Lecture-08: Ferromagnets and Ising Model

Lecture-08: Ferromagnets and Ising Model 1 Ferromagnets and Ising models Magnetic materials contain molecules with individual magnetic moments, that tend to align the external magnetic field felt by the molecule. Magnetic moments of different molecules interact with each other. In many materials, the energy is lower when the moments align. A simple mathematical model for considering a number of particles with interacting moments is the Ising model, which describes the magnetic moments by Ising spins localized at the vertices of a certain region of a d-dimensional cubic lattice L. The cubic lattice L is determined by the vertices [L]d and the d d d d edges ((i, j) 2 [L] × [L] : ∑k=1 jik − jkj = 1) between the nearest neighbors. At each coordinate i 2 [L] , the configuration of a particle is an Ising spin si 2 X = f−1,1g. We have shown an example configuration of Ising spins for L = 5 and d = 2 in Figure 1. 1.1 Energy function Let N = Ld, then the N particle system configuration s is given by assigning the values of spins for each N N of the N particles as s = (s1,s2,...,sN). The space of configuration is X = f−1,1g . The energy of an N particle configuration s is given by E(s) = − ∑ sisj − B ∑ si. (1) (i,j) i2[L]d where the sum over (i, j) runs over all the unordered pairs of sites i, j 2 [L]d which are nearest neighbors and B is the applied external magnetic field. Determining the free energy density f (b) in the thermodynamic limit for this model is a non-trivial task. In 1924, Ernst Ising solved the d = 1 case and showed the absence of phase transitions. In 1948, Lars Onsager solved the d = 2 case, exhibiting the first soluble ‘finite-dimensional’ model with a second-order phase transition. In higher dimensions, the problem is unsolved, although many important features of the solution are well understood. 1.2 Temperature limits The two limiting cases that can be considered are at high and low temperatures. At high temperature when b ! 0, the energy no longer matters and the Boltzmann distribution is uniform over all configurations s 2 XN. That is, 1 m (s) = . b 2N At low temperature when b ! ¥, the Boltzmann distribution concentrates onto the ground state(s). In the absence of external magnetic field, i.e B = 0, the two degenerate ground states are given by, d d s+ = (si = 1 : i 2 [L] ), s− = (si = −1 : i 2 [L] ). If the magnetic field is set to some non-zero value, one of the two configuration dominates. The ground state is s+ if B > 0 and the ground state is s− if B < 0. To analyse this behavior upon the application of a magnetic field, We define a rescaled magnetic field x = bB, with b ! 0 or b ! ¥, keeping x fixed. With this, we will subsequently study some of the qualitative properties of the resultant model. 1 Figure 1: A configuration of a two-dimensional Ising model with L = 5. There is an Ising spin si on each vertex i, shown by an arrow pointing up if si = 1 and pointing down if si = −1. The energy (1) is given by the sum of two types of contributions: (i) a term −sisi for each edge (i, j) of the graph, such that the energy is minimized when the two neighboring spins si and sj point in the same direction; and (ii) a term −Bsi for each site i, due to the coupling to an external magnetic field. The configuration shown here has an energy −8 + 9B. Definition 1.1 (Expected Spin). Consider a ferromagnetic Ising model whose energy function is given by, E(s) = − ∑ sisj − B ∑ si. (ij) i2L Let us assume that the spins are non-interacting, then, the total energy of the system due to all possible configurations is given by, E(s) = −B ∑ si. (2) i2L The corresponding partition function is Z(b) = ∑exp(−bE(s)), s Z(b) = ∑ exp(bB ∑ si). (3) s1=s2=...=sN =±1 i2L The partition function for a single spin with energy E(si) = −Bsi is therefore, Zi(b) = ∑ exp(−bE(si)), si=±1 bB −bB Zi(b) = e + e , Zi(b) = 2cosh(bB). (4) At each site, the probability of an up spin or a down spin is given by Boltzmann’s distribution, exp(−bE(si)) P(si) = . (5) Zi(b) Therefore, the average value of a single spin in a region L is hsii = ∑ P(si)si, si=±1 ebB − e−bB hsii = , Zi(b) hsii = tanh(bB). (6) From Fig. 1, it can be observed that at low temperatures (b ! ¥) all the spins are either up or down 2 Figure 2: Variation of average spin in a region with temperature (b = 1/T) depending on the external magnetic field (B). However, at high temperatures (b ! 0) the spins are random due to which the average spin hsii ! 0. Summarising, the expected value of any spin in a region L, ( tanh(x), for b ! 0 hsii = tanh(Nx), for b ! ¥ Definition 1.2 (Average Magnetization). The extent of alignment in a region due to an external magnetic field (B) is given by average magnetization, 1 MN(b, B) = ∑ hsii N i2L MN(b, B) is an odd function of B due to the symmetry between up and down directions, as a consequence, MN(b,0) = 0. Definition 1.3 (Spontaneous Magnetization). The alignment in a region due to an external magnetic field can be analysed using spontaneous magnetization, M+(b) = lim lim MN(b, B). B!0+ N!¥ A few remarks about spontaneous magnetization, + + 1. At low temperatures (b ! ¥) with B ! 0 , the alignment of spins is s = (si = 1, 8i) due to which M+(¥) = 1 2. At high temperatures (b ! 0), the alignment of spins are random due to which M+(0) = 0 3. The critical temperature Tc = 1/bc is the one at which a transition in the phase of the system occurs 4. Spontaneous magnetization is always zero in the high temperature phase 8d (Paramagnetic Phase) 5. In one-dimensional systems (d = 1), a phase transition occurs at Tc = 0 =) M+(¥) = 0, 8b < bc 6. For d ≥ 2, the critical temperature is non-zero, and M+(b) > 0, 8b > bc(d) 3 1.3 The one-dimensional case Figure 3: Illustration of an one-dimensional model with sites and their respective spins (arrows) Consider a one-dimensional system (d = 1) of N spins, with energy (E(s)) N−1 N E(s) = − ∑ sisi+1 − B ∑ si. i=1 i=1 The partial partition function where the configurations of all spins s1,··· ,sp have been summed over, at fixed sp+1: p p zp(b, B,sp+1) = ∑ exp b ∑ sisi+1 + bB ∑ si . (7) s1,...,sp i=1 i=1 Expressing (7) recursively, p−1 p−1 zp(b, B,sp+1) = ∑exp bspsp+1 + bBsp ∑ exp b ∑ sisi+1 + bB ∑ si , sp s1,...,sp−1 i=1 i=1 zp(b, B,sp+1) = ∑ T(sp+1,sp)zp(b, B,sp), sp=±1 where T(sp+1,sp) := exp[bsp+1sp + bBsp] is a 2 × 2 transfer matrix: eb+bB e−b−bB T = . e−b+bB eb−bB The partition function of the system with N spins can be written using (7) as ZN(b, B) = ∑ zN−1(b, B,sN)exp(bBsN). sN Introducing the scalar product between two vectors as (a,b) = a1b1 + a2b2, the partition function can now be expressed in matrix form as, N−1 ZN(b, B) = (yL, T yR), (8) exp(bB) 1 where, y = and y = . The eigenvalues of the transfer matrix T are, L exp(−bB) R 1 q b 2b 2 −2b l1,2 = e cosh(bB) ± e sinh (bB) + e . (9) Let y1 and y2 be the corresponding eigenvectors, then, yR = u1y1 + u2y2. (10) 4 Using (9) and (10) in (8) we have, lN−1 0 u 0 y Z (b, B) = y , 1 1 1 , N L N−1 u 0 l2 0 2 y2 N−1 l1 u1 0 y1 ZN(b, B) = yL, N−1 , 0 l2 u2 y2 N−1 N−1 ZN(b, B) = u1(yL,y1)l1 + u2(yL,y2)l2 . (11) Definition 1.4 (Free Entropy Density). It is given by 1 f(b, B) = lim fN(b, B), N!¥ N 1 f(b, B) = lim log ZN(b, B) N!¥ N However, for finite b, in the large N limit, the partition function is dominated by the largest eigenvalue l1, and therefore f(b, B) = logl1. (12) Definition 1.5 (Expected Spin). Using the transfer matrix we can compute the expected value of a spin; hsii = ∑ P(si)si, s N−1 N si exp b ∑j=1 sjsj+1 + bB ∑j=1 sj = , Z(b) N−1 N ! = ∑ zi−1(b, B,i)si exp b ∑ sjsj+1 + bB ∑ sj , s1,s2,...,sN j=i j=i 1 i−1 N−i hsii = (yL, T sˆ T yR), (13) ZN(b, B) 1 0 where, sˆ = . 0 −1 Definition 1.6 (Average Magnetization). It can be computed by averaging over alignments in a region. We know that, 1 N MN(b, B) = ∑hsii, N i=1 Simplifying (13) and using (11), N N−1 N−1 ! 1 u1(yL,y1)l − u2(yL,y2)l M (b, B) = 1 2 , N N ∑ N−1 N−1 i=1 u1(yL,y1)l1 + u2(yL,y2)l2 In the thermodynamic limit, sinh(bB) 1 ¶f lim MN(b, B) = = (b, B).

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