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 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.
The transverse field Ising and XY models (the simplest quantum spin models) provide the organising principle for the rich variety of interconnected subjects which are covered in this book. From a generic introduction to in-depth discussions of the subtleties of the transverse field Ising and related models, it includes the essentials of quantum dynamics and quantum information. A wide range of relevant topics has also been provided: quantum phase transitions, various measures of quantum information, the effects of disorder and frustration, quenching dynamics and the Kibble–Zurek scaling relation, the Kitaev model, topological phases of quantum systems, and bosonisation. In addition, it also discusses the experimental studies of transverse field models (including the first experimental realisation of quantum annealing) and the recent realisation of the transverse field Ising model using tunable Josephson junctions. Further, it points to the obstacles still remaining to develop a successful quantum computer.
The past decade has seen a substantial rejuvenation of interest in the study of quantum phase transitions, driven by experiments on cuprate superconductors, heavy fermion materials, organic conductors, and related compounds. Although quantum phase transitions in simple spin systems, like the Ising model in a transverse field, were studied in the early 1970s, much of the subsequent theoretical work examined a particular example: the metal–insulator transition. While this is a subject of considerable experimental importance, the greatest theoretical progress was made for the case of the Anderson transition of non-interacting electrons, which is driven by localization of the electronic states in the presence of a random potential. The critical properties of this transition of noninteracting electrons constituted the primary basis upon which most condensed matter physicists have formed their intuition on the behavior of the systems near a quantum phase transition. However, it is clear that strong electronic interactions play a crucial role in the systems of current interest noted earlier, and simple paradigms for the behavior of such systems near quantum critical points are not widely known. It is the purpose of this book to move interactions to center stage by describing and classifying the physical properties of the simplest interacting systems undergoing a quantum phase transition. The effects of disorder will be neglected for the most part but will be considered in the concluding chapters. Our focus will be on the dynamical properties of such systems at nonzero temperature, and it will become apparent that these differ substantially from the noninteracting case.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
Phase transitions and their associated critical phenomena are of fundamental importance and play a crucial role in the development of statistical physics for both classical and quantum systems. Phase transitions embody diverse aspects of physics and also have numerous applications outside physics, e.g., in chemistry, biology, and combinatorial optimization problems in computer science. Many problems can be reduced to a system consisting of a large number of interacting agents, which under some circumstances (e.g., changes of external parameters) exhibit collective behavior; this type of scenario also underlies phase transitions. The theoretical understanding of equilibrium phase transitions was put on a solid footing with the establishment of the renormalization group. In contrast, non-equilibrium phase transition are relatively less understood and currently a very active research topic. One important milestone here is the Kibble-Zurek (KZ) mechanism, which provides a useful framework for describing a system with a transition point approached through a non-equilibrium quench process. I developed two efficient Monte Carlo techniques for studying phase transitions, one is for classical phase transition and the other is for quantum phase transitions, both are under the framework of KZ scaling. For classical phase transition, I develop a non-equilibrium quench (NEQ) simulation that can completely avoid the critical slowing down problem. For quantum phase transitions, I develop a new algorithm, named quasi-adiabatic quantum Monte Carlo (QAQMC) algorithm for studying quantum quenches. I demonstrate the utility of QAQMC quantum Ising model and obtain high-precision results at the transition point, in particular showing generalized dynamic scaling in the quantum system. To further extend the methods, I study more complex systems such as spin-glasses and random graphs. The techniques allow us to investigate the problems efficiently. From the classical perspective, using the NEQ approach I verify the universality class of the 3D Ising spin-glasses. I also investigate the random 3-regular graphs in terms of both classical and quantum phase transitions. I demonstrate that under this simulation scheme, one can extract information associated with the classical and quantum spin-glass transitions without any knowledge prior to the simulation.
When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.
This part of the book is fully devoted to the physics of (1 + 1)-dimensional quantum or twodimensional classical models. As usual, I shall distinguish between classical and quantum systems only if it is necessary. Those models where the difference between quantum and classical is not relevant will be called ‘two-dimensional’; those where for some reason or other I want to emphasize the quantum aspect will be called ‘one-dimensional’. The reason I spend so much time discussing two-dimensional physics is that here one finds a kind of paradise for strong interactions and nonperturbative effects. Those effects which are tricky or even impossible to achieve in higher dimensions appear in almost every two-dimensional model. To our greatest satisfaction there are theoretical tools at our disposal which allow us to solve many of the corresponding problems and describe these phenomena. The availability of these strong mathematical tools is a unique feature of two-dimensional physics. Even if it turns out to be impossible to generalize these tools for higher dimensions, I hope that the reader will be rewarded for the time spent by the pleasure obtained from contemplating their beauty. The world we are about to enter has certain distinct features which are worth mentioning in the introduction. The first one concerns the second-order phase transitions. In (1 + 1)-dimensional systems such transitions may occur only at zero temperature.