Ernst Ising. Student of Wilhelm Lenz in Hamburg. PhD Thesis work on linear chains of coupled magnetic moments. This is known as the Ising model.

Transcription

1 The Ising model

2 Ernst Ising May 10, 1900 in Köln-May in Peoria (IL) Student of Wilhelm Lenz in Hamburg. PhD Thesis work on linear chains of coupled magnetic moments. This is known as the Ising model. The name Ising model was coined by Rudolf Peierls in his 1936 publication On Ising s model of ferromagnetism. He survived World War II but it removed him from research. He learned in years after the publication of his model - that his model had become famous. Lars Onsager solved the Ising model (zero field) in two dimensions in S. G. Brush, History of the Lenz-Ising Model, Rev. Mod. Phys 39, (1962)

3 A general Ising model is defined as H = i Ising model H i s i J i. j s i s j Ji, 1 j,ks i s j s k +... i, j i, j,k coupling to a field pair interactions 3-body interactions It has the following general properties Ising model in 2D No phase transition at d=1 for T>0 For J 1 ijk=0, phase transition(s) for J i j < i j For d>4, mean field results are exact Lower critical dimension is dl =1 and the upper critical dimension is du=4. Thermodynamics of the Ising model can be obtained from ( ) F F = k B T ln[tr e βh ] for example s i = H i T

4 Ising model in 1D Define h βh and K βj. The partition function is given by can be calculated exactly. Z(h,K,N) = e h N i=1 s i +K i s i s i+1 {s} Ising model in 1D In the following, we will take a look at boundary conditions, thermodynamics and correlations.

5 Ising model in 1D: Periodic boundaries Periodic boundary conditions are defined by s N+1 = s 1 Ising model in 1D with PBC N-1 N N+1... We assume that there is no external field (h=0). Then, we have note PBC Z =... e K N 1 i=1 s i s i+1 +Ks N s 1 s 1 s 2 s N We can solve this. Define η i = s i s i+1 where i = 1,...,N 1. Then, we have η i = } +1 when si=si+1-1 when si=-si+1 Substitution to the partition function gives Z = (2cosh K) N + (2sinh K) N

6 Ising model in 1D: Free boundaries Ising model in 1D with free boundary conditions N-1 N... Again we assume that there is no external field (h=0). Then, we have Z =... e K N 1 i=1 s i s i+1 s 1 s 2 s N Using the same transformation as before, i.e., η i = s i s i+1 where i = 1,...,N 1 we have that is, η i = Z = 2(2coshK) N 1 } +1 when si=si+1-1 when si=-si+1 We have the partition function now. Next, we take a look at free energy and thermodynamics.

7 Ising model in 1D: Free energy Since we have the partition function, we also have the free energy For PBC: thermodynamic limit F = k B T N { ln(2cosh K) + ln[1 + (tanh K) N ] } Nk B T ln(2cosh K) For free (or open) boundary conditions: [ F = k B T N ln2 + N 1 ] ln(cosh K) N thermodynamic limit Nk B T ln(2cosh K) The difference between boundary conditions becomes negligible at the thermodynamic limit. The more general way is do this with transfer matrix. Works also for nonzero field.

8 Ising model in 1D: Pair correlation function The two-point spin-spin correlation function is defined as G(i, j) (s i s i )(s j s j ) = s i s j s i s j If the system is spatially homogeneous (has translational invariance), then s i = s j s Above Tc we have s = 0 G(i, j) = s i s j What does G(i,j) measure? At T=0 The probability for spins i and j to have the same value is P i j = δ si s j = 1 2 (1 + a i s j ) Around Tc = [G(i, j) + s i s j ] Above Tc we have P i j = 1 [1 + G(i, j)] 2

9 Pair correlation function For a translationally invariant system we have G(i,i + j) = G(i + j i) = G(i) The result (homework exercise, see e.g., Goldenfeld) is How about the other limit, T-> 0? Then, obviously G(i)=1. This defines perfect long-range order. G(i) = (tanh K) i At T=0 Definition for the correlation length: G(i) = e i/ξ For the 1D Ising model we have ξ = [ln(coth K)] 1 As T->0, ξ e 2K This is not a power law but an essential singularity! Around Tc Definition, the correlation function exponent η : G(i) i 2 d η Now, G(i) = e i/ξ = 1 i ξ +... constant

10 2D magnetization

11 Phase equilibrium Phase diagram of the Ising model at finite temperature (d>1): H coexistence line Coexistence in the Ising model Tc disordered T(~1/J) domain wall L -d M H=0 system size, L Tc T

12 Landau theory of phase transitions

13 Overview Introduction Methods Results Recap Next time Overview and new concepts Description of a continuous phase transition using a field, NOT microscopic properties of particles. Reminder: phase transitions Change of a system from one phase (state) to another at a minute change in the external physical conditions. They are divided into two classes: New concepts in a nutshell Symmetry Order parameter Free energy expansion Spontaneous symmetry breaking Symmetries Rotational, translational, etc. Translational Statistical symmetry 1) First-order transitions 2) Continuous transitions Discrete symmetry Continuous symmetry StatPhys Intro to Landau theory Dr. Karttunen pressure solid liquid vapor temperature critical point crossing lines: 1st order transitions

14 Overview Introduction Methods Results Recap Next time StatPhys Intro to Landau theory The Big Picture: usefulness of new concepts Ferromagnets Liquid crystal theory Superconductivity Superfluidity Applications of the theory: Materials modeling, the so-called phase field models Landau-type approach is extremely useful in modeling Even biophysics: Lipid rafts This stage: Thermodynamics Basics of phase transitions Ising model The concept of free energy Limitations: Thermodynamics Basics of phase transitions Ising model The concept of free energy Extensions: Thermodynamics Basics of phase transitions Ising model The concept of free energy Dr. Karttunen

15 Digression: Lev Davidovich Landau Lev Davidovich Landau, Jan in Baku Apr Moscow Nobel Prize 1962 for pioneering theories in condensed matter physics Graduated from Leningrad University at the age of 19. He started at the age of 14! After graduating from Leningrad he spent time in Denmark with Bohr. Collaborated and interacted also with Pauli, Peierls and Teller. For his travels he got a Rockefeller fellowship! His work covers basically all of theoretical physics from fluids to quantum field theory. Was imprisoned by Stalin for a year after being accused to be a German spy. Was freed after Piotr Kapitza threatened to stop his own work unless Landau was released On Jan he suffered a major car accident and was unable to continue his work. For the same reason he was not able to attend the Nobel Prize seremonies. More reading: Akhiezer, Recollections of Lev Davidovich Landau, Physics Today 47, (1994). Ginzburg, Landau's attitude towards physics and physicists, Physics Today 42, (1989). Khalatnikov, Reminiscences of Landau, Physics Today 42, (1989).

16 Landau and Lifshitz started in 1930 s and the 10 volume series was completed in 1979 by Lifshitz.They received the 1962 Lenin Prize for the Course of Theoretical Physics.

17 Landau s revolutionary ideas Superfluidity: Landau considered the quantized states of the motion of the whole liquid instead of single atoms. That was a revolutionary idea and using it Landau was able to explain superfluidity. Superconductivity: Even before the BCS theory, Ginzburg and Landau suggested a phenomenological theory of superconductivity based on Landau's earlier theory of continuous phase transitions. When it was published, the GL theory received only limited attention. This changed dramatically in 1959, when L.P. Gorkov showed rigorously that close to Tc the GL theory and the BCS theory become equivalent. Furthermore, two years before Gorkov, A. Abrikosov predicted the possibility of two different kinds of superconductors by using the GL theory! Landau s theory of phase transitions. If we sum up the leading ideas we end up with two things: the importance of symmetry and symmetry breaking, and the existence of an order parameter.

18 Landau theory The phenomenological Landau theory of continuous phase transitions stresses the importance of overall general symmetry properties and analyticity over microscopic details in determining the macroscopic properties of a system. Those generic properties were also used in the superconductivity and superfluidity problems! The Landau theory is based on the following assumptions: 1.It is possible to define an order parameter. 2.It is possible to describe the system with a free energy. 3.The free energy must be consistent with the high temperature symmetry properties of the system. Mathematically speaking, the Hamiltonian must commute with the symmetry group of the high temperature phase (note: discrete & continuous). 4. The free energy must be analytic. In addition, the expansion coefficients must be regular functions of the temperature.

19 Symmetry 1.It is possible to define an order parameter. 2.It is possible to describe the system with a free energy. 3.The free energy must be consistent with the high temperature symmetry properties of the system. 4. The free energy must be analytic. In addition, the expansion coefficients must be regular functions of the temperature. The order parameter characterizes the system the following way: Ψ = 0 Ψ in the disordered state (above Tc), is small and finite in the ordered state (T<Tc).

20 Order parameters system liquid-gas ferromagnetic superconducting liquid crystal binary mixture (methanol-n-hexane) helix-coil XY-model BaTiO3 crystal liquid crystal order parameter density magnetization condensate wave function degree of molecular alignemnt concentration of either substance number of helix base pairs magnetization (Mx,My) polarization density wave director

21 Symmetry 1.It is possible to define an order parameter. 2.It is possible to describe the system with a free energy. 3.The free energy must be consistent with the high temperature symmetry properties of the system. 4. The free energy must be analytic. In addition, the expansion coefficients must be regular functions of the temperature. Close to Tc the free energy can be expanded in powers of the order parameter free energy F(Ψ) = order parameter. must be small a 2n Ψ 2n n=0 expansion coefficients are phenomenological parameters that depend on T and microscopics Order parameter must be small for the expansion to converge.