ISING MODEL Lecture 12 Page 1 of 100
Page 2 of 100 Ernst Ising (1900-1996) Ernst Ising was born May 10, 1900 in Cologne, Germany. In 1919 he began studying mathematics and physics at the University of Gottingen, and continued his studies in Bonn and Humburg. In Humburg his teacher, Wilhelm Lenz, suggested to E.Ising that he turned to theoretical physics and he began investigating a model of ferromagnetism which Lenz had introduced in 1920. E.Ising graduated with a Ph.D. in physics in 1924. In his doctoral thesis he studied the spacial case of a linear chain of magnetic moments, which are only able to take two positions, up and down, and which are coupled by interactions between nearest neighbors, a model which came to be known as the famous Ising Model. E.Ising got a teaching position at a high school,but was dismissed when Hitler came into power.till 1938, he was a teacher and headmaster of a Jewish school. In 1938, this school was destroyed by the Nazis and E.Ising and his wife traveled to Luxembourg with the plan to emigrate to the US. After the Germans invaded Luxembourg in in 1940, for the next four years, they worked at menial jobs, struggling to survive. They finally got to the US in 1947 and from 1948 till 1976 E.Ising was Professor of Physics at Bradley University, Peotria (IL). It was not until 1949 that E.Ising found out from the scientific literature that his model had become widely known. Today, each year about 800 papers are published that used the it Ising Model. S.Kobe, Ernst Ising - Physicist and Teacher. J. Stat. Phys., 88, pp.991-995 (1997).
Paramagnetism. Paramagnetism refers to materials which become magnetized in a magnetic field. Their magnetism disappears when the field is removed. Page 3 of 100 B = 0 B 0 µ i α B U i = µ i B cos(α) M=Σ µ i B
Thermodynamics. If V - the system volume and N - number of particles remain constant, the total differential of the entropy S is defined by: ds = du T + B T dm, Page 4 of 100 where T is the temperature, U is the internal energy, B is the magnetic field and M is the magnetization. Let U depends only on T: U = U(T). In this case: (1/T) M = 0, and so: (B/T) = 0. U It follows that the magnetization is defined by M = f(b/t). It is convenient to write the equation of state of a paramagnetic in the form: M = M L( µ B k B T ), where M = N µ.
Paul Langevin (student of Pierre Curie), Magnétisme et Théorie des électrons, Annales de Chimie et de Physique 8e série 5, 70-127 (1905). U = µ B i cos(α i ) Page 5 of 100 The magnetization of a configuration is defined by: M = µ i cos(α i ) The mean value of the magnetization is given by:... cos(α j ) exp( µ B k B T cos(α i))dcos(α i ) j < M >= µ... exp( µ B k B T cos(α i))dcos(α i ) cos(α) exp( µ B k < M >= µ N B T cos(α))dcos(α) exp( µ B k B T cos(α))dcos(α) [ < M >= µ N cth( µ B k B T ) k ] BT µ B The magnetization is defined by: < M >= M L( µ B k B T ).
The Langevin function is defined by: L(x) = cth(x) 1 x. 1.5 1 0.5 L(x) 0 0.5 Page 6 of 100 1 1.5 30 20 10 0 10 20 30 x Langevin function. L(x) x 3, x 1; L(x) 1 1 x, x Curie law. x = µ B k B T, if x 1: χt M B T 1 T, where χ T is the magnetic susceptibility.
Ferromagnetism. Ferromagnetism refers to materials that can retain their magnetic properties when the magnetic field is removed. Pierre Weiss extended Langevin s theory of paramagnetism to ferromagnetism. P.Weiss, L Hypothése du champ moléculaire et la propriété ferromagnétique. Journal de Phisique et le Radium, 6, 661-690 (1907). Internal-field hypothesis : B tot = B + c M, where c is the Weiss coefficient which depend on a material. Page 7 of 100 Weiss preserved Langevin s equation, but he replaced the external field B by B tot. The magnetization is defined by the equation: M = M L( µ (B + c M)). k B T
Graphic analysis. x = µ (B + c M). k B T k B T x 1 B = L(x) c µ M c M 0.6 0.4 B 0 B=0 B=0 0.2 y 0 Page 8 of 100 0.2 0.4 0.6 3 2 1 0 1 2 3 x
Page 9 of 100 If B = 0 (x = µ c M) in the vicinity of the point x = 0: k B T k B T c µ M x = L (0) x + 1 6 L (0) x 3. The trivial solution is x 0 = 0 (M = 0). k B T x 2 0 = 6 c µ M L (0). L (0) The Langevin function is asymmetric and L (0) < 0, therefore the nontrivial solution exist if: T < T C = L (0) µ M c k B, where T C is the Curie temperature. 1 0.9 0.8 0.7 0.6 M 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T / T C
In the frame of the Weiss approximation the susceptibility is defined by: 1 χ T T T c. This is the Curie-Weiss law. 0.5 0.4 C / N k B 0.3 0.2 Page 10 of 100 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 T / T c Figure 1: The specific heat of a Weiss ferromagnet as a function of temperature. The magnetization is defined by: M M (T C T) 1 2.
The only ferromagnetic elements are iron (Fe), cobalt (Co), nickel (Ni), gadolinium (Gd), and dysposium (Dy) (in addition, there are many composite substances). In general, the spontaneous magnetization is defined by: M M (T C T) β. Element Curie temp., K β Fe 1043 0.33 0.37 Co 1388 0.33 0.37 Page 11 of 100 Ni 627 0.33 0.37 Gd 293 0.33 0.37 Dt 85 0.33 0.37 β = 1 and is independent on a material. 3
Wilhelm Lenz, A contribution to the Understandig of Magnetic Phenomena in solid Materials.Physicalische Zeitschrift, 21, 613-615 (1920). In the frame of the model a magnetic dipole was allowed to perform turnovers. Two directions were singled out: α = 0 and α = π. If an external magnetic field of strength B is applied along the axis the energy of the system is defined by: U = µ B i σ i, where σ i = ±1. Page 12 of 100 The magnetization is given by: For µ B k B T < M >= M e µ B k BT e µ B k B T 1, this becomes: that is, the Curie Law. e µ B k BT + e µ B k B T < M >= µ M B = M th( µ B k B T ). 1 k B T,
Ising (Lenz-Ising) model. Particle locations are rigidly fixed in the lattice. The interaction energy between two particles is postulated to be: { J σi σ U ij = j if i and j are nearest neighbors 0 otherwise. Page 13 of 100 The energy of the configuration is defined by: U = 1 N 2 J N γ ij σ i σ j µ B σ i, i,j=1 i=1 where: { 1 if i and j are nearest neighbors γ ij = 0 otherwise.
Page 14 of 100 The properties of the system can be determined from the partition function: Z = ( N 1... exp 2 K N γ ij σ i σ j + µ b σ i ), σ 1 =±1 σ N =±1 where K = J k B T and b = B k b T. i,j=1 The value σ = 1 corresponds to a spin state with the spin in some preassigned direction, and σ = 1 corresponds to a spin in the opposite direction. The spin is considered to be either up or down. The energy per particle is given by: < U > N = k BT 2 N ln(z) T i=1 and the magnetization per particle by: < M > N = 1 N ln(z). b
Binary alloys. Page 15 of 100 The grand partition function is defined by: Ξ = ( µ1 n 1 + µ 2 n 2 U ) m exp, k b T n 1 +n 2 =N m where m is the number of configurations for fixed n i, and U m is the energy of the system in the state m. The energy is given by: where: U m = 1 2 N γ ij v(σ i, σ j ), i,j=1 v(σ i, σ j ) = 1 4 ɛ 11 (1+σ i ) (1+σ j )+ 1 4 ɛ 22 (1 σ i ) (1 σ j )+ 1 2 ɛ 12 (1 σ i σ j ).
Page 16 of 100 The grand partition function of alloy is reduced to the partition function for the Ising problem : with: Ξ = e N α k BT Z µ B = 1 ( ) 4 2 µ 1 2 µ 2 q (ɛ 11 ɛ 22 ), J = 1 ( ) 4 ɛ 11 + ɛ 22 2 ɛ 12 ), α = 1 ( ) 8 4 µ 1 + 4 µ 2 q (ɛ 11 + ɛ 22 + 2 ɛ 12 ) Here q is the number of nearest neighbors. and < n 1 > N = 1 2 (1 + M µ ) < n 2 > N = 1 2 (1 M µ ) n 1 n 2 N Experiment data shows β 0.33. M (T c T) β.
Lattice gas. Home Page Page 17 of 100 For lattice gas ɛ 12 = 0 and ɛ 22 = 0. The interaction potential between particles is defined by: + if the two particles occupy the same site, φ ij = -4 J if the two particles are nearest neighbors, 0 otherwise. The density is given by: ρ = 1 2 (1 + M µ ). ρ l = 1 2 (1 + M µ ) and ρ g = 1 2 (1 M µ ). Experiment data shows β 1 3. ρ l ρ g M (T c T) β. It is believed that the critical exponent is independent on specific characteristics such as the form of interactions. This phenomenon is called universality.
Mean-field approximation. Z = σ 1 =±1... σ N =±1 where K = J k B T and b = B k b T. ( N 1 exp 2 K i,j=1 γ ij σ i σ j + µ b N σ i ), i=1 σ 2 σ 1 σ 3 σ 5 Page 18 of 100 σ 4 N σ i γ ij σ j = σ i q < σ >. j=1 The partition function is defined by: Z =... exp (( q N 2 K < σ > +µ b) σ i ). σ 1 =±1 σ N =±1 i
The partition function: [ ( Z = exp ( q N. i)] 2 K < σ > +µ b) σ σ 1 =±1 Z = [ ch( q 2 K < σ > +µ b) ] N. The magnetization per particle: < σ >= 1 N ln(z) ( q J = µ th b 2 < σ > k B T + µ B ) k B T The mean-field statistical approximation is equivalent to Internalfield hypothesis of Weiss. Page 19 of 100 The mean-field approximation is exact for the model: Z = σ 1 =±1... σ N =±1 ( 1 exp 2 K N N σ i σ j ). i,j=1 This approximation should be exact in the infinite dimension space.
Page 20 of 100 1D Ising model. The partition function for the linear chain of N spins with periodic boundary condition σ N+1 = σ 1 : Z = σ 1 =±1... σ N =±1 ( N exp [ 1 2 K σ i σ i+1 + µ b ) 2 (σ i + σ i+1 )], i=1 where K = J k B T and b = B k b T. Z =... P(σ 1, σ 2 ) P(σ 2, σ 3 )... where: σ 1 =±1 σ N 1 =±1 σ N =±1 ( 1 P(σ, σ ) = exp 2 K σ σ + µ b ) 2 (σ + σ ), P(σ N 1, σ N ) P(σ N, σ 1 ) We can consider P(σ, σ ) as a matrix element of the following matrix: ( ) ( ) P = exp 1 2 K + µ b exp 12 ( ) ( K ) exp 12 K 1 exp 2 K µ b P(σ N 1, σ N ) P(σ N, σ 1 ) = P 2 (σ N 1, σ 1 ). σ N =±1
Page 21 of 100 The partition function: Z = σ 1 =±1 ) Z = Tr (P N = ) P N (σ 1, σ 1 ) = Tr (P N ( Tr(P)) N = (λ1 + λ 2 ) N. where λ 1 and λ 2 are eigenvalues of the matrix P. ( ) ( ) 1 exp 2 K + µ b λ exp 12 K ( ) ( ) exp 12 K 1 exp 2 K µ b λ = 0 ( ) λ 1,2 = e K ch(µ b) ± sh 2 (µ b) + e 4 K Hence: ln(z) lim N N = ln(λ 1) and the magnetisation is defined by: M M = sh( µ B k B T ) sh 2 ( µ B k B T ) + exp( 4 J k B T ). One-dimensional Ising model does not exhibit ferromagnetism. E.Ising, Beitrag zur Theorie des Ferromagnetismus. Zeitschrift ürphisik 31, 252-258 (1925).
2D Ising model. Rudolf Peierls had proved that spontaneous magnetisation appears in the two-dimensional case. R.E.Peierls, On Ising s model of ferromagnetism. Proc. of the Cambridge Phil. Soc. 32, 477-481 (1936). Page 22 of 100 The total energy is J k B T N l+2 J k B T L, where N l is the number of links on the lattice and L is the length of the perimeter. R.Peierls concluded that the number of magnets enclosed by boundaries decreases with decreasing temperature. It was shown that for sufficiently low temperatures, less than half of the magnets are enclosed by boundaries, that is, the system displays spontaneous magnetization.
Kramers-Wennier duality. H.A.Kramers, G.H.Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I.Phys. Rev., 60, 252-262 (1941). 6 5 4 3 2 1 1 2 3 4 5 6 Page 23 of 100 The low temperature expansion : k Z = e J N q2 B T Q(k) e 2 J k B T k At high temperatures: Z = σ 1 =±1... k σ N =±1 n.n. e J k B T σ i σ j. The product over i and j corresponds to nearest neighbors. e K σ i σ j = ch(k) (1 + th(k) σ i σ j ) Z = 2 N ch N q 2 ( J k B T ) S(k) th k ( J k B T ). k
In the case of square lattice: Q(k) = S(k). Hence the partition function of one lattice at a high temperature is related to the partition function of its dual at a low temperature and vice versa. If x = e 2 J k BT the transformation is given by: x 1 x 1 + x. The fixed point of this transformation is defined by: Page 24 of 100 The solution of this equation: x C = 1 x C 1 + x C.. J x C = e 2 k B T C = 2 1 The exact value of the critical temperature (Curie point) is: k B T C J = 2.2691
Larse Onsager had derived the partition function for zero-field 2D Ising model in a closed form. L.Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition. Phys. Rev. 65, 117-149 (1944). The free energy is given by: F ( N k B T = ln J ) ch(2 k B T ) + 1 π 2 π where: 0 ( sh k 2 = 2 ( ch 2 J k B T J k B T 1 ( ln[ 1+ 2 )] 1 k 2 sin 2 (ψ) dψ, ) ). Page 25 of 100 C.N.Yang, The Spontaneous Magnetization of a Two-Dimensional Ising-Model. Phys. Rev. 85, 808-816 (1952). where β = 1 (exact result). 8 M (T C T) β,
3D Ising model. Up to now the three-dimensional Ising model with or without external magnetic field is an unsolved problem. Low-temperature and high-temperature expansions of the partition function are used to study thermodynamics properties. Although the expansions were not expected to give reliable values in the neighborhood of the transition point, it was thought possible to estimate properties of the critical point by various extrapolation methods. M (T C T) β. Page 26 of 100 The early estimations give 0.303 β 0.318 (β = 5 16?). The improved result β = 0.327. The experimental result β 0.33.
Ensemble averaging. Ensemble averages are defined by: < A >= Ω A(X) ρ(x) dx, where ρ(x) is the probability density and X = {q i }. The Gibbs distribution function is defined by: ρ(x) = const exp( U(q i}) k B T ). Page 27 of 100 The Monte Carlo method in the statistical theory is used to evaluate numerically the average of any functional A (to develop a quadrature formula): < A > 1 M M j=1 A(X j ) Nodes (configurations X i ) are independent of a functional and are defined only by the probability density.
Monte Carlo method. 1 0.8 ρ (x) 0.6 0.4 0.2 0 1 0 1 2 3 4 5 6 x Page 28 of 100 The Metropolis algorithm (random walks) ( N.Metropolis et al, J. Chem. Phys., v.21, p.1087-1092 (1953)). Two states i and j are linked by a transition probability w(i j) which is the probability of going from state i to state j. The transition probability satisfies the sufficient (unnecessary) condition of detailed balance (microscopic reversibility): ρ(x i ) w(i j) = ρ(x j ) w(j i) The simplest solution of this condition is defined by: w(i j) = min [ 1, ρ(x ] j ρ(x i
Page 29 of 100 - Choose a spin at random. - Imagin you flip this spin to its opposite state to give a new configuration of the system. - Calculate the energy change U. - If U < 0, accept the new configuration. - If U > 0, accept the new configuration if e U k BT > z, where z is a random number between 0 and 1. Otherwise retain the original configuration. - Choose another spin of the lattice at random and repeat the above steps. U i = s i (J S + µ B)
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G.F.Newell, E.W.Montroll. On the Theory of the Ising Model of Ferromagnetism. Rev. Mod. Phys., 25, 353-389 (1953). S.G.Brush. History of the Lenz-Ising Model. Rev. Mod. Phys., 39, 883-893 (1967). M.Niss. History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena. Arch. Hist. Exact Sci., 59, 267-318 (2005). M.E.Fisher. The Nature of Critical Points. Lectures in Theoretical Physics, v. YIIc. Univ. of Colorado Press, USA (1965) Page 31 of 100