The X-Y model in one- and two-dimensions is investigated by means of the path integral. The low temperature properties are shown to be described by the phase mode. In particu-lar, the quantum model in the limit S--'fco is found to be well expressed by the planer model. The spin-correlation functions are calculated explicitly. As a result it is found that at zero temperature the quasi long-range order in one-dimension and the long-range order in two-dimen-sions appear. The spontaneous magnetization is explicitly given at zero temperature in two-dimensions. At finite temperatures the dynamical behaviour is the same as that studied by Nelson and Fisher. The divergence of the susceptibility at T=O in one-dimension is shown to be different from that of the classical model. Considerable progress has been made recently 111 understanding the nature of the phase transition in the two-dimensional planer model without long-range or-cler.l)~81 It has been revealed that below the transition temperature there exists a topological order in which bound pairs of vortex and antivortex are formed, the
The X-Y model in one- and two-dimensions is investigated by means of the path integral. The low temperature properties are shown to be described by the phase mode. In particu-lar, the quantum model in the limit S--'fco is found to be well expressed by the planer model. The spin-correlation functions are calculated explicitly. As a result it is found that at zero temperature the quasi long-range order in one-dimension and the long-range order in two-dimen-sions appear. The spontaneous magnetization is explicitly given at zero temperature in two-dimensions. At finite temperatures the dynamical behaviour is the same as that studied by Nelson and Fisher. The divergence of the susceptibility at T=O in one-dimension is shown to be different from that of the classical model. Considerable progress has been made recently 111 understanding the nature of the phase transition in the two-dimensional planer model without long-range or-cler.l)~81 It has been revealed that below the transition temperature there exists a topological order in which bound pairs of vortex and antivortex are formed, the
Phase diagram, critical properties and thermodynamic functions of the two-dimensional field-free quantum-spin-1/2 XXZ model has been calculated globally using a numerical renormalisation group theory. The nearest-neighbour spin-spin correlations and entanglement properties, as well as internal energy and specific heat are calculated globally at all temperatures for the whole range of exchange interaction anisotropy, from XY limit to Ising limits, for both antiferromagnetic and ferromagnetic cases. We show that there exists long-range (quasi-long-range) order at low-temperatures, and the low-lying excitations are gapped (gapless) in the Ising-like easy-axis (XY-like easy-plane) regime. Besides, we identify quantum phase transitions at zero-temperature.
We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
We consider the zero-temperature properties of four different spin 1/2 models on two-dimensional lattices: the XY ferromagnet, the XY antiferromagnet, the Heisenberg antiferromagnet, and the Dzyaloshinsky–Moriya models. Most of this article is a review of previously published work, but a few previously unpublished results are included. The relation between three of the models on bipartite lattices is described. The properties of the XY ferromagnet in two dimensions, especially those derived from extrapolation of finite lattice results, are reviewed. A numerical factor by which spin-wave and finite-lattice estimates of the long-range order parameter differ is discussed. For frustrated models on the triangular lattice the possibility of a chirality phase transition instead of, or in addition to, a magnetic phase transition is considered.
We study the quantum transition at T=0 in the spin-1/2 Ising spin glass in a transverse field in two dimensions. The world line path integral representation of this model corresponds to an effective classical system in 2+1 dimensions, which we study by Monte Carlo simulations. Values of the critical exponents are estimated by a finite-size scaling analysis. We find that the dynamical exponent, z, and the correlation length exponent, \ensuremath{\nu}, are given by z=1.5\ifmmode\pm\else\textpm\fi{}0.05 and \ensuremath{\nu}=1.0\ifmmode\pm\else\textpm\fi{}0.1. Both the linear and nonlinear susceptibility are found to diverge at the critical point.
Pairing and topological phases in cold atoms with long range interactions Chapter 1 Introduction The systematic study and classification of phase transitions has become a popular research area in physics over the past decades. The most familiar phenomena associated to the concept of phase transition are the ones giving rise to macroscopic changes in the properties of strongly correlated many-body systems due to thermal fluctuations. The most prominent examples of this kind of behaviour are the ferromagnetic-paramagnetic transition in lattice spin models and the liquid-gas transition. On the other hand, when dealing with quantum-mechanical systems changes of matter from one state to another can arise as the result of the variation of parameters other than temperature. In particular, when a given many-body quantum problem is studied at zero temperature, thereby suppressing thermal fluctuations, it is the interplay between kinetic energy and interactions that can cause the system to jump into different phases as the model parameters are varied. Phase transitions occurring at zero temperature are widely known as quantum phase transitions. In the framework of the physics of phase transitions, the role of dimensionality is a noticeable one. The major challenge in this perspective is the treatment of low dimensional systems, where standard mean-field treatments and ordinary perturbation theory are known to fail due to the enhancement of fluctuations. The goal of describing collective behaviours in reduced dimensionality is usually pursued in one-dimensional (1D) systems, which turn out to be simple enough to be studied in an effective way through both numerical and analytical techniques, while exhibiting a rich and deeply fascinating phenomenology. As a matter of fact, since particles confined to one dimension cannot avoid interactions with each other, the typical low energy excitations of the system are represented by collective density waves, which substitute nearly free quasiparticle excitations characterizing the behaviour of three-dimensional Fermi liquids in the description of the effects of the interplay between kinetic fluctuations and interactions. On the other hand, the tools employed in the activity of tackling 1D manybody problems are efficient and well developed. The analytical techniques aremainly based on the bosonization technique, which allows for the reformulation of strongly correlated 1D systems onto free bosonic theories, which can then be analyzed by means of standard path integral techniques. Meanwhile, the density matrix renormalization group (DMRG) algorithm represents the state-of-theart numerical machinery in the study of 1D physics, since it allows to efficiently extract the low energy properties of the model Hamiltonian and consequently characterize the corresponding phase diagram by means of the behaviour of properly chosen order parameters. As a final remark pointing out the relevance of the research effort in the direction of a better understanding of 1D many-body quantum systems, it is worth noticing that 1D systems do not represent only a playground for theorists, but have been experimentally realized by means of setups with cold atoms trapped in optical lattices, where the experimentalist is able to tune the model parameters by varying the characteristic features of the trapping laser beams and meanwhile has a comfortable access to the most relevant observables.