Analytical and Numerical Studies of 2D XY Models with Ring Exchange
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Notes on Statistical Field Theory
Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual).......................... -
Correlated Disorder in the 3D XY Model
2005:202 CIV MASTER’S THESIS Correlated Disorder in the 3D XY Model A model for 4He in aerogel PER BURSTRÖM MASTER OF SCIENCE PROGRAMME Engineering Physics Luleå University of Technology Department of Applied Physics and Mechanical Engineering Division of Physics 2005:202 CIV • ISSN: 1402 - 1617 • ISRN: LTU - EX - - 05/202 - - SE Abstract Experiments with superfluid 4He inserted into aerogels have revealed some sur- prising results. Most porous materials, with porosities down to as low as 30%, generally do not change the behaviour of 4He at the superfluid (λ) phase transi- tion. This is also supported by theoretical work about the nature of the disorder structure in most materials. However, when inserted in silica aerogels of vari- ous porosities, the superfluid density exponent ζ is distinctly changed from the undisturbed value of ζ ≈ 0:6740:003 to approximately 0:75−0:81. This is sur- prising as these aerogels are extremely porous, effectively thread-like structures of SiO2 making up 0.1-10% of the material, with the rest being incipient voids of air or, in this case, Helium. The reason for the changes in scaling parameters is believed to be that the disorder structure of aerogel is correlated over cer- tain length scales in the material. The scope of this thesis was to investigate a model for correlated aerogel disorder and its scaling properties via Monte Carlo simulations at the λ transition. i Contents 1 Introduction 1 2 Theoretical concepts 2 2.1 Derivation of the XY model . 2 2.2 Statistical mechanics using Monte Carlo methods . 2 2.3 The Metropolis algorithm . -
Trends in Mathematical Crystallisation May 2016 Organiser: Stefan Adams, Markus Heydenreich, Frank Den Hollander, Sabine Jansen
Trends in mathematical crystallisation May 2016 Organiser: Stefan Adams, Markus Heydenreich, Frank den Hollander, Sabine Jansen Description and aim: The workshop concentrates on the mathematical theory of crystallisation. More precisely, we focus on five topics that are closely connected with crystallisation, and approach these topics from different mathematical angles. These five topics are: (1) Spatial symmetry breaking at positive temperature (2) Crystallisation and surface effects at zero temperature (3) Continuum particle systems (Gibbs point fields, metastability, and Percolation) (4) Elasticity (variational analysis & gradient fields and quasi-crystals and liquid crystals (5) Differential/integral geometry We expect participants from analysis, statistical physics and probability. Our primary goal is to discuss the different approaches towards crystallisation in these disciplines. Vision: Crystallisation is the (natural or artificial) process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallisation is also a chemical solid-liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid crystalline phase occurs. In chemical engineering crystallisation occurs in a crystalliser. Crystallisation is therefore an aspect of precipitation, obtained through a variation of the solubility conditions of the solute in the solvent, as compared to precipitation due to chemical reaction. Crystallisation poses major challenges for mathematicians - it is one of the challenging open interdisciplinary problems involving analysts, probabilists, applied mathematicians, physicists and material scientists. There are many open questions, among which are the following. Most of these have been discussed at the workshop 'Facets of mathematical crystallization' - Lorentz Center Leiden, September 2014. A key challenge of the workshop is to address and clarify their potential further. -
Investigation of Zero-Temperature Transverse-Field Ising Models With
INVESTIGATION OF ZERO-TEMPERATURE TRANSVERSE-FIELDISINGMODELSWITH LONG-RANGEINTERACTIONS Untersuchung langreichweitiger Isingmodelle im transversalen Feld bei Temperatur T 0K Æ Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades dr. rer. nat. vorgelegt von sebastian fey Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 2020-05-07 Vorsitzender des Promotionsorgans: Prof. Dr. Georg Kreimer Gutachter: Prof. Dr. Kai P. Schmidt Prof. Dr. Martin Eckstein ABSTRACT In this thesis I aim to investigate ground-state properties of a quantum-mechanical long- range interacting spin model at temperature T 0K. Paradigmatic models such as the Æ Ising model are mostly limited to nearest-neighbor interactions. However, their long- range counterparts often display a drastically different behavior. Long-range interactions can induce an effective dimensionality into the system, leading to continuously varying critical exponents of quantum phase transitions in ferromagnetic systems and the ap- pearance of multiplicative logarithmic corrections. For antiferromagnetic interactions frustration can result in the appearance of new phases. During the last decades several studies of such models have been performed with various methods. Exact diagonalization and Quantum Monte-Carlo calculations are yet limited to finite system sizes. Density- matrix renormalization-group methods allow handling infinite sizes but results are only available for (quasi-)one-dimensional models. In this thesis, a method is presented which allows the computation of quantitative results for gapped quantum-many-body systems with long-range interactions in the bulk limit based on a perturbative approach. Perturbative continuous unitary transformations are combined with Monte-Carlo methods for an evaluation of nested infinite sums and Padé extrapolations to extract critical behavior. -
The Absence of Phase Transition for the Classical XY-Model On
February 28, 2018 19:54 Phase Transitions paper Phase Transitions Vol. 00, No. 00, Month 200x, 1–7 RESEARCH ARTICLE The absence of phase transition for the classical XY-model on Sierpi´nski pyramid with fractal dimension D=2 Michelle Przedborski and Boˇzidar Mitrovi´c∗ Department of Physics, Brock University, St. Catharines, Ontario, Canada L2S 3A1 () For the spin models with continuous symmetry on regular lattices and finite range of interac- tions the lower critical dimension is d=2. In two dimensions the classical XY-model displays Berezinskii-Kosterlitz-Thouless transition associated with unbinding of topological defects (vortices and antivortices). We perform a Monte Carlo study of the classical XY-model on Sierpi´nski pyramids whose fractal dimension is D = log4/log2=2 and the average coordination number per site is ≈ 7. The specific heat does not depend on the system size which indicates the absence of long range order. From the dependence of the helicity modulus on the cluster size and on boundary conditions we draw a conclusion that in the thermodynamic limit there is no Berezinskii-Kosterlitz-Thouless transition at any finite temperature. This conclusion is also supported by our results for linear magnetic susceptibility. The lack of finite temperature phase transition is presumably caused by the finite order of ramification of Sierpi´nski pyramid. Keywords: XY-model; fractals; Sierpi´nski pyramid; Monte Carlo simulations 1. Introduction One of the powerful predictions of the renormalization group theory of critical phenomena is universality according to which the critical behavior of a system is determined by: (1) symmetry group of the Hamiltonian, (2) spatial dimension- ality d and (3) whether or not the interactions are short-ranged [1]. -
Physics 682: Quantum Magnetism Homework 3
3/4/2004. Due: Thursday, March 18, 2004 (in class). Physics 682: Quantum Magnetism Problems with stars are not for credit and will NOT be graded. Homework 3 Exercise 1: Isotropic ferromagnet Consider a three-dimensional classical isotropic ferromagnet described by a micro- scopic Hamiltonian: X ~ ~ H = − JijSiSj, (1) i,j ~2 2 where Si = S is a classical spin at site i of a simple cubic lattice, exchange constants ~ Jij = J are non zero for nearest neighbors only. Let us assume that spin S = (S1,S2,...,SN ) has N-components (N = 1, e.g., corresponds to Ising model). Let us assume that the ordering phase transition is described by Landau free energy 2 4 2 f = αtm + bm + g(∂µ ~m) . a) Extract the values of parameters α, b, and g from an RPA approximation to the theory (1). Hint: See ex. 5-6 from Homework 2. b) Write down Ginzburg-Levanyuk criterion of an applicability of the mean field approximation in the vicinity of critical temperature. Use the values of parameters found in a). c) Is there any small parameter which guarantees the existence of the range where mean field theory works well? Will fluctuation region expand or shrink if one increases (decreases) the dimensionality N of the order parameter? Exercise 2: Real space renormalization group for one-dimensional Ising model Consider partition function for one-dimensional Ising model. N X X ZN (K) = exp K σiσi+1, (2) {σi=±1} i=1 where K = J/T . Summing over Ising spins on odd sites express this partition ˜ ˜ function in terms of ZN/2(K). -
Arxiv:1611.06019V1 [Math-Ph]
CORRELATION INEQUALITIES FOR CLASSICAL AND QUANTUM XY MODELS COSTANZA BENASSI, BENJAMIN LEES, AND DANIEL UELTSCHI Abstract. We review correlation inequalities of truncated functions for the classical and quantum XY models. A consequence is that the critical temperature of the XY model is necessarily smaller than that of the Ising model, in both the classical and quantum cases. We also discuss an explicit lower bound on the critical temperature of the quantum XY model. 1. Setting and results The goal of this survey is to recall some results of old that have been rather neglected in recent years. We restrict ourselves to the cases of classical and quantum XY models. Correlation inequalities are an invaluable tool that allows to obtain the monotonicity of spontaneous magnetisation, the existence of infinite volume limits, and comparisons be- tween the critical temperatures of various models. Many correlation inequalities have been established for the planar rotor (or classical XY) model, with interesting applications and consequences in the study of the phase diagram and the Gibbs states [4, 16, 19, 20, 17, 18, 8]. Some of these inequalities can also be proved for its quantum counterpart [10, 25, 22, 2]. Let Λ be a finite set of sites. The classical XY model (or planar rotor model) is a model for interacting spins on such a lattice. The configuration space of the system is defined as 1 ΩΛ = σx x∈Λ : σx S x Λ : each site hosts a unimodular vector lying on a unit circle.{{ It is} convenient∈ to represent∀ ∈ } the spins by means of angles, namely 1 σx = cos φx (1) 2 σx = sin φx (2) with φx [0, 2π]. -
Arxiv:1012.0653V3
Quantum phase transitions in transverse field spin models: From Statistical Physics to Quantum Information Amit Dutta∗ Department of Physics, Indian Institute of Technology, Kanpur 208 016, India Uma Divakaran† Theoretische Physik, Universit¨at des Saarlandes, 66041 Saarbr¨ucken, Germany Diptiman Sen‡ Centre for High Energy Physics, Indian Institute of Science, Bangalore 560 012, India Bikas K. Chakrabarti§ Theoretical Condensed Matter Physics Division and Centre for Applied Mathematics and Computational Science, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700 064, India Thomas F. Rosenbaum¶ Department of Physics and the James Franck Institute, University of Chicago, Chicago, IL 60637 Gabriel Aeppli∗∗ London Centre for Nanotechnology and Department of Physics and Astronomy, UCL, London, WC1E 6BT, United Kingdom We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of some paradigmatic models, namely, the spin-1/2 Ising and XY models in a transverse field in one and higher spatial dimensions. Beginning with a brief overview of quantum phase transitions, we introduce the model Hamiltonians and discuss the equivalence between the quantum phase transition in such a model and the finite temperature phase transition in a higher dimensional classical model. We then provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies when possible. We also discuss Kitaev models, and some other exactly solvable spin systems in this context. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in details. We discuss novel phenomena like Griffiths-McCoy singularities associated with quantum phase transitions of low- dimensional transverse Ising models with random interactions and transverse fields. -
Approaching the Kosterlitz-Thouless Transition for the Classical XY Model with Tensor Networks
PHYSICAL REVIEW E 100, 062136 (2019) Approaching the Kosterlitz-Thouless transition for the classical XY model with tensor networks Laurens Vanderstraeten,1,* Bram Vanhecke,1 Andreas M. Läuchli,2 and Frank Verstraete1 1Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium 2Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria (Received 6 September 2019; published 26 December 2019) We apply variational tensor-network methods for simulating the Kosterlitz-Thouless phase transition in the classical two-dimensional XY model. In particular, using uniform matrix product states (MPS) with non-Abelian O(2) symmetry, we compute the universal drop in the spin stiffness at the critical point. In the critical low- temperature regime, we focus on the MPS entanglement spectrum to characterize the Luttinger-liquid phase. In the high-temperature phase, we confirm the exponential divergence of the correlation length and estimate the critical temperature with high precision. Our MPS approach can be used to study generic two-dimensional phase transitions with continuous symmetries. DOI: 10.1103/PhysRevE.100.062136 I. INTRODUCTION ρ(T ), which discontinuously drops [7] from a finite value in the critical phase to zero in the gapped phase; from the In contemporary theoretical physics the interplay between renormalization analysis, the size of the drop is known to be symmetry and dimensionality is widely appreciated for giving given by rise to fascinating physical phenomena. Historically, one of 2T the crucial results in establishing this viewpoint was the phase lim ρ(T ) = c , (1) → − π transition in the two-dimensional classical XY model. This T Tc model is introduced by placing continuous angles {θ } on i but the value of T is not known exactly. -
Ising Model and XY Model
Ising model and XY model 1 Ising Model The Ising model is one of the simplest and most fundamental models of statistical mechanics. It can be used to describe such diverse phenomena as magnets; liquid/gas coexistence; alloys of two metals; and many others, even outside of physics. Each such system can be described by elementary variables si, usually called “spins” with 2 possible values si = ±1. The 2 values stand for, e.g., an elementary magnet pointing up or down; a piece of liquid or gas; an atom of metal A or B; etc.). These variables usually live on the sites i of some lattice. One associates an energy X E = − J si sj (1) neighbors i,j with each configuration of spins, where J is some constant. Each state of the system occurs with probability given by the Boltzmann factor 1 E p = exp(− ) , (2) Z kBT where T is temperature, kB the Boltzmann constant, and Z a normalization factor. Even though the Ising model is drastically simplified from realistic situations, it is able to describe, often quantitatively, the occurence of order at low temperatures and disorder at high temperatures, and especially the phase transition between those situation, in which spin correlations over very large length scales become essential. The Ising model can be solved exactly only in the simplest cases (in one spatial dimension, and on a two-dimensional square lattice). In most cases of practical interest, one has to resort to either analytical approximations like series expansions for high or for low temperature, or to numerical techniques like Markov Chain Monte Carlo simulations. -
Kosterlitz–Thouless Physics: a Review of Key Issues Accepted for Publication 7 October 2015 Published 28 January 2016
Reports on Progress in Physics REVIEW Related content - Berezinskii–Kosterlitz–Thouless transition Kosterlitz–Thouless physics: a review of key and two-dimensional melting V N Ryzhov, E E Tareyeva, Yu D Fomin et issues al. - Depinning and nonequilibrium dynamic phases of particle assemblies driven over To cite this article: J Michael Kosterlitz 2016 Rep. Prog. Phys. 79 026001 random and ordered substrates: a review C Reichhardt and C J Olson Reichhardt - Quantum phase transitions Matthias Vojta View the article online for updates and enhancements. Recent citations - Isolating long-wavelength fluctuation from structural relaxation in two-dimensional glass: cage-relative displacement Hayato Shiba et al - Disappearance of the Hexatic Phase in a Binary Mixture of Hard Disks John Russo and Nigel B. Wilding - Berezinskii—Kosterlitz—Thouless transition and two-dimensional melting Valentin N. Ryzhov et al This content was downloaded from IP address 128.227.24.141 on 25/02/2018 at 11:47 IOP Reports on Progress in Physics Reports on Progress in Physics Rep. Prog. Phys. Rep. Prog. Phys. 79 (2016) 026001 (59pp) doi:10.1088/0034-4885/79/2/026001 79 Review 2016 Kosterlitz–Thouless physics: a review of key © 2016 IOP Publishing Ltd issues RPPHAG J Michael Kosterlitz 026001 Department of Physics, Brown University, Providence, RI 02912, USA J M Kosterlitz E-mail: [email protected] Received 1 June 2015, revised 5 October 2015 Kosterlitz–Thouless physics: a review of key issues Accepted for publication 7 October 2015 Published 28 January 2016 Printed in the UK Abstract This article reviews, from a very personal point of view, the origins and the early work on ROP transitions driven by topological defects such as vortices in the two dimensional planar rotor model and in 4Helium films and dislocations and disclinations in 2D crystals. -
Arxiv:1101.4337V2 [Cond-Mat.Stat-Mech] 18 Mar 2011 Havior in Agreement with the KTHNY Predictions
Characterization of the Melting Transition in Two Dimensions at Vanishing External Pressure Using Molecular Dynamics Simulations Daniel Asenjo1;4, Fernando Lund1, Sim´onPoblete2, Rodrigo Soto1, and Marcos Sotomayor3 1Departamento de F´ısica and CIMAT, Facultad de Ciencias F´ısicas y Matem´aticas, Universidad de Chile, Santiago, Chile 2Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany 3 Howard Hughes Medical Institute and Neurobiology Department, Harvard Medical School, Boston, MA, USA 4Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, United Kingdom. A molecular dynamics study of a two dimensional system of particles interacting through a Lennard-Jones pairwise potential is performed at fixed temperature and vanishing external pressure. As the temperature is increased, a solid-to-liquid transition occurs. When the melting temperature Tc is approached from below, there is a proliferation of dislocation pairs and the elastic constant approaches the value predicted by the KTHNY theory. In addition, as Tc is approached from above, the relaxation time increases, consistent with an approach to criticality. However, simulations fail to produce a stable hexatic phase using systems with up to 90,000 particles. A significant jump in enthalpy at Tc is observed, consistent with either a first order or a continuous transition. The role of external pressure is discussed. I. INTRODUCTION On the numerical side, molecular dynamics and Monte- Carlo simulations19,20 of systems with a small number of particles (N) broadly detected a transition where the Melting of an infinite solid in two dimensions has number of dislocations proliferates, but failed to pro- been described as a process driven by a proliferation vide clear evidence for the nature of the observed tran- of thermally excited dislocation pairs in the Kosterlitz- sition.