exploited thoroughly over the last fifty years, but especially in the last two decades to understand quantum phase transitions and their connection to information processing, non-equilibrium dynamics, and quantum annealing.

The Ising model in transverse magnetic field is perhaps the simplest quantum spin model, yet the elements of site disorder, random magnetic fields, and tunable quantum fluctuations permit a rich array of ground states, and the rare ability to emphasize their classical or quantum character. We briefly review in this chapter experimental results from chains to crystals of spins, with magnetic, glassy, and spin liquid ground states.

Singlet Ground State Magnets

In a crystalline lattice, the crystal fields often take a form such that the ground states of magnetic ions are singlets separated from the higher lying spin multiplets. Sufficiently strong magnetic dipolar or exchange couplings between different sites can mix the singlets with the multiplets enough so as to induce non-zero expectation values for the magnetic dipoles at individual sites. Examples of this phenomenon are generally found among metals and insulators based on rare earths. Such metals were a subject of extensive research in the 1960's and 1970's, and the book by MacKintosh and Jensen [408] gives an excellent account of both the data as well as mean field approaches. With respect to insulating materials, LiTbF4 emerged as a model system in the 1970's. It is an example of an ideal Ising dipolar coupled ferromagnet; here the long-range nature of the dipolar interactions leads to an upper critical dimension [149] (at which mean field theory with logarithmic corrections calculable using RG methods becomes an exact description of thermal phase transitions) of three rather than four.

LiTbF4 belongs to the LiREF4 isostructural series of ionic salts. Li and F carry valences +1 and −1 respectively, leaving RE, which can be any rare earth atom or the nonmagnetic element Y, with a valence of +3. Figure 14.1 shows the body-centered tetragonal crystal structure for the family.