Recently quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. Quantum magnets provide a perfect playground for studying these phase transitions since they can be triggered by many control parameters such as frustration, lattice dimerization, and magnetic field. Most previous studies have focused on the magnetic properties in pure systems. In these systems, responses to the triggering parameters are found to be uniform, leading to homogeneous phases. However little progress has been made so far on the phase transitions and properties in disordered quantum magnets because they are more complicated systems, and few theoretical tools can be applied.
Recently quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. Quantum magnets provide a perfect playground for studying these phase transitions since they can be triggered by many control parameters such as frustration, lattice dimerization, and magnetic field. Most previous studies have focused on the magnetic properties in pure systems. In these systems, responses to the triggering parameters are found to be uniform, leading to homogeneous phases. However little progress has been made so far on the phase transitions and properties in disordered quantum magnets because they are more complicated systems, and few theoretical tools can be applied.
In recent years, quantum phase transitions have attracted the interest of both theorists and experimentalists in condensed matter physics. These transitions, which are accessed at zero temperature by variation of a non-thermal control parameter, can influence the behaviour of electronic systems over a wide range of the phase diagram. Quantum phase transitions occur as a result of competing ground state phases. The cuprate superconductors which can be tuned from a Mott insulating to a d -wave superconducting phase by carrier doping are a paradigmatic example. This review introduces important concepts of phase transitions and discusses the interplay of quantum and classical fluctuations near criticality. The main part of the article is devoted to bulk quantum phase transitions in condensed matter systems. Several classes of transitions will be briefly reviewed, pointing out, e.g., conceptual differences between ordering transitions in metallic and insulating systems. An interesting separate class of transitions is boundary phase transitions where only degrees of freedom of a subsystem become critical; this will be illustrated in a few examples. The article is aimed at bridging the gap between high-level theoretical presentations and research papers specialized in certain classes of materials. It will give an overview on a variety of different quantum transitions, critically discuss open theoretical questions, and frequently make contact with recent experiments in condensed matter physics.
Quantum magnets represent an ideal playground for the controlled realization of novel quantum phases and of quantum phase transitions.The Hamiltonian of the system can be indeed manipulated by applying a magnetic field or pressure on the sample.When doping the system with non-magnetic impurities, novel inhomogeneous phases emerge from the interplay between geometric randomness and quantum fluctuations.In this paper we review our recent work on quantum phase transitions and novel quantum phases realized in disordered quantum magnets.The system inhomogeneity is found to strongly affect phase transitions by changing their universality class, giving the transition a novel, quantum percolative nature.Such transitions connect conventionally ordered phases to unconventional, quantum disordered ones -quantum Griffiths phases, magnetic Bose glass phases -exhibiting gapless spectra associated with low-energy localized excitations.
The study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.
This chapter has been co-authored with T. Senthil, and adapted from the Ph.D. thesis of T. Senthil, submitted to Yale University (1997, unpublished). The last two chapters of this book move beyond the study of regular Hamiltonians that have the full translational symmetry of an underlying crystalline lattice and consider the physically important case of disordered systems described by Hamiltonians with couplings that vary from point to point in space. By the standards of the regular systems we have already discussed, the quantum phase transitions of disordered systems are very poorly understood, and only a few well-established results are available. A large amount of theoretical effort has been expended toward unraveling the complicated phenomena that occur, and they remain active topics of current research. The aims of our discussion here are therefore rather limited: we highlight some important features that are qualitatively different from those of nondisordered systems, make general remarks about insights that can be drawn from our understanding of the finite- T crossovers in Part II, and discuss the properties of some simple solvable models. In keeping with the general strategy of this book, we introduce some basic concepts by studying the effects of disorder on the magnetic ordering transitions of quantum Ising/rotor models studied in Part II; we also make some remarks in Section 21.4 on the effects of disorder on the ordering transitions of Fermi liquids considered in Chapter 18. Models with much stronger disorder and frustrating interactions that have new phases not found in ordered systems are considered in Chapter 22.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
Quantum Phase Transitions is the first book to describe in detail the fundamental changes that can occur in the macroscopic nature of matter at zero temperature due to small variations in a given external parameter. The subject plays a central role in the study of the electrical and magnetic properties of numerous important solid state materials. The author begins by developing the theory of quantum phase transitions in the simplest possible class of non-disordered, interacting systems - the quantum Ising and rotor models. Particular attention is paid to their non-zero temperature dynamic and transport properties in the vicinity of the quantum critical point. Several other quantum phase transitions of increasing complexity are then discussed and clarified. Throughout, the author interweaves experimental results with presentation of theoretical models, and well over 500 references are included. The book will be of great interest to graduate students and researchers in condensed matter physics.