DOCSLIB.ORG

Statistical Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Statistical Field Theory Statistical Field theory Thierry Giamarchi November 8, 2003 Contents 1 Introduction 5 2 Equilibrium statistical mechanics 7 2.1 Basics; correlation functions . ............................... 7 2.2Externalfield;linearresponse................................ 8 2.3 Continuous systems: functional integral . ....................... 11 2.4Examples:elasticsystems.................................. 17 3 Calculation of Functional Integrals 29 3.1Perturbationtheory..................................... 29 3.2Variationalmethod...................................... 31 3.3Saddlepointmethod..................................... 32 4 Quantum problems; functional integral 35 4.1Singleparticle......................................... 35 4.2 Correlation functions; Matsubara frequencies ....................... 39 4.3 Many degrees of freedom: example of the elastic system . ............... 42 4.4Linkwithclassicalproblems................................. 44 5 Perturbation; Linear response 47 5.1Method............................................ 47 5.2Analyticalcontinuation................................... 50 5.3Fluctuationdissipationtheorem............................... 52 5.4Kuboformula......................................... 53 5.5 Scattering, Memory function . ............................... 57 6 Examples 59 6.1 Compressibility . ....................................... 59 6.2 Conductivity of quantum crystals; Hall effect ....................... 61 6.3Commensuratesystems................................... 66 7 Out of equilibrium classical systems 69 7.1Overdampeddynamics.................................... 69 7.2Linkwiththermodynamics................................. 70 7.3MSRFormalism....................................... 72 3 Chapter 1 Introduction The material discussed in these notes can also be found in more specialized books. Among them • Statistical mechanics and functional integrals [1, 2, 3, 4] • Many body physics [5, 3] These notes are still in a larval stage. This is specially true for chapters 3 and 7. Despite the at- tempts to unify the notations and eliminates mistakes, factors of 2 and other errors of signs, it is clear that many of those remain. I’ll try to update the notes from time to time. If you find these notes useful and feel like helping to improve them, do not hesitate to email to me ([email protected]) if you have found errors in those notes or simply to let me know if there are parts that you would like to see improved or expanded. 5 Chapter 2 Equilibrium statistical mechanics 2.1 Basics; correlation functions In statistical mechanics one is interested in getting basic quantities such as the partition function. Space is meshed with a network on which some variable describe the state of the system. To be more specific let us consider an Ising model, on a square lattice. On each site i exists a spin σi = ±. The state of the system is describe by all spin variables σ1,σ2,...σN , where N grows as the volume of the system. The set of all these variables is a configuration that we denote in the following as {σ}.To compute physical quantities one needs in addition an Hamiltonian describing the energy of the system for a given configuration. To be specific let us again consider a Ising type Hamiltonian H0 = − Ji,jσiσj (2.1) i,j A physical quantity such as the partition function results from the sum over all configurations of the system Z = e−βH0 = ··· e−βH0[{σ}] (2.2) {σ} σ1=± σ1=± σN =± The difficulty if of course to make the sum over the humongous (thermodynamics) number of variables. From Z or F = −T log(Z) many simple physical quantities can be directly extracted (specific heat, etc.). In addition to these rather global quantities, one is often interested in correlation functions, mea- suring thermodynamics averages of local observables or products of local observables. Some examples are σi1 (2.3) σi1σi2 (2.4) where the thermodynamics average stands for −βH O[σα,...,σβ]e 0 {σ} O[α,...,β] = −βH (2.5) {σ} e 0 Where O is any operator. Note that the correlation functions depends on the spatial positions α,...,β, the average having been made over the microscopic variables σ. 7 In fact the correlation functions can also be obtained from a partition function. If one adds to the Hamiltonian a source term − βHs = hiOi (2.6) i one introduces the partition function, that now depends on a thermodynamics number of external sources −β[H0+Hs] Z[h1,h2,...,hN ]= e (2.7) {σ} and the associated free energy F [h1,h2,...,hN ]. It is then easy to see that 1 ∂Z[h1,h2,...,hN ] [Oα ...Oβ] = (2.8) Z[h =0] ∂hα ...∂hβ {h}=0 Of course this is a rather formal relation since computing Z[h1,h2,...,hN ] is in general a formidable task. Similar formulas can be derived from the free energy. In particular one can obtain from the free energy ∂F[h1,h2,...,hN ] Oα = −β (2.9) ∂hα {h}=0 It is easy to see that differentiating the free energy leads to the so-called connected correlations ∂F[h1,h2,...,hN ] − β =[OαOβ−OαOβ] (2.10) ∂hα∂hβ {h}=0 Each time one differentiate an average one gets two terms one coming from the numerator one from the denominator. Thus ∂hγ ... =[...Oγ−...Oγ] (2.11) It is thus easy to obtain the higher derivatives. The ingredients that we have illustrated on this example of the Ising model are totally general. It is important to understand that what is generally needed is to be able to perform the sums over the microsocpic degrees of freedom. 2.2 External field; linear response Among the various correlation functions some are of special importance. Let us assume that a physical variable of the system can be coupled to an external field, in a way similar to (2.6). In our Ising example this is the case if one puts the system in a magnetic field. The natural variable associated with such a perturbation is the magnetization of the system. One would have two way to compute it. Adding the magnetic field to the Hamiltonian H0 leads to Hh = −h σi (2.12) i From H = H0 + Hh one can compute the free energy F [h]. Standard thermodynamics tells us that the total magnetization M of the system is simply given by dF [h] M = − (2.13) dh This relation comes immediately from (2.8) using O = − i σi. M describes the response of the system to the external field h. Even for this simple case (h is space independent) is is quite complicated to get the magnetization. To generalize this notion of response, let us consider an operator Oi (that can a priori depend on the position) that couples to an external field hi (that can also depend on space) trough the Hamiltonian Hp = − hiOi (2.14) i Let us furthermore assume that Oi is such that in the absence of perturbation Hp one has Oi0 =0, where 0 denotes averages performed with H0 alone. If it is not the case one can simply subtract the average value from Oi. In that case one can measure the response of the system, that is given by Oi[{h}]=Oi (2.15) The notation Oi[{h}] means that the value Oi at point i depends in fact on the value of the external field at all points, i.e. of the whole set of values of h at each spatial point. For even a simple H0 it is hopeless to expect to compute the full response Oi. A simpler case, but of considerable interest, is to compute the linear response of the system. Indeed if the external field is small one can expand (2.15), in powers of the external field. The lowest order term is linear in the external field (hence the term linear response). One has d Oi[{h}]=0+a χijhj (2.16) j which is the most general linear term that one can write. The factor ad is here for convenience (see below). At that stage this merely defines χ. If we assume in addition that H0 is invariant by translation, then χij can only depend on the distance between i and j (that we denote symbolically by i − j). A more transparent way to write the convolution (2.16) is to go to Fourier space. Let us recall the convention that we use for fourier transform on the lattice. d −iq·r hq = a e i hi (2.17) i 1 iq·r hi = e i hq (2.18) Ω q=2πn/L where we have introduced a lattice spacing a (a can be set to 1 at that stage since it is purely formal). The volume of the system is Ω = Nad and one introduces a distance r = ai. The sum over q runs from q = −π/a to q =+π/a, i.e. over N values of n. This corresponds to the standard Brillouin zone. In that case iq·ri e = Nδq,0 (2.19) i where δi,j is the discrete δ. For a continuous system ri → r, one has the continuous Fourier transform h(q)= ddre−iq·rh(r) (2.20) Ω 1 h(r)= eiq·rh(q) (2.21) Ω q=2πn/L where Ω = Ld is the volume of the system. The relation between the lattice variables and the continuous ones is thus hi → h(r) (2.22) hq → h(q) (2.23) For the continuous case iq·r dre =Ωδq,0 (2.24) still with a discrete δ. In the limit where the volume of the system goes to infinity, one can use 1 → 1 d d d q (2.25) Ω q (2π) d Ωδq,0 → (2π) δ(q) (2.26) where δ(q) is the Dirac δ. Using these definitions for the Fourier transform one can rewrite (2.16) as d 1 −iqri+iq rj Oq = a e χi−jhq (2.27) N ij q using ri = rj + rl one has d 1 i q−q r −iqr O ( ) j l q = a e e χrl hq (2.28) N jl q d −iqrl = a e χrl hq (2.29) l = χqhq (2.30) Thus in Fourier space the relation between the response and the external field is truly linear, each q mode being independent.
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.
    [Show full text]
  • 2D Potts Model Correlation Lengths
    KOMA Revised version May D Potts Mo del Correlation Lengths Numerical Evidence for = at o d t Wolfhard Janke and Stefan Kappler Institut f ur Physik Johannes GutenbergUniversitat Mainz Staudinger Weg Mainz Germany Abstract We have studied spinspin correlation functions in the ordered phase of the twodimensional q state Potts mo del with q and at the rstorder transition p oint Through extensive Monte t Carlo simulations we obtain strong numerical evidence that the cor relation length in the ordered phase agrees with the exactly known and recently numerically conrmed correlation length in the disordered phase As a byproduct we nd the energy moments o t t d in the ordered phase at in very go o d agreement with a recent large t q expansion PACS numbers q Hk Cn Ha hep-lat/9509056 16 Sep 1995 Introduction Firstorder phase transitions have b een the sub ject of increasing interest in recent years They play an imp ortant role in many elds of physics as is witnessed by such diverse phenomena as ordinary melting the quark decon nement transition or various stages in the evolution of the early universe Even though there exists already a vast literature on this sub ject many prop erties of rstorder phase transitions still remain to b e investigated in detail Examples are nitesize scaling FSS the shap e of energy or magnetization distributions partition function zeros etc which are all closely interrelated An imp ortant approach to attack these problems are computer simulations Here the available system sizes are necessarily
    [Show full text]
  • Full and Unbiased Solution of the Dyson-Schwinger Equation in the Functional Integro-Differential Representation
    PHYSICAL REVIEW B 98, 195104 (2018) Full and unbiased solution of the Dyson-Schwinger equation in the functional integro-differential representation Tobias Pfeffer and Lode Pollet Department of Physics, Arnold Sommerfeld Center for Theoretical Physics, University of Munich, Theresienstrasse 37, 80333 Munich, Germany (Received 17 March 2018; revised manuscript received 11 October 2018; published 2 November 2018) We provide a full and unbiased solution to the Dyson-Schwinger equation illustrated for φ4 theory in 2D. It is based on an exact treatment of the functional derivative ∂/∂G of the four-point vertex function with respect to the two-point correlation function G within the framework of the homotopy analysis method (HAM) and the Monte Carlo sampling of rooted tree diagrams. The resulting series solution in deformations can be considered as an asymptotic series around G = 0 in a HAM control parameter c0G, or even a convergent one up to the phase transition point if shifts in G can be performed (such as by summing up all ladder diagrams). These considerations are equally applicable to fermionic quantum field theories and offer a fresh approach to solving functional integro-differential equations beyond any truncation scheme. DOI: 10.1103/PhysRevB.98.195104 I. INTRODUCTION was not solved and differences with the full, exact answer could be seen when the correlation length increases. Despite decades of research there continues to be a need In this work, we solve the full DSE by writing them as a for developing novel methods for strongly correlated systems. closed set of integro-differential equations. Within the HAM The standard Monte Carlo approaches [1–5] are convergent theory there exists a semianalytic way to treat the functional but suffer from a prohibitive sign problem, scaling exponen- derivatives without resorting to an infinite expansion of the tially in the system volume [6].
    [Show full text]
  • 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)..........................
    [Show full text]
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan
    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
    [Show full text]
  • 10 Basic Aspects of CFT
    10 Basic aspects of CFT An important break-through occurred in 1984 when Belavin, Polyakov and Zamolodchikov [BPZ84] applied ideas of conformal invariance to classify the possible types of critical behaviour in two dimensions. These ideas had emerged earlier in string theory and mathematics, and in fact go backto earlier (1970) work of Polyakov [Po70] in which global conformal invariance is used to constrain the form of correlation functions in d-dimensional the- ories. It is however only by imposing local conformal invariance in d =2 that this approach becomes really powerful. In particular, it immediately permitted a full classification of an infinite family of conformally invariant theories (the so-called “minimal models”) having a finite number of funda- mental (“primary”) fields, and the exact computation of the corresponding critical exponents. In the aftermath of these developments, conformal field theory (CFT) became for some years one of the most hectic research fields of theoretical physics, and indeed has remained a very active area up to this date. This chapter focusses on the basic aspects of CFT, with a special emphasis on the ingredients which will allow us to tackle the geometrically defined loop models via the so-called Coulomb Gas (CG) approach. The CG technique will be exposed in the following chapter. The aim is to make the presentation self- contained while remaining rather brief; the reader interested in more details should turn to the comprehensive textbook [DMS87] or the Les Houches volume [LH89]. 10.1 Global conformal invariance A conformal transformation in d dimensions is an invertible mapping x x′ → which multiplies the metric tensor gµν (x) by a space-dependent scale factor: gµ′ ν (x′)=Λ(x)gµν (x).
    [Show full text]
  • The Reality of Casimir Friction
    Review The Reality of Casimir Friction Kimball A. Milton 1,*, Johan S. Høye 2 and Iver Brevik 3 1 Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA 2 Department of Physics, Norwegian University of Science and Technology, Trondheim 7491, Norway; [email protected] 3 Department of Energy and Process Engineering, Norwegian University of Science and Technology, Trondheim 7491, Norway; [email protected] * Correspondence: [email protected]; Tel.: +1-405-325-3961 x 36325 Academic Editor: Sergei D. Odintsov Received: 17 March 2016; Accepted: 21 April 2016; Published: date Abstract: For more than 35 years theorists have studied quantum or Casimir friction, which occurs when two smooth bodies move transversely to each other, experiencing a frictional dissipative force due to quantum electromagnetic fluctuations, which break time-reversal symmetry. These forces are typically very small, unless the bodies are nearly touching, and consequently such effects have never been observed, although lateral Casimir forces have been seen for corrugated surfaces. Partly because of the lack of contact with observations, theoretical predictions for the frictional force between parallel plates, or between a polarizable atom and a metallic plate, have varied widely. Here, we review the history of these calculations, show that theoretical consensus is emerging, and offer some hope that it might be possible to experimentally confirm this phenomenon of dissipative quantum electrodynamics. Keywords: Casimir friction; dissipation; quantum fluctuations; fluctuation-dissipation theorem 1. Introduction The essence of quantum mechanics is fluctuations in sources (currents) and fields. This underlies the great successes of quantum electrodynamics such as explaining the Lamb shift [1] and the anomalous magnetic moment of the electron [2].
    [Show full text]
  • Hamiltonian Perturbation Theory on a Lie Algebra. Application to a Non-Autonomous Symmetric Top
    Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top. Lorenzo Valvo∗ Michel Vittot Dipartimeno di Matematica Centre de Physique Théorique Università degli Studi di Roma “Tor Vergata” Aix-Marseille Université & CNRS Via della Ricerca Scientifica 1 163 Avenue de Luminy 00133 Roma, Italy 13288 Marseille Cedex 9, France January 6, 2021 Abstract We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra V, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamil- tonian system that preserves a subalgebra B of V, when we add a perturbation the subalgebra B will no longer be preserved. We show how to transform the perturbed dynamical system to preserve B up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement. A dynamical system on some set V is a flow: a one-parameter group of mappings associating to a given element F 2 V (the initial condition) another element F (t) 2 V, for any value of the parameter t. A flow on V is determined by a linear mapping H from V to itself. However, the flow can be rarely computed explicitly. In perturbation theory we aim at computing the flow of H + V, where the flow of H is known, and V is another linear mapping from V to itself. In physics, the set V is often a Lie algebra: for instance in classical mechanics [4], fluid dynamics and plasma physics [20], quantum mechanics [24], kinetic theory [18], special and general relativity [17].
    [Show full text]
  • 1 Introduction 2 Classical Perturbation Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Classical and Quantum Perturbation Theory for two Non–Resonant Oscillators with Quartic Interaction Luca Salasnich Dipartimento di Matematica Pura ed Applicata, Universit`a di Padova, Via Belzoni 7, I–35131 Padova, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy Istituto Nazionale per la Fisica della Materia, Unit`a di Milano, Via Celoria 16, I–20133 Milano, Italy Abstract. We study the classical and quantum perturbation theory for two non–resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the ”exact” quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections. Key words: Classical Perturbation Theory; Quantum Mechanics; General Mechanics 1 Introduction Nowadays there is considerable renewed interest in the transition from classical mechanics to quantum mechanics, a powerful motivation behind that being the problem of the so–called quantum chaos [1–3]. An important aspect is represented by the semiclassical quantization formula of the (regular) energy levels for quasi–integrable systems [4–6], the so–called torus quantization, initiated by Einstein [7] and completed by Maslov [8]. It has been recently shown [9,10] that, for perturbed non–resonant harmonic oscillators, the algorithm of classical perturbation theory can be used to formulate the quantum mechanical perturbation theory as the semiclassically quantized classical perturbation theory equipped with the quantum corrections in powers of ¯h ”correcting” the classical Hamiltonian that appears in the classical algorithm.
    [Show full text]
  • Perturbation Theory and Exact Solutions
    PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC­ TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?.
    [Show full text]
  • 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
    [Show full text]
  • Green-Kubo Formula for Open Systems
    Green-Kubo formula for open systems Abhishek Dhar Anupam Kundu (CEA, Grenoble) Onuttom Narayan (UC, Santa Cruz) Keiji Saito (Tokyo University) Raman Research Institute SNBose, Kolkata, India November 2010 (RRI) July 2010 1 / 31 Outline Introduction to the problem. Green-Kubo formula for thermal conductivity. Anomalous heat conduction in low-dimensional systems. Conductance for open systems: Landauer formula, Fluctuation theorem. Exact linear response formula for conductance. Frequency dependent conductance. Summary. Classical treatment. (RRI) July 2010 2 / 31 Response to small temperature difference J T TL R For small ∆T = TL − TR and system size L : ∆T Fourier0s law implies : j ∼ κ L The thermal conductivity κ is expected to be an intrinsic material property. (RRI) July 2010 3 / 31 Computing κ j L κ = ∆T Computing κ is difficult since this is a nonequilibrium problem. In equilibrium we know that e−βH(x,v) Prob[x, v] = P (x, v) = 0 Z Hence we can find the expectation value of any physical observable, say A(x, v). Thus Z Z hAi = dx dv A(x, v) P0(x, v) . For a system with one end at T and the other end at T + ∆T , in general, one does not know Prob(x, v) even in the nonequilibrium steady state. Hence difficult to find j = hj(x, v)i. (RRI) July 2010 4 / 31 Green-Kubo linear response theory For small deviations from equilibrium caused by small changes of Hamiltonian, one can do perturbation theory. Start with equilibrium phase space distribution P0(x, v) corresponding to Hamiltonian H. In this case hJi = 0.
    [Show full text]