Statistical Field Theory University of Cambridge Part III Mathematical Tripos
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Arxiv:1910.05796V2 [Math-Ph] 26 Oct 2020 5
Towards a conformal field theory for Schramm-Loewner evolutions Eveliina Peltola [email protected] Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relation- ship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies. 1. Introduction 2 2. Schramm-Loewner evolution in statistical mechanics and conformal field theory4 A. Schramm-Loewner evolution (SLE) 5 B. SLE in critical models – the Ising model6 C. Conformal field theory (CFT) 7 D. Martingale observables for interfaces 9 3. Multiple SLE partition functions and applications 12 A. Multiple SLEs and their partition functions 12 B. Definition of the multiple SLE partition functions 13 C. Properties of the multiple SLE partition functions 16 D. Relation to Schramm-Loewner evolutions and critical models 18 4. Fusion and operator product expansion 20 A. Fusion and operator product expansion in conformal field theory 21 B. Fusion: analytic approach 22 C. Fusion: systematic algebraic approach 23 D. Operator product expansion for multiple SLE partition functions 26 arXiv:1910.05796v2 [math-ph] 26 Oct 2020 5. From OPE structure to products of random distributions? 27 A. Conformal bootstrap 27 B. Random tempered distributions 28 C. Bootstrap construction? 29 Appendix A. Representation theory of the Virasoro algebra 31 Appendix B. Probabilistic construction of the pure partition functions 32 Appendix C. -
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).......................... -
The Ising Model
The Ising Model Today we will switch topics and discuss one of the most studied models in statistical physics the Ising Model • Some applications: – Magnetism (the original application) – Liquid-gas transition – Binary alloys (can be generalized to multiple components) • Onsager solved the 2D square lattice (1D is easy!) • Used to develop renormalization group theory of phase transitions in 1970’s. • Critical slowing down and “cluster methods”. Figures from Landau and Binder (LB), MC Simulations in Statistical Physics, 2000. Atomic Scale Simulation 1 The Model • Consider a lattice with L2 sites and their connectivity (e.g. a square lattice). • Each lattice site has a single spin variable: si = ±1. • With magnetic field h, the energy is: N −β H H = −∑ Jijsis j − ∑ hisi and Z = ∑ e ( i, j ) i=1 • J is the nearest neighbors (i,j) coupling: – J > 0 ferromagnetic. – J < 0 antiferromagnetic. • Picture of spins at the critical temperature Tc. (Note that connected (percolated) clusters.) Atomic Scale Simulation 2 Mapping liquid-gas to Ising • For liquid-gas transition let n(r) be the density at lattice site r and have two values n(r)=(0,1). E = ∑ vijnin j + µ∑ni (i, j) i • Let’s map this into the Ising model spin variables: 1 s = 2n − 1 or n = s + 1 2 ( ) v v + µ H s s ( ) s c = ∑ i j + ∑ i + 4 (i, j) 2 i J = −v / 4 h = −(v + µ) / 2 1 1 1 M = s n = n = M + 1 N ∑ i N ∑ i 2 ( ) i i Atomic Scale Simulation 3 JAVA Ising applet http://physics.weber.edu/schroeder/software/demos/IsingModel.html Dynamically runs using heat bath algorithm. -
Scale Without Conformal Invariance in Membrane Theory
Scale without conformal invariance in membrane theory Achille Mauria,∗, Mikhail I. Katsnelsona aRadboud University, Institute for Molecules and Materials, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Abstract We investigate the relation between dilatation and conformal symmetries in the statistical mechanics of flex- ible crystalline membranes. We analyze, in particular, a well-known model which describes the fluctuations of a continuum elastic medium embedded in a higher-dimensional space. In this theory, the renormalization group flow connects a non-interacting ultraviolet fixed point, where the theory is controlled by linear elas- ticity, to an interacting infrared fixed point. By studying the structure of correlation functions and of the energy-momentum tensor, we show that, in the infrared, the theory is only scale-invariant: the dilatation symmetry is not enhanced to full conformal invariance. The model is shown to present a non-vanishing virial current which, despite being non-conserved, maintains a scaling dimension exactly equal to D − 1, even in presence of interactions. We attribute the absence of anomalous dimensions to the symmetries of the model under translations and rotations in the embedding space, which are realized as shifts of phonon fields, and which protect the renormalization of several non-invariant operators. We also note that closure of a sym- metry algebra with both shift symmetries and conformal invariance would require, in the hypothesis that phonons transform as primary fields, the presence of new shift symmetries which are not expected to hold on physical grounds. We then consider an alternative model, involving only scalar fields, which describes effective phonon-mediated interactions between local Gaussian curvatures. -
Qualitative Picture of Scaling in the Entropy Formalism
Entropy 2008, 10, 224-239; DOI: 10.3390/e10030224 OPEN ACCESS entropy ISSN 1099-4300 www.mdpi.org/entropy Article Qualitative Picture of Scaling in the Entropy Formalism Hans Behringer Faculty of Physics, University of Bielefeld, D-33615 Bielefeld, Germany E-mail: [email protected] Received: 30 June 2008; in revised form: 27 August 2008 / Accepted: 2 September 2008 / Published: 5 September 2008 Abstract: The properties of an infinite system at a continuous phase transition are charac- terised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems are deduced starting from the well-established scaling re- lations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling are suppressed in the microcanonical ensemble compared to the corresponding corrections in the canonical ensemble. Keywords: Critical phenomena, microcanonical entropy, scaling relations, Gaussian and spherical model 1. Introduction In a mathematically idealised way continuous phase transitions are described in infinite systems al- though physical systems in nature or in a computer experiment are finite. The singular behaviour of physical quantities such as the response functions is related to scaling properties of the thermodynamic potential which is used for the formulation [1, 2]. Normally the Gibbs free energy is chosen for the description although other potentials also contain the full information about an equilibrium phase transi- tion. The Hankey-Stanley theorem relates the scaling properties of the Gibbs free energy to the scaling properties of any other energy-like potential that is obtained from the Gibbs free energy by a Legendre transformation [3]. -
Arxiv:0807.3472V1 [Cond-Mat.Stat-Mech] 22 Jul 2008 Conformal Field Theory and Statistical Mechanics∗
Conformal Field Theory and Statistical Mechanics∗ John Cardy July 2008 arXiv:0807.3472v1 [cond-mat.stat-mech] 22 Jul 2008 ∗Lectures given at the Summer School on Exact methods in low-dimensional statistical physics and quantum computing, les Houches, July 2008. 1 CFT and Statistical Mechanics 2 Contents 1 Introduction 3 2 Scale invariance and conformal invariance in critical behaviour 3 2.1 Scaleinvariance ................................. 3 2.2 Conformalmappingsingeneral . .. 6 3 The role of the stress tensor 7 3.0.1 An example - free (gaussian) scalar field . ... 8 3.1 ConformalWardidentity. 9 3.2 An application - entanglement entropy . ..... 12 4 Radial quantisation and the Virasoro algebra 14 4.1 Radialquantisation.............................. 14 4.2 TheVirasoroalgebra .............................. 15 4.3 NullstatesandtheKacformula . .. 16 4.4 Fusionrules ................................... 17 5 CFT on the cylinder and torus 18 5.1 CFTonthecylinder .............................. 18 5.2 Modularinvarianceonthetorus . ... 20 5.2.1 Modular invariance for the minimal models . .... 22 6 Height models, loop models and Coulomb gas methods 24 6.1 Heightmodelsandloopmodels . 24 6.2 Coulombgasmethods ............................. 27 6.3 Identification with minimal models . .... 28 7 Boundary conformal field theory 29 7.1 Conformal boundary conditions and Ward identities . ......... 29 7.2 CFT on the annulus and classification of boundary states . ......... 30 7.2.1 Example................................. 34 CFT and Statistical Mechanics 3 7.3 -
Arxiv:1511.03031V2
submitted to acta physica slovaca 1– 133 A BRIEF ACCOUNT OF THE ISING AND ISING-LIKE MODELS: MEAN-FIELD, EFFECTIVE-FIELD AND EXACT RESULTS Jozef Streckaˇ 1, Michal Jasˇcurˇ 2 Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Saf´arikˇ University, Park Angelinum 9, 040 01 Koˇsice, Slovakia The present article provides a tutorial review on how to treat the Ising and Ising-like models within the mean-field, effective-field and exact methods. The mean-field approach is illus- trated on four particular examples of the lattice-statistical models: the spin-1/2 Ising model in a longitudinal field, the spin-1 Blume-Capel model in a longitudinal field, the mixed-spin Ising model in a longitudinal field and the spin-S Ising model in a transverse field. The mean- field solutions of the spin-1 Blume-Capel model and the mixed-spin Ising model demonstrate a change of continuous phase transitions to discontinuous ones at a tricritical point. A con- tinuous quantum phase transition of the spin-S Ising model driven by a transverse magnetic field is also explored within the mean-field method. The effective-field theory is elaborated within a single- and two-spin cluster approach in order to demonstrate an efficiency of this ap- proximate method, which affords superior approximate results with respect to the mean-field results. The long-standing problem of this method concerned with a self-consistent deter- mination of the free energy is also addressed in detail. More specifically, the effective-field theory is adapted for the spin-1/2 Ising model in a longitudinal field, the spin-S Blume-Capel model in a longitudinal field and the spin-1/2 Ising model in a transverse field. -
The Quantized Fermionic String
Class 2: The quantized fermionic string Canonical quantization Light-cone quantization Spectrum of the fermionic string, GSO projection . The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 . One new feature is the existence of two sectors: bosonic and fermionic, which have to be studied separately. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. Summary The fermionic string is quantized analogously to the bosonic string, although now we'll find the critical dimension is 10 The super-Virasoro constraints, which allow to eliminate the ghosts, are implemented and analyzed in essentially the same way as in the bosonic string. To remove the tachyon one has to perform the so-called GSO projection, which guarantees space-time supersymmetry of the ten-dimensional theory. There are two possible space-time supersymmetric GSO projections which result in the Type IIA and Type IIB superstring. -
Euclidean Field Theory
February 2, 2011 Euclidean Field Theory Kasper Peeters & Marija Zamaklar Lecture notes for the M.Sc. in Elementary Particle Theory at Durham University. Copyright c 2009-2011 Kasper Peeters & Marija Zamaklar Department of Mathematical Sciences University of Durham South Road Durham DH1 3LE United Kingdom http://maths.dur.ac.uk/users/kasper.peeters/eft.html [email protected] [email protected] 1 Introducing Euclidean Field Theory5 1.1 Executive summary and recommended literature............5 1.2 Path integrals and partition sums.....................7 1.3 Finite temperature quantum field theory.................8 1.4 Phase transitions and critical phenomena................ 10 2 Discrete models 15 2.1 Ising models................................. 15 2.1.1 One-dimensional Ising model................... 15 2.1.2 Two-dimensional Ising model................... 18 2.1.3 Kramers-Wannier duality and dual lattices........... 19 2.2 Kosterlitz-Thouless model......................... 21 3 Effective descriptions 25 3.1 Characterising phase transitions..................... 25 3.2 Mean field theory.............................. 27 3.3 Landau-Ginzburg.............................. 28 4 Universality and renormalisation 31 4.1 Kadanoff block scaling........................... 31 4.2 Renormalisation group flow........................ 33 4.2.1 Scaling near the critical point................... 33 4.2.2 Critical surface and universality................. 34 4.3 Momentum space flow and quantum field theory........... 35 5 Lattice gauge theory 37 5.1 Lattice action................................. 37 5.2 Wilson loops................................. 38 5.3 Strong coupling expansion and confinement.............. 39 5.4 Monopole condensation.......................... 39 3 4 1 Introducing Euclidean Field Theory 1.1. Executive summary and recommended literature This course is all about the close relation between two subjects which at first sight have very little to do with each other: quantum field theory and the theory of criti- cal statistical phenomena. -
Lecture 8: the Ising Model Introduction
Lecture 8: The Ising model Introduction I Up to now: Toy systems with interesting properties (random walkers, cluster growth, percolation) I Common to them: No interactions I Add interactions now, with significant role I Immediate consequence: much richer structure in model, in particular: phase transitions I Simulate interactions with RNG (Monte Carlo method) I Include the impact of temperature: ideas from thermodynamics and statistical mechanics important I Simple system as example: coupled spins (see below), will use the canonical ensemble for its description The Ising model I A very interesting model for understanding some properties of magnetic materials, especially the phase transition ferromagnetic ! paramagnetic I Intrinsically, magnetism is a quantum effect, triggered by the spins of particles aligning with each other I Ising model a superb toy model to understand this dynamics I Has been invented in the 1920's by E.Ising I Ever since treated as a first, paradigmatic model The model (in 2 dimensions) I Consider a square lattice with spins at each lattice site I Spins can have two values: si = ±1 I Take into account only nearest neighbour interactions (good approximation, dipole strength falls off as 1=r 3) I Energy of the system: P E = −J si sj hiji I Here: exchange constant J > 0 (for ferromagnets), and hiji denotes pairs of nearest neighbours. I (Micro-)states α characterised by the configuration of each spin, the more aligned the spins in a state α, the smaller the respective energy Eα. I Emergence of spontaneous magnetisation (without external field): sufficiently many spins parallel Adding temperature I Without temperature T : Story over. -
Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography
Recent advances in percolation theory and its applications Abbas Ali Saberi aDepartment of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran Abstract Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model. Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, di- mensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive per- colation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. In the past decade, after the invention of stochastic L¨ownerevolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals. -
Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales Michel Bauer a, Denis Bernard a 1, Kalle Kyt¨ol¨a b [email protected]; [email protected]; [email protected] a Service de Physique Th´eorique de Saclay CEA/DSM/SPhT, Unit´ede recherche associ´ee au CNRS CEA-Saclay, 91191 Gif-sur-Yvette, France. b Department of Mathematics, P.O. Box 68 FIN-00014 University of Helsinki, Finland. Abstract A statistical mechanics argument relating partition functions to martingales is used to get a condition under which random geometric processes can describe interfaces in 2d statistical mechanics at critical- ity. Requiring multiple SLEs to satisfy this condition leads to some natural processes, which we study in this note. We give examples of such multiple SLEs and discuss how a choice of conformal block is related to geometric configuration of the interfaces and what is the physical meaning of mixed conformal blocks. We illustrate the general arXiv:math-ph/0503024v2 21 Mar 2005 ideas on concrete computations, with applications to percolation and the Ising model. 1Member of C.N.R.S 1 Contents 1 Introduction 3 2 Basics of Schramm-L¨owner evolutions: Chordal SLE 6 3 A proposal for multiple SLEs 7 3.1 Thebasicequations ....................... 7 3.2 Archprobabilities......................... 8 4 First comments 10 4.1 Statistical mechanics interpretation . 10 4.2 SLEasaspecialcaseof2SLE. 10 4.3 Makingsense ........................... 12 4.4 A few martingales for nSLEs .................. 13 4.5 Classicallimit........................... 14 4.6 Relations with other work . 15 5 CFT background 16 6 Martingales from statistical mechanics 19 6.1 Tautological martingales .