DOCSLIB.ORG

RANDOM WALKS from STATISTICAL PHYSICS II Two Dimensons and Conformal Invariance Wald Lectures 2011 Joint Statistical Meetings

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
RANDOM WALKS from STATISTICAL PHYSICS II Two Dimensons and Conformal Invariance Wald Lectures 2011 Joint Statistical Meetings RANDOM WALKS FROM STATISTICAL PHYSICS II Two Dimensons and Conformal Invariance Wald Lectures 2011 Joint Statistical Meetings Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago August 3, 2011 1 / 32 CRITICAL PHENOMENA IN STATISTICAL PHYSICS I Study systems at or near parameters at which a phase transition occurs I Parameter ¯ = C=T where T = temperature I Large ¯ (low temperature) | long range correlation. I Small ¯ (high temperature) | short range correlation I Critical value ¯c at which sharp transition occurs I Belief: systems at criticality \in the scaling limit" exhibit fractal-like behavior (power-law correlations) with nontrivial critical exponents. I The exponents depend on dimension. 2 / 32 TWO DIMENSIONS I Belavin, Polyakov, Zamolodchikov (1984) | critical systems in two dimensions in the scaling limit exhibit some kind of \conformal invariance". I A number of theoretical physicists (Nienhuis, Cardy, Duplantier, Saleur, ...) made predictions about critical exponents using nonrigorous methods | conformal ¯eld theory and Coulomb gas techniques. I Exact rational values for critical exponents | predictions strongly supported by numerical simulations I While much of the mathematical framework of conformal ¯eld theory was precise and rigorous (or rigorizable), the nature of the limit and the relation of the ¯eld theory to the lattice models was not well understood. 3 / 32 SELF-AVOIDING WALK (SAW) I Model for polymer chains | polymers are formed by monomers that are attached randomly except for a self-avoidance constraint. 2 ! = [!0;:::;!n];!j 2 Z ; j!j = n j!j ¡ !j¡1j = 1; j = 1;:::; n !j 6= !k ; 0 · j < k · n: I Critical exponent º: a typical SAW has diameter about j!jº. I If no self-avoidance constraint º = 1=2; for 2-d SAW Flory predicted º = 3=4. 4 / 32 N z w 0 N Each SAW from z to w gets measure e¡¯j!j: Partition function X Z = Z(N; ¯) = e¡¯j!j: ¯ small | typical path is two-dimensional ¯ large | typical path is one-dimensional ¯c | typical path is (1=º)-dimensional 5 / 32 Choose ¯ = ¯c ; let N ! 1; divide by Z(N; ¯) and hope to get a probability measure on curves connecting boundary points of the square. z w 6 / 32 N z w 0 N Similarly, if we ¯x D ½ C, we can consider walks restricted to the domain D z w Predict that these probability measures are conformally invariant. 7 / 32 SIMPLE RANDOM WALK N z w 0 N I Simple random walk | no self-avoidance constraint. Criticality: each walk ! gets weight (1=4)j!j. I Scaling limit is Brownian motion which is conformally invariant (L¶evy). 8 / 32 LOOP-ERASED RANDOM WALK Start with simple random walks and erase loops in chronological order to get a path with no self-intersections. Limit should be a measure on paths with no self-intersections. z w 9 / 32 CRITICAL PERCOLATION Color vertices of the triangular lattice in the upper half plane black or white independently each with probability 1=2. Put a boundary condition of black on negative real axis and white on positive real axis. The percolation exploration process is the boundary between black and white. 10 / 32 CARDY'S FORMULA The probability of a black crossing at criticality (in the limit as lattice space goes to zero) was predicted to be a conformal invariant. 1 1 A1 D A 3 x 1 Cardy used conformal ¯eld theory to predict the value. it is most easily given for an equilateral triangle. 11 / 32 STRATEGY I Make precise the conformal invariance assumption and other properties expected of scaling limit. I Find all possible limits satisfying these assumptions. I For a given discrete process, identify which is the correct limit. I Prove the discrete converges to continuous. Nonrigorous approaches in mathematical physics using conformal ¯eld theory had some of the properties of this strategy. 12 / 32 ASSUMPTIONS ON SCALING LIMIT Probability measure ¹D (z; w) on curves connecting boundary points of a domain D. f z w f(z) f(w) I Conformal invariance: If f is a conformal transformation f ± ¹D (z; w) = ¹f (D)(f (z); f (w)): I For simply connected D, it su±ces to know ¹H(0; 1) (Riemann mapping theorem). 13 / 32 I Domain Markov property Given γ[0; t], the conditional distribution on γ[t; 1) is the same as ¹Hnγ(0;t](γ(t); 1): γ (t) I Satis¯ed on discrete level by SAW, LERW, percolation exploration, Ising exploration ... (but not by simple random walk) 14 / 32 LOEWNER DIFFERENTIAL EQUATION I Developed by Loewner in studying Bieberbach conjecture I Let D be a simply connected proper domain in C containing 0. Riemann mapping theorem says there is a unique conformal transformation f : D ! D with f (0) = 0; f 0(0) > 0. P I 1 j f (z) = j=1 aj z I Bieberbach conjecture: jaj j · j a1 I Su±ces to prove for slit domains D = C n γ[t; 1) where γ(s) ! 1 as s ! 1 I Di®erential equation studies how coe±cients change as function of t I Loewner proved ja3j · 3 a1. BC proved by de Branges (proof uses Loewner di® eq) 15 / 32 LOEWNER EQUATION IN UPPER HALF PLANE I Let γ : (0; 1) ! H be a simple curve with γ(0+) = 0 and γ(t) ! 1 as t ! 1. I gt : H n γ(0; t] ! H g γ(t) t 0 Ut I Can reparametrize if necessary so that 2t g (z) = z + + ¢ ¢ ¢ ; z ! 1 t z I gt satis¯es 2 @t gt (z) = ; g0(z) = z: gt (z) ¡ Ut Moreover, Ut = gt (γ(t)) is continuous. 16 / 32 (Schramm) Suppose γ is a random curve satisfying conformal invariance and Domain Markov property. Then Ut must be a random continuous curve satisfying I For every s < t, Ut ¡ Us is independent of Ur ; 0 · r · s and has the same distribution as Ut¡s . ¡1 I c Uc2t has the same distribution as Ut . p Therefore, Ut = · Bt where Bt is a standard (one-dimensional) Brownian motion. The (chordal) Schramm-Loewner evolution with parameter · p (SLE·) is the solution obtained by choosing Ut = · Bt . 17 / 32 (Rohde-Schramm) Solving the Loewner equation with a Brownian input gives a random curve. The qualitative behavior of the curves varies greatly with · I 0 < · · 4 | simple (non self intersecting) curve I 4 < · < 8 | self-intersections (but not crossing); not plane-¯lling I 8 · · < 1 | plane-¯lling (Be®ara) For · < 8, the Hausdor® dimension of the paths is · 1 + : 8 18 / 32 The fundamental tools for studying SLE are those of stochastic calculus (It^ointegral and formula, martingales, Girsanov transformation) For which · does SLE have double points? Equivalent to ask, for which · does SLE hit the real line? Let x > 0 and Xt = Xt (x) = gt (x) ¡ Ut . Then SLE hits [x; 1) if and only if Xt reaches zero in ¯nite time. Xt satis¯es 2 p dXt = dt + · dBt : Xt Bessel equation. Well known that Xt reaches zero if and only if · > 4. 19 / 32 iθ What is the probability that z = re lies to the left of the SLE· curve? I Function only of θ, f (θ) with f (0) = 0; f (¼) = 1. I Let Mt be the probability given γ(0; t]. Then Mt = f (arg(gt (z) ¡ Ut )) is a martingale, i.e., if s < t, E [Mt j γ(0; s]] = Ms : I It^o'sformula gives a di®erential equation for f f 00(θ) + 2(1 ¡ 2a) f 0(θ) cot θ = 0; a = 2=·: I Z θ c dx f (θ) = 2¡4a : 0 sin x 20 / 32 WHICH · FOR WHICH MODEL? How does the measure of a path change when we perturb the boundary? (· · 4) w D D'\D γ z nc o 1fγ ½ Dg exp ¤(D0; γ; D0 n D) 2 J(D0; γ; D0 n γ) is a conformal invariant given by the measure (using a certain Brownian loop measure) of loops in D0 that intersect both D0 n D and γ. 21 / 32 I For SAW, perturbing the domain does not change the measure. Expect c = 0. I For LERW, shrinking the domain loses some simple random walks whose loop-erasure is γ. Expect c < 0. I c is the central charge which is the parameter used in conformal ¯eld theory to distinguish models. p (3· ¡ 8) (6 ¡ ·) (13 ¡ c) § (13 ¡ c)2 ¡ 144 c = ;· = : 2· 3 Each c < 1 corresponds to two values ·; ·0 with ··0 = 16. c = 1 corresponds to the double root · = ·0 = 4. 22 / 32 BROWNIAN PATHS I The ¯rst major problem solved with SLE was the Brownian intersection exponents. One example goes back to a conjecture of Mandelbrot. Consider a \Brownian island" formed by taking taking a Brownian motion (random walk), conditioning to end at the same place it began, and ¯lling in the bounded holds. Mandelbrot noted that simulations of this coastline indicated that it should have dimension 4=3. I L. showed that the dimension could be calculated in terms of a particular value of the intersection exponents. I L, Schramm, and Werner showed how the locality property of SLE6 could be used to calculate this exponent and veri¯ed Mandelbrot's conjecture. 23 / 32 CRITICAL PERCOLATION I Smirnov proved that the scaling limit of critical percolation on the triangular lattice satis¯es Cardy's formula. Using this and the work of LSW he established that the scaling limit of percolation is SLE6. I · = 6 is the only value of · for which SLE satis¯es the locality property | something that would be expected of the scaling limit of percolation.
Load more

Recommended publications
  • 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].
    [Show full text]
  • Statistical Field Theory University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] –1– Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very di↵erent perspectives. They have titles like “Critical Phenomena”, “Phase Transitions”, “Renormalisation Group” or, less helpfully, “Advanced Statistical Mechanics”. Here are some that I particularly like Nigel Goldenfeld, Phase Transitions and the Renormalization Group • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. Mehran Kardar, Statistical Physics of Fields • The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. John Cardy, Scaling and Renormalisation in Statistical Physics • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent perspective than these lectures, with more focus on renormalisation in real space. Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter • Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html.
    [Show full text]
  • Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point
    entropy Article Thermodynamic Derivation of Scaling at the Liquid–Vapor Critical Point Juan Carlos Obeso-Jureidini , Daniela Olascoaga and Victor Romero-Rochín * Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México 01000, Mexico; [email protected] (J.C.O.-J.); [email protected] (D.O.) * Correspondence: romero@fisica.unam.mx; Tel.: +52-55-5622-5096 Abstract: With the use of thermodynamics and general equilibrium conditions only, we study the entropy of a fluid in the vicinity of the critical point of the liquid–vapor phase transition. By assuming a general form for the coexistence curve in the vicinity of the critical point, we show that the functional dependence of the entropy as a function of energy and particle densities necessarily obeys the scaling form hypothesized by Widom. Our analysis allows for a discussion of the properties of the corresponding scaling function, with the interesting prediction that the critical isotherm has the same functional dependence, between the energy and the number of particles densities, as the coexistence curve. In addition to the derivation of the expected equalities of the critical exponents, the conditions that lead to scaling also imply that, while the specific heat at constant volume can diverge at the critical point, the isothermal compressibility must do so. Keywords: liquid–vapor critical point; scaling hypothesis; critical exponents Citation: Obeso-Jureidini, J.C.; 1. Introduction Olascoaga, D.; Romero-Rochín, V. The full thermodynamic description of critical phenomena in the liquid–vapor phase Thermodynamic Derivation of transition of pure substances has remained as a theoretical challenge for a long time [1].
    [Show full text]
  • Indicators of Conformal Field Theory: Entanglement Entropy and Multiple
    Indicators of Conformal Field Theory: entanglement entropy and multiple-point correlators Pranay Patil, Ying Tang, Emanuel Katz, and Anders W. Sandvik Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA The entanglement entropy (EE) of quantum systems is often used as a test of low-energy de- scriptions by conformal field theory (CFT). Here we point out that this is not a reliable indicator, as the EE often shows the same behavior even when a CFT description is not correct (as long as the system is asymptotically scale-invariant). We use constraints on the scaling dimension given by the CFT with SU(2) symmetry to provide alternative tests with two- and four-point correlation functions, showing examples for quantum spin models in 1+1 dimensions. In the case of a critical amplitude-product state expressed in the valence-bond basis (where the amplitudes decay as a power law of the bond length and the wave function is the product of all bond amplitudes), we show that even though the EE exhibits the expected CFT behavior, there is no CFT description of this state. We provide numerical tests of the behavior predicted by CFT for the correlation functions in the critical transverse-field Ising chain and the J-Q spin chain, where the conformal structure is well understood. That behavior is not reproduced in the amplitude-product state. I. INTRODUCTION tuned to criticality for particular weightings of the sin- glet covers which contribute to it, and one may then ask whether a state similar to the ground state of the Heisen- The emergence of conformal invariance at a quantum berg Hamiltonian could be constructed this way.
    [Show full text]
  • Strong Violation of Critical Phenomena Universality: Wang-Landau Study of the Two-Dimensional Blume-Capel Model Under Bond Randomness
    Strong violation of critical phenomena universality: Wang-Landau study of the two-dimensional Blume-Capel model under bond randomness The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Malakis, A. et al. “Strong violation of critical phenomena universality: Wang-Landau study of the two-dimensional Blume- Capel model under bond randomness.” Physical Review E 79.1 (2009): 011125. © 2009 The American Physical Society. As Published http://dx.doi.org/10.1103/PhysRevE.79.011125 Publisher American Physical Society Version Final published version Citable link http://hdl.handle.net/1721.1/51744 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. PHYSICAL REVIEW E 79, 011125 ͑2009͒ Strong violation of critical phenomena universality: Wang-Landau study of the two-dimensional Blume-Capel model under bond randomness A. Malakis,1 A. Nihat Berker,2,3,4 I. A. Hadjiagapiou,1 and N. G. Fytas1 1Department of Physics, Section of Solid State Physics, University of Athens, Panepistimiopolis, GR 15784 Zografos, Athens, Greece 2College of Sciences and Arts, Koç University, Sarıyer 34450, Istanbul, Turkey 3Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 4Feza Gürsey Research Institute, TÜBİTAK-Bosphorus University, Çengelköy 34684, Istanbul, Turkey ͑Received 24 September 2008; published 27 January 2009͒ We study the pure and random-bond versions of the square lattice ferromagnetic Blume-Capel model, in both the first-order and second-order phase transition regimes of the pure model.
    [Show full text]
  • Critical Behavior and Crossover Effects in the Properties of Binary and Ternary Mixtures and Verification of the Dynamic Scaling Conception
    Critical Behavior and Crossover Effects in the Properties of Binary and Ternary Mixtures and Verification of the Dynamic Scaling Conception Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultaten¨ der Georg-August-Universitat¨ zu Gottingen¨ vorgelegt von Dipl. Phys. Ireneusz Iwanowski aus Sorau (Zary)˙ Gottingen¨ 2007 D7 Referent: Prof. Dr. Werner Lauterborn Korreferent: Prof. Dr. Christoph Schmidt Tag der mundlichen¨ Prufung:¨ 29.11.07 ...one of the strongest motives that lead men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of one’s own ever-shifting desires. A finely tempered nature longs to escape from the personal life into the world of objective perception and thought. ALBERT EINSTEIN (1879-1955) Dla rodzicow´ Anny i Kazimierza Iwanowskich Contents 1 Introduction 1 2 Thermodynamics of Critical Phenomena 5 2.1 Phase transitions .............................. 5 2.2 Critical fluctuations ............................. 7 2.3 Critical phenomena ............................. 8 2.3.1 Critical exponents ......................... 8 2.3.2 Static scaling hypothesis ...................... 9 2.3.3 Dynamic scaling hypothesis and critical slowing down ...... 10 2.3.4 Renormalization of critical exponents ............... 12 2.3.5 Critical opalescence and equal volume criterions ......... 12 2.4 Phase diagrams ............................... 12 2.4.1 Phase diagrams for binary liquids at constant pressure ...... 12 2.4.2 Phase diagrams of ternary mixtures at constant pressure ..... 13 3 Experimental Methods 19 3.1 General aspects ............................... 19 3.2 Dynamic light scattering .......................... 20 3.2.1 Electromagnetic scattering theory ................. 20 3.2.2 Spectrum of scattered field - hydrodynamic considerations .... 22 3.2.3 Data evaluation of dynamic light scattering ...........
    [Show full text]
  • Dynamic Critical Behavior in Models of Ferromagnetic Gadolinium A
    Dynamic Critical Behavior in Models of Ferromagnetic Gadolinium A. K. Murtazaev and V. A. Mutailamov* Institute of Physics, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, 367003 Russia * e-mail: [email protected] A numerical technique combining Monte Carlo and molecular dynamics simulations is used for the first time to examine the complex critical dynamics of models of ferromagnetic gadolinium in which both strong exchange interactions and relativistic effects of several different types are taken into account. A finitesize scaling technique is used to calculate the corresponding dynamic critical exponents. The role played by isotropic dipole–dipole interaction in the critical behavior of gadolinium is evaluated. The results obtained provide an explanation for the anomalous dynamic critical behavior of gadolinium. 1. INTRODUCTION Analysis of dynamic critical properties is a major problem in statistical physics and theory of phase transitions [1–3]. The substantial progress achieved to date in this field became possible mainly through theoretical and experimental studies. Nevertheless, the development of a rigorous and consistent theory of dynamic critical phenomena based on microscopic Hamiltonians remains a challenging problem in the modern theory of phase transitions and critical phenomena [1, 4]. Current knowledge of critical dynamics has mostly been gained in mode-coupling and dynamic scaling theories [1–3], which were developed independently on the basis of essentially different ideas. However, many predictions of these theories are in good agreement. In [3], universality classes of dynamic critical behavior were proposed in the framework of the dynamic scaling theory. It was shown that the dynamic universality classes depend not only on the space dimension d, the number n of order-parameter components, the interaction range, and the symmetry of the Hamiltonian, but also on the energy and order-parameter conservation laws [3].
    [Show full text]
  • Field Theory Techniques on Spin Systems Physics 230A, Spring 2007, Hitoshi Murayama
    Field Theory Techniques on Spin Systems Physics 230A, Spring 2007, Hitoshi Murayama 1 Introduction Solid state systems are based on crystals that have discretized positions for electronic degrees of freedom. In other words, the atomic distance provides a short-distance cutoff to the system below which you have to change the description of the system to that of nuclei and electrons instead of atoms. The intrinsic inability of the atomic description below the Bohr radius is not a problem, as long as we are aware of its limitation. Any physical theories have their limited applicability we should be aware of and we can live with. Despite the intrinsic distance scale in the problem, the atomic scale, many solid state systems exhibit interesting phenomena at distances much larger than atoms. This is particularly true when the system is close to a second- order phase transition. Imagine a ferromagnet described by the Heisenberg model, J X H = − 2 ~si · ~sj: (1) ~ hi;ji Here and below, the sum over hi; ji refers to nearest neighbors on the lattice. The parameter J has the dimension of energy, describing the strength of the spin-spin interaction. The origin of the interaction is actually the Coulomb replusion between electrons which prefers the orbital wave function to be anti- symmetric than symmetric to separate their positions, requiring symmetric spin wave function to be consistent with the overall anti-symmetry of fermion wave functions. Therefore J is expected to be of the order of electronic energies, i.e., electronvolt. At a high temperature, all spins are random due to their thermal fluctu- ations.
    [Show full text]
  • Analyzing the Two Dimensional Ising Model with Conformal Field Theory
    Analyzing the two dimensional Ising model with conformal field theory Paolo Molignini Theoretisches Proseminar FS2013, Departement Physik, ETH Z¨urich [email protected] March 10, 2013 Abstract The purpose of this report is to explain the topics discussed during my talk as part of the the- oretical proseminar titled \Conformal field theory and string theory", which was organized by Prof. Matthias Gaberdiel at the Federal Institute of Technology (ETH) in Z¨urich during the spring semester 2013. In the following analysis the mathematics of conformal field theory and their ap- plication to the statistical mechanical Ising model of ferromagnetism is presented. This approach allows to highlight the connections between phenomena at criticality and quantum field theory, in particular the model of a free fermion. Starting from a classical picture, the Ising model is quantized and a series of clever transformations are applied to obtain a system whose mathematical struc- ture can be linked to that of quantum field theory. The symmetries of the derived Hamiltonian and in particular conformal invariance can then be exploited to extract the values of the critical exponents of physical quantities - such as the magnetization - matching the empirical values. In this way conformal field theory allows for an effective connection between theoretical predictions and phenomenological measurements. Contents 0 Outline 3 1 Classical statistical mechanics 4 1.1 The partition function in classical statistics . .4 1.2 The partition function in quantum statistics . .5 1.3 The Ising model . .5 1.4 Generalizations . .8 1.5 Critical phenomena and phase transitions . .9 1.6 Scaling and universality .
    [Show full text]
  • Chapter 1 Critical Phenomena
    Chapter 1 Critical Phenomena The aim of this introductory chapter is to introduce the concept of a phase transition and to motivate the subject of statistical field theory. Here we introduce the concept of universality as applied to critical phenomena and define some of the notation used throughout these lectures. 1.1 Collective Phenomena: from Particles to Fields It is rare in physics to find examples of interacting many-particle systems which admit to a full and accessible microscopic description. More useful is a hydrodynamic description of the collective long-wavelength behaviour which surrenders information at the micro- scopic scale. A familiar example is the Navier-Stokes equation of fluid dynamics. The averaged variables appropriate to these length and time scales are no longer the discrete set of particle degrees of freedom but rather slowly varying continuous fields describing the collective motion of a macroscopic set of particles. Familiar examples include magnetic spin-waves, and vibrational or phonon modes of an atomic lattice. The most striking consequence of interactions among particles is the appearance of new phases of matter whose collective behaviour bears little resemblance to that of a few particles. How do the particles then transform from one macroscopic state to another? Formally, all macroscopic properties can be deduced from the free energy or the partition function. However, since phase transitions typically involve dramatic changes in response functions they must correspond to singularities in the free energy. Since the canonical partition function of a finite collection of particles is always analytic, phase transitions can only be associated with infinitely many particles, i.e.
    [Show full text]
  • Critical Phenomena at the Antiferromagnetic Phase Transition
    Critical Phenomena at the Antiferromagnetic Phase Transition of Azurite P. T. Cong, B. Wolf, R. S. Manna, A. Brühl, S. Köhler and M. Lang Physikalisches Institut, J.W. Goethe-Universität Frankfurt, SFB/TR49, Frankfurt(M), Germany We report on high-resolution acoustic, specific-heat and thermal expansion measurements in the vicinity of the antiferromagnetic phase transition at TN = 1.88 K on a high-quality single crystal of the natural mineral azurite. A detailed investigation of the critical contribution to the various quantities at TN is presented. The set of critical exponents and amplitude ratios of the singular contributions above and below the transition indicate that the system can be reasonably well described by a three-dimensional Heisenberg antiferromagnet. PACS number: 75.40.Cx, M 72.55.+s,71.27.+a Keywords: low-dimensional spin systems, critical behavior Email: [email protected] Fax: +49-69-798-47250 1 I. INTRODUCTION The natural mineral azurite, Cu3(CO3)2(OH)2, has been considered as a model substance for the 1D diamond chain, where anomalous magnetic properties result from the interplay of strong quantum fluctuations, low dimensionality and frustrating interactions [1, 2]. In a recent computational study a reasonable description was given for not too small energy scales (i.e., T > 2 K and magnetic fields across the plateau region B ≤ 40 T), covering a broad range of experimental results and placing azurite in the highly-frustrated parameter regime of the 1D distorted diamond chain [3]. However, early studies imply 3D long-range antiferromagnetic (AFM) order at TN ~ 1.88 K, indicating that parameters such as inter-chain coupling and magnetic anisotropy play an important role and affect the magnetic properties of azurite in the low-temperature region [4].
    [Show full text]
  • Chapter 7 Phase Transitions
    Chapter 7 Phase transitions In the last few chapters we have been preoccupied with various methods of calculating par- tition functions, both for ideal and for interacting systems. In principle, once we know a partition function we can calculate any thermodynamic properties that we might be inter- ested in. As we discussed a particular perturbative approach to calculating the partition function for a dilute gas { and then had to patch it up via Maxwell's construction in order to describe both a gas and a liquid state { you may be wondering: if it is even clear that a single partition function can actually describe multiple phases? In fact, perhaps you have an even deeper concern. Phase transitions are characterized by discontinuities in the free energy or its derivatives, and the free energy can be written as Z = e−βF = Tr e−βH ; where I'm using the trace notation to indicated a schematic sum over all of the degrees of freedom in the Hamiltonian, H. But most H that we study are non-singular functions of their degrees of freedom, so the partition function is just a sum of a large number of terms, each of which is exp (some analytic function). How could such a sum be non-analytic, and can a single partition function even describe a phase transition (let alone multiple phases)? A deep understanding of the answer to these questions, together with an understanding of how near critical points whole classes of disparate systems behave in the same way, has only been understood for, say, the last 50 years1.
    [Show full text]