Cambridge University Press eBooks

Quantum Phase Transitions in Transverse Field Spin Models

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti, Uma Divakaran, T. F. Rosenbaum, Diptiman Sen

Published Jan 28, 2015• 337 citations

Abstract

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.

DOI: 10.1017/cbo9781107706057

Related papers

Quantum phase transitions in transverse field spin models: from statistical physics to quantum information

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

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.

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Simulating dynamics of the two-dimensional transverse-field Ising model: a comparative study of large-scale classical numerics

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Concluding Comments

Amit Dutta, G. Aeppli, Bikas K. Chakrabarti +3 more

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.

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