DOCSLIB.ORG

Non-Equilibrium Phase Transitions an Introduction

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Non-Equilibrium Phase Transitions an Introduction Directed Percolation Other Universality Classes Non-equilibrium phase transitions An Introduction Lecture III Haye Hinrichsen University of Würzburg, Germany March 2006 Directed Percolation Other Universality Classes Third Lecture: Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes Phenomenological Scaling Theory – Scale Invariance Scaling hypothesis: In the scaling regime the large-scale properties of absorbing phase transitions are invariant under scale transformations (zoom in – zoom out) ∆ → λ ∆ (∆ = p − pc) ρ → λβρ 0 P → λβ P −ν⊥ ξ⊥ → λ ξ⊥ −νk ξk → λ ξk 0 with the four exponents β, β , ν⊥, νk. This is called simple scaling (opposed to multiscaling). Directed Percolation Other Universality Classes Using Scale Invariance If you have an expression... multiply the distance from criticality p − pc by λ. multiply all quantities that measure local activity by λβ 0 multiply all quantities that activate sites locally by λβ multiply all lengths by λ−ν⊥ multiply all times by λ−νk ...then this expression should be asymptotically invariant. Directed Percolation Other Universality Classes Using Scale Invariance 1st Example: Decay of the density at criticality: −α β −νk −α ρ ∼ t ⇒ λ ρ ∼ (λ t) ⇒ α = β/νk The same more formally: ρ(t) = λ−βρ(λ−νk t) Set λ = t1/νk . ⇒ ρ(t) = t−β/ν⊥ ρ(1) Directed Percolation Other Universality Classes Using Scale Invariance 2nd Example: Decay of the density for ∆ 6= 0: ρ(∆, t) = λ−βρ(λ∆, λ−νk t) Set again λ = t1/νk . ⇒ ρ(∆, t) = t−β/ν⊥ ρ(∆t1/νk , 1) ρ(∆, t) = t−β/ν⊥ f (∆t1/νk ) This is a scaling form, f (ξ) is called scaling function. Directed Percolation Other Universality Classes Using Scale Invariance – Scaling function ρ(∆, t) = t−β/ν⊥ f (∆t1/νk ) The scaling function f (ξ) has certain asymptotic properties: f (ξ) → const for ξ → 0 f (ξ) ∼ ξβ for ξ → ∞, ∆ > 0 f (ξ) → 0 for ξ → ∞, ∆ < 0 Check scaling and visualize f (ξ) by a data collapse, plotting ρ(∆, t)tβ/ν⊥ versus ∆t1/νk . Directed Percolation Other Universality Classes Using Scale Invariance – Data Collapse ρ(∆, t) = t−β/ν⊥ f (∆t1/νk ) 0 10 0 10 -1 10 -1 10 α -2 ρ(t) 10 ρ(t) t -2 10 -3 10 -3 10 -4 10 0 1 2 3 4 -4 -3 -2 -1 0 1 10 10 10 10 10 10 10 10 10 10 10 ν t t τ || Directed Percolation Other Universality Classes Using Scale Invariance 3rd Example: Two-point correlation function: c(r, τ, ∆) = hs(r1, t1)s(r2, t2)i Here r = |r2 − r1| and τ = t2 − t1 c(λ−ν⊥ r, λ−νk τ, λ∆) ' λ−2βc(r, τ, ∆) Setting λ = τ 1/νk we obtain: c(r, ∆t, ∆) = τ −2β/νk g(r/τ ν⊥/νk , ∆τ 1/νk ) , Directed Percolation Other Universality Classes Universality of Scaling Functions The form of scaling functions (apart from metric factors) is universal, e.g., the same in all models for DP. Scaling functions are as universal as critical exponents. Like critical exponents scaling functions cannot always be expressed exactly in a closed form. A universality class is characterized by values of the critical exponents and specific scaling functions. Directed Percolation Other Universality Classes Using Scale Invariance There is much more about scaling: Empty-interval probability distribution External field and fluctuations Finite-size scaling Pair connectedness function Spreading from a seed Early-time regime / critical initial slip ...see lecture notes. Directed Percolation Other Universality Classes Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes Phenomenological Langevin Equation Let ρ(x, t) be a mesoscopic density of active sites. The Langevin equation for DP reads: 2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t) Here ξ(x, t) is a density-dependent Gaussian noise with the correlations hξ(x, t)i = 0 , hξ(x, t)ξ(x0, t0)i = Γ ρ(x, t) δd (x − x0) δ(t − t0) The amplitude of ξ(x, t) is proportional to pρ(x, t), Directed Percolation Other Universality Classes DP Langevin Equation: Form of the Noise Why ξ(x, t) ∝ pρ(x, t)? The noise describes local fluctuations of the coarse-grained density ρ(x, t) Directed Percolation Other Universality Classes DP Langevin Equation: Form of the Noise Each active site creates offspring (+1) or dies (-1), hence generates bounded noise. Central limit theorem: Total noise amplitude is proportional to the square-root of the number of noise sources. ⇒ amplitude[ξ(x, t)] ∝ pρ(x, t) Directed Percolation Other Universality Classes Langevin Equation – Static Mean Field Approximation 2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t) Stationary state: ∂t ρ(x, t) = 0 0 absorbing state ⇒ ρstat = a/b active state The parameter a plays the role of ∆ = p − pc. The critical point is ac = 0. The density exponent is βMF = 1. Directed Percolation Other Universality Classes Langevin Equation – Scale Invariance Apply scale transformation ρ → λβρ −ν⊥ ξ⊥ → λ ξ⊥ −νk ξk → λ ξk to the Langevin equation 2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t) The result reads: β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t) Directed Percolation Other Universality Classes Langevin Equation – Mean Field Critical Exponents To determine mean field exponents require scale invariance: β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t) The scaling factors drop out if: a = 0 βMF = 1 MF ν⊥ = 1/2 MF νk = 1 d = 4 Directed Percolation Other Universality Classes Langevin Equation – Upper Critical Dimension β+ν β 2β 2 β+2ν⊥ 2 λ k ∂t ρ(x, t) = λ aρ(x, t) − λ bρ (x, t) + λ D∇ ρ(x, t) 1 (β+dν +ν ) + λ 2 ⊥ k ξ(x, t) ⇑ The dimension enters only here. d noise d>4 irrelevant mean field correct d=4 marginal mean field + log corrections d<4 relevant full field theory needed dc = 4 is called upper critical dimension. Directed Percolation Other Universality Classes Langevin Equation – Higher Order Terms 2 2 ∂t ρ(x, t) = aρ(x, t) − bρ (x, t) + D∇ ρ(x, t) + ξ(x, t) Higher-order terms such as ρ3, ∇4ρ, (∇ρ)2,... would be irrelevant under rescaling. This is the origin of universality. Directed Percolation Other Universality Classes Directed Percolation – Field theory Backbone of a two-point function: ⇒ Lecture by U. C. Täuber (Monday 11:30) Directed Percolation Other Universality Classes DP conjecture Janssen / Grassberger Any stochastic model with the properties... short-range interactions two-state model single absorbing state spontaneous removal and offspring production no quenched disorder no unconventional symmetries ...belongs to the DP universality class. Directed Percolation Other Universality Classes Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes Parity-conserving Class Reaction-diffusion schemes: DP-class A → 0 A → 2A 2A → A + diffusion PC-class 2A → 0 A → 3A + diffusion PC-processes conserve the number of particles modulo 2. Example: Branching Annihilating Random Walk with even number of offspring (BARW): Directed Percolation Other Universality Classes PC Transitions belong to a Different Class Parity-conserving class : 2A → 0 , A → 3A and diffusion Two different dynamical sectors. Absorbing (subcritical) phase governed√ by 2A → 0 decays algebraically, in 1d as 1/ t. Field theory (Cardy, Täuber) with two critical dimensions: dc = 2 for annihilation, dc = 4/3 for branching. Different set of critical exponents in 1d: 0 β = β νk ν⊥ DP ≈ 0.276 ≈ 1.734 ≈ 1.097 PC 0.92(2) 3.22(6) 1.83(3) Directed Percolation Other Universality Classes Visualizing Scale Invariance: 1/z Plot x/t versus ln t, where z = νk/ν⊥ is the expected dynamical exponent: Directed Percolation Other Universality Classes Visualizing Scale Invariance in the Critical BARW: x → λ−ν⊥ x t → λ−νk t ρ → λβρ Directed Percolation Other Universality Classes Parity-Conserving Class 2A → 0 ,A → 3A and diffusion Is it only parity conservation that drives the transition away from DP? Directed Percolation Other Universality Classes Parity-Conserving Class Directed Percolation Other Universality Classes Outline 1 Directed Percolation Scaling Theory Langevin Equation 2 Other Universality Classes Parity-conserving Class Voter Universality Class DP with long-range interactions Directed Percolation Other Universality Classes Voter Universality Class Voter model: If you can’t make up your mind... with two states: Z2-symmetry. simply adopt the opinion of a randomly chosen nearest neighbour. Directed Percolation Other Universality Classes Voter Model in One Dimension In 1d the voter model is like Glauber-Ising dynamics at T = 0: In 1d the kinks between Z2-symmetric domains can be interpreted as particles: 2A → 0 Directed Percolation Other Universality Classes Voter Model in Two Dimensions Glauber−Ising model at T=0 Classical voter model Directed Percolation Other Universality Classes Voter Phase Transitions (Z2-symmetric transitions) Take the voter model and add interfacial noise: Directed Percolation Other Universality Classes Voter Phase Transitions In 1d voter phase transitions belong to the parity-conserving class.
Load more

Recommended publications
  • Universal Scaling Behavior of Non-Equilibrium Phase Transitions
    Universal scaling behavior of non-equilibrium phase transitions HABILITATIONSSCHRIFT der FakultÄat furÄ Naturwissenschaften der UniversitÄat Duisburg-Essen vorgelegt von Sven LubÄ eck aus Duisburg Duisburg, im Juni 2004 Zusammenfassung Kritische PhÄanomene im Nichtgleichgewicht sind seit Jahrzehnten Gegenstand inten- siver Forschungen. In Analogie zur Gleichgewichtsthermodynamik erlaubt das Konzept der UniversalitÄat, die verschiedenen NichtgleichgewichtsphasenubÄ ergÄange in unterschied- liche Klassen einzuordnen. Alle Systeme einer UniversalitÄatsklasse sind durch die glei- chen kritischen Exponenten gekennzeichnet, und die entsprechenden Skalenfunktionen werden in der NÄahe des kritischen Punktes identisch. WÄahrend aber die Exponenten zwischen verschiedenen UniversalitÄatsklassen sich hÄau¯g nur geringfugigÄ unterscheiden, weisen die Skalenfunktionen signi¯kante Unterschiede auf. Daher ermÄoglichen die uni- versellen Skalenfunktionen einerseits einen emp¯ndlichen und genauen Nachweis der UniversalitÄatsklasse eines Systems, demonstrieren aber andererseits in ubÄ erzeugender- weise die UniversalitÄat selbst. Bedauerlicherweise wird in der Literatur die Betrachtung der universellen Skalenfunktionen gegenubÄ er der Bestimmung der kritischen Exponen- ten hÄau¯g vernachlÄassigt. Im Mittelpunkt dieser Arbeit steht eine bestimmte Klasse von Nichtgleichgewichts- phasenubÄ ergÄangen, die sogenannten absorbierenden PhasenubÄ ergÄange. Absorbierende PhasenubÄ ergÄange beschreiben das kritische Verhalten von physikalischen, chemischen sowie biologischen
    [Show full text]
  • Three-Dimensional Phase Transitions in Multiflavor Lattice Scalar SO (Nc) Gauge Theories
    Three-dimensional phase transitions in multiflavor lattice scalar SO(Nc) gauge theories Claudio Bonati,1 Andrea Pelissetto,2 and Ettore Vicari1 1Dipartimento di Fisica dell’Universit`adi Pisa and INFN Largo Pontecorvo 3, I-56127 Pisa, Italy 2Dipartimento di Fisica dell’Universit`adi Roma Sapienza and INFN Sezione di Roma I, I-00185 Roma, Italy (Dated: January 1, 2021) We investigate the phase diagram and finite-temperature transitions of three-dimensional scalar SO(Nc) gauge theories with Nf ≥ 2 scalar flavors. These models are constructed starting from a maximally O(N)-symmetric multicomponent scalar model (N = NcNf ), whose symmetry is par- tially gauged to obtain an SO(Nc) gauge theory, with O(Nf ) or U(Nf ) global symmetry for Nc ≥ 3 or Nc = 2, respectively. These systems undergo finite-temperature transitions, where the global sym- metry is broken. Their nature is discussed using the Landau-Ginzburg-Wilson (LGW) approach, based on a gauge-invariant order parameter, and the continuum scalar SO(Nc) gauge theory. The LGW approach predicts that the transition is of first order for Nf ≥ 3. For Nf = 2 the transition is predicted to be continuous: it belongs to the O(3) vector universality class for Nc = 2 and to the XY universality class for any Nc ≥ 3. We perform numerical simulations for Nc = 3 and Nf = 2, 3. The numerical results are in agreement with the LGW predictions. I. INTRODUCTION The global O(N) symmetry is partially gauged, obtain- ing a nonabelian gauge model, in which the fields belong to the coset SN /SO(N ), where SN = SO(N)/SO(N 1) Global and local gauge symmetries play a crucial role c − in theories describing fundamental interactions [1] and is the N-dimensional sphere.
    [Show full text]
  • Universal Scaling Behavior of Non-Equilibrium Phase Transitions
    Universal scaling behavior of non-equilibrium phase transitions Sven L¨ubeck Theoretische Physik, Univerit¨at Duisburg-Essen, 47048 Duisburg, Germany, [email protected] December 2004 arXiv:cond-mat/0501259v1 [cond-mat.stat-mech] 11 Jan 2005 Summary Non-equilibrium critical phenomena have attracted a lot of research interest in the recent decades. Similar to equilibrium critical phenomena, the concept of universality remains the major tool to order the great variety of non-equilibrium phase transitions systematically. All systems belonging to a given universality class share the same set of critical exponents, and certain scaling functions become identical near the critical point. It is known that the scaling functions vary more widely between different uni- versality classes than the exponents. Thus, universal scaling functions offer a sensitive and accurate test for a system’s universality class. On the other hand, universal scaling functions demonstrate the robustness of a given universality class impressively. Unfor- tunately, most studies focus on the determination of the critical exponents, neglecting the universal scaling functions. In this work a particular class of non-equilibrium critical phenomena is considered, the so-called absorbing phase transitions. Absorbing phase transitions are expected to occur in physical, chemical as well as biological systems, and a detailed introduc- tion is presented. The universal scaling behavior of two different universality classes is analyzed in detail, namely the directed percolation and the Manna universality class. Especially, directed percolation is the most common universality class of absorbing phase transitions. The presented picture gallery of universal scaling functions includes steady state, dynamical as well as finite size scaling functions.
    [Show full text]
  • Sankar Das Sarma 3/11/19 1 Curriculum Vitae
    Sankar Das Sarma 3/11/19 Curriculum Vitae Sankar Das Sarma Richard E. Prange Chair in Physics and Distinguished University Professor Director, Condensed Matter Theory Center Fellow, Joint Quantum Institute University of Maryland Department of Physics College Park, Maryland 20742-4111 Email: [email protected] Web page: www.physics.umd.edu/cmtc Fax: (301) 314-9465 Telephone: (301) 405-6145 Published articles in APS journals I. Physical Review Letters 1. Theory for the Polarizability Function of an Electron Layer in the Presence of Collisional Broadening Effects and its Experimental Implications (S. Das Sarma) Phys. Rev. Lett. 50, 211 (1983). 2. Theory of Two Dimensional Magneto-Polarons (S. Das Sarma), Phys. Rev. Lett. 52, 859 (1984); erratum: Phys. Rev. Lett. 52, 1570 (1984). 3. Proposed Experimental Realization of Anderson Localization in Random and Incommensurate Artificial Structures (S. Das Sarma, A. Kobayashi, and R.E. Prange) Phys. Rev. Lett. 56, 1280 (1986). 4. Frequency-Shifted Polaron Coupling in GaInAs Heterojunctions (S. Das Sarma), Phys. Rev. Lett. 57, 651 (1986). 5. Many-Body Effects in a Non-Equilibrium Electron-Lattice System: Coupling of Quasiparticle Excitations and LO-Phonons (J.K. Jain, R. Jalabert, and S. Das Sarma), Phys. Rev. Lett. 60, 353 (1988). 6. Extended Electronic States in One Dimensional Fibonacci Superlattice (X.C. Xie and S. Das Sarma), Phys. Rev. Lett. 60, 1585 (1988). 1 Sankar Das Sarma 7. Strong-Field Density of States in Weakly Disordered Two Dimensional Electron Systems (S. Das Sarma and X.C. Xie), Phys. Rev. Lett. 61, 738 (1988). 8. Mobility Edge is a Model One Dimensional Potential (S.
    [Show full text]
  • Arxiv:1908.10990V1 [Cond-Mat.Stat-Mech] 29 Aug 2019
    High-precision Monte Carlo study of several models in the three-dimensional U(1) universality class Wanwan Xu,1 Yanan Sun,1 Jian-Ping Lv,1, ∗ and Youjin Deng2, 3 1Anhui Key Laboratory of Optoelectric Materials Science and Technology, Key Laboratory of Functional Molecular Solids, Ministry of Education, Anhui Normal University, Wuhu, Anhui 241000, China 2Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 3CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China We present a worm-type Monte Carlo study of several typical models in the three-dimensional (3D) U(1) universality class, which include the classical 3D XY model in the directed flow representation and its Vil- lain version, as well as the 2D quantum Bose-Hubbard (BH) model with unitary filling in the imaginary-time world-line representation. From the topology of the configurations on a torus, we sample the superfluid stiff- ness ρs and the dimensionless wrapping probability R. From the finite-size scaling analyses of ρs and of R, we determine the critical points as Tc(XY) = 2.201 844 1(5) and Tc(Villain) = 0.333 067 04(7) and (t/U)c(BH) = 0.059 729 1(8), where T is the temperature for the classical models, and t and U are respec- tively the hopping and on-site interaction strength for the BH model. The precision of our estimates improves significantly over that of the existing results.
    [Show full text]
  • Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu the Ames Laboratory
    Physics and Astronomy Publications Physics and Astronomy 2004 Crossover Between Mean-Field and Ising Critical Behavior in a Lattice-Gas Reaction-Diffusion Model Da-Jiang Liu The Ames Laboratory N. Pavlenko Universitat Hannover James W. Evans Iowa State University, [email protected] Follow this and additional works at: http://lib.dr.iastate.edu/physastro_pubs Part of the Chemistry Commons, and the Physics Commons The ompc lete bibliographic information for this item can be found at http://lib.dr.iastate.edu/physastro_pubs/456. For information on how to cite this item, please visit http://lib.dr.iastate.edu/howtocite.html. This Article is brought to you for free and open access by the Physics and Astronomy at Iowa State University Digital Repository. It has been accepted for inclusion in Physics and Astronomy Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Crossover Between Mean-Field and Ising Critical Behavior in a Lattice- Gas Reaction-Diffusion Model Abstract Lattice-gas models for CO oxidation can exhibit a discontinuous nonequilibrium transition between reactive and inactive states, which disappears above a critical CO-desorption rate. Using finite-size-scaling analysis, we demonstrate a crossover from Ising to mean-field behavior at the critical point, with increasing surface mobility of adsorbed CO or with decreasing system size. This behavior is elucidated by analogy with that of equilibrium Ising-type systems with long-range interactions. Keywords critical behavior, lattice-gas reaction-diffusion model, Ising, mean field Disciplines Chemistry | Physics Comments This article is published as Liu, Da-Jiang, N.
    [Show full text]
  • Directed Percolation and Turbulence
    Emergence of collective modes, ecological collapse and directed percolation at the laminar-turbulence transition in pipe flow Hong-Yan Shih, Tsung-Lin Hsieh, Nigel Goldenfeld University of Illinois at Urbana-Champaign Partially supported by NSF-DMR-1044901 H.-Y. Shih, T.-L. Hsieh and N. Goldenfeld, Nature Physics 12, 245 (2016) N. Goldenfeld and H.-Y. Shih, J. Stat. Phys. 167, 575-594 (2017) Deterministic classical mechanics of many particles in a box statistical mechanics Deterministic classical mechanics of infinite number of particles in a box = Navier-Stokes equations for a fluid statistical mechanics Deterministic classical mechanics of infinite number of particles in a box = Navier-Stokes equations for a fluid statistical mechanics Transitional turbulence: puffs • Reynolds’ original pipe turbulence (1883) reports on the transition Univ. of Manchester Univ. of Manchester “Flashes” of turbulence: Precision measurement of turbulent transition Q: will a puff survive to the end of the pipe? Many repetitions survival probability = P(Re, t) Hof et al., PRL 101, 214501 (2008) Pipe flow turbulence Decaying single puff metastable spatiotemporal expanding laminar puffs intermittency slugs Re 1775 2050 2500 푡−푡 − 0 Survival probability 푃 Re, 푡 = 푒 휏(Re) ) Re,t Puff P( lifetime to N-S Avila et al., (2009) Avila et al., Science 333, 192 (2011) Hof et al., PRL 101, 214501 (2008) 6 Pipe flow turbulence Decaying single puff Splitting puffs metastable spatiotemporal expanding laminar puffs intermittency slugs Re 1775 2050 2500 푡−푡 − 0 Splitting
    [Show full text]
  • Block Scaling in the Directed Percolation Universality Class
    OPEN ACCESS Recent citations Block scaling in the directed percolation - 25 Years of Self-organized Criticality: Numerical Detection Methods universality class R. T. James McAteer et al - The Abelian Manna model on various To cite this article: Gunnar Pruessner 2008 New J. Phys. 10 113003 lattices in one and two dimensions Hoai Nguyen Huynh et al View the article online for updates and enhancements. This content was downloaded from IP address 170.106.40.139 on 26/09/2021 at 04:54 New Journal of Physics The open–access journal for physics Block scaling in the directed percolation universality class Gunnar Pruessner1 Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK E-mail: [email protected] New Journal of Physics 10 (2008) 113003 (13pp) Received 23 July 2008 Published 7 November 2008 Online at http://www.njp.org/ doi:10.1088/1367-2630/10/11/113003 Abstract. The universal behaviour of the directed percolation universality class is well understood—both the critical scaling and the finite size scaling. This paper focuses on the block (finite size) scaling of the order parameter and its fluctuations, considering (sub-)blocks of linear size l in systems of linear size L. The scaling depends on the choice of the ensemble, as only the conditional ensemble produces the block-scaling behaviour as established in equilibrium critical phenomena. The dependence on the ensemble can be understood by an additional symmetry present in the unconditional ensemble. The unconventional scaling found in the unconditional ensemble is a reminder of the possibility that scaling functions themselves have a power-law dependence on their arguments.
    [Show full text]
  • QCD Critical Point: Fluctuations, Hydrodynamics and Universality Class
    QCD critical point: fluctuations, hydrodynamics and universality class M. Stephanov U. of Illinois at Chicago and RIKEN-BNL QCD critical point: fluctuations, hydrodynamics and universality class – p.1/21 QCD critical point hot QGP T, GeV 3 1 E critical 0.1 point 2 nuclear CFL vacuum matter quark matter 0 1 µB , GeV 1 Lattice at µ =0 → crossover. 2 Sign problem. Models: 1st order transition. 1 + 2 = 3 : critical point E. (As in water at p = 221bar, T = 373◦C – critical opalescence.) Where is point E? Challenge to theory and experiment. QCD critical point: fluctuations, hydrodynamics and universality class – p.2/21 Location of the CP (theory) Source (T, µB ), MeV Comments Label MIT Bag/QGP none only 1st order — Asakawa,Yazaki ’89 (40, 1050) NJL, CASE I NJL/I “ (55, 1440) NJL, CASE II NJL/II Barducci, et al ’89-94 (75, 273)TCP composite operator CO Berges, Rajagopal ’98 (101, 633)TCP instanton NJL NJL/inst Halasz, et al ’98 (120, 700)TCP random matrix RM Scavenius, et al ’01 (93,645) linear σ-model LSM “ (46,996) NJL NJL Fodor, Katz ’01 (160, 725) lattice reweighting I Hatta, Ikeda, ’02 (95, 837) effective potential (CJT) CJT Antoniou, Kapoyannis ’02 (171, 385) hadronic bootstrap HB Ejiri, et al ’03 (?,420) lattice Taylor expansion Fodor, Katz ’04 (162, 360) lattice reweighting II QCD critical point: fluctuations, hydrodynamics and universality class – p.3/21 Location of the CP (theory) 200 T HB lat. Taylor exp. lat. reweighting I 150 lat. reweighting II NJL/inst RM 100 CO LSM CJT NJL/II NJL 50 NJL/I 0 0 200 400 600 800 1000 1200 1400 1600 µB Slope: Allton et al ’02 (Taylor exp.), de Forcrand, Philipsen ’02 (Imµ): 2 T (µ ) µ B ≈ 1 − .00(5 − 8) B T (0) T (0) Allton et al ’03: .013.
    [Show full text]
  • [Cond-Mat.Stat-Mech] 26 Jul 1999
    Ordered phase and scaling in Zn models and the three-state antiferromagnetic Potts model in three dimensions Masaki Oshikawa Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan (July 20, 1999) the spontaneously broken Zn symmetry and the high- Based on a Renormalization-Group picture of Zn symmet- temperature disordered phase. The intermediate phase is ric models in three dimensions, we derive a scaling law for the O(2) symmetric and corresponds to the low-temperature Zn order parameter in the ordered phase. An existing Monte phase of the XY model. Carlo calculation on the three-state antiferromagnetic Potts On three dimensional (3D) case, Blankschtein et al.3 model, which has the effective Z6 symmetry, is shown to be in 1984 proposed an RG picture of the Z models, to consistent with the proposed scaling law. It strongly supports 6 the Renormalization-Group picture that there is a single mas- discuss the STI model. They suggested that the transi- sive ordered phase, although an apparently rotationally sym- tion between the ordered and disordered phases belongs metric region in the intermediate temperature was observed to the (3D) XY universality class, and that the ordered numerically. phase reflects the symmetry breaking to Z6 in a large enough system. It means that there is no finite region of 75.10.Hk, 05.50+q rotationally symmetric phase which is similar to the or- dered phase of the XY model. Unfortunately, their paper is apparently not widely known in the related fields. It might be partly because their discussion was very brief I.
    [Show full text]
  • Disorder Driven Itinerant Quantum Criticality of Three Dimensional
    Disorder driven itinerant quantum criticality of three dimensional massless Dirac fermions J. H. Pixley, Pallab Goswami, and S. Das Sarma Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742- 4111 USA (Dated: August 31, 2018) Progress in the understanding of quantum critical properties of itinerant electrons has been hin- dered by the lack of effective models which are amenable to controlled analytical and numerically exact calculations. Here we establish that the disorder driven semimetal to metal quantum phase transition of three dimensional massless Dirac fermions could serve as a paradigmatic toy model for studying itinerant quantum criticality, which is solved in this work by exact numerical and approx- imate field theoretic calculations. As a result, we establish the robust existence of a non-Gaussian universality class, and also construct the relevant low energy effective field theory that could guide the understanding of quantum critical scaling for many strange metals. Using the kernel polynomial method (KPM), we provide numerical results for the calculated dynamical exponent (z) and corre- lation length exponent (ν) for the disorder-driven semimetal (SM) to diffusive metal (DM) quantum phase transition at the Dirac point for several types of disorder, establishing its universal nature and obtaining the numerical scaling functions in agreement with our field theoretical analysis. PACS numbers: 71.10.Hf,72.80.Ey,73.43.Nq,72.15.Rn I. INTRODUCTION lengths (ξl and ξt) are infinite. Even though the QCP occurs strictly at zero temperature, its effects on physical Phase transitions are ubiquitous in the natural world properties manifest over a broad range of temperatures ranging from the solidification of water to the thermal (T ) and energies (E), such that suppression of magnetism in iron or superconductivity in 1 E, k T ξ− , metals.
    [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]