MIT OpenCourseWare http://ocw.mit.edu 8.334 Statistical Mechanics II: Statistical Physics of Fields Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.Z         VI.H Critical behavior of the two-dimensional Ising model To understand the singularity in the free energy of the two dimensional Ising model in eq.(VI.73), we start with the simpler expression obtained by the unrestricted sum over phantom lo ops in sec.VI. F. Specializing eq.(VI.53) to d = 2, fG = ln  2 cosh2 K  − 21 dq(2xπdq)2 y ln [1 − 2t (cos qx + cos qy)] . (VI.75) Apart from the argument of the logarithm, this expression is simila r t o the exact result. Is it possible that such similar functional forms lead to distinct singular behaviors? The singularity results from the vanishing of the argument of the lo garithm at tc = 1 /4. In the vicinity of this point we make an expansion as in eq.(VI.50 ), AG(t, q) = (1 − 4t) + tq2 + O(q 4) ≈ tc q 2 + 4 δt , (VI.76) tc where δt = tc − t. The singular part of eq.(VI.75) can be obtained by focusing of the behavio r of the integrand as q 0, and replacing the sq uare Brillouin zone for the range → of the integral with a circle of radius Λ ≈ 2π, fsing. =1 Z Λ 2πqdq ln q 2 + 4 δt − 2 0 4π2 tc (VI.77) 1  δt   q2 + 4δt/tc Λ = − 8πq 2 + 4 tc ln e 0 . Only the expression evaluated at q = 0 is singular, and 1 δt δt fsing. = ln . (VI.78) 2π tc tc The resulting heat capacity, CG ∝ ∂2fG/∂2t, diverges as 1/δt. Since eq.(VI.75) is not valid for t > tc, we cannot obtain the behavior of heat capacity on the low temperature side. For the exa ct result of eq.(VI.73), the argument of the logarithm is 2 2A ∗ (t, q) =  1 + t2 − 2t  1 − t (cos qx + cos qy) . (VI.79) The minimum value of this expression, for q = 0, is 2 22 2A ∗ (t, 0) =  1 + tc 2 − 4tc  1 − t2 c  =  1 − tc 2 + 4tc − 4tc  1 − t2 c  =  1 − tc − 2tc 2 . (VI.80) 111 " #     Since this expression (and hence the argument of the lo garithm) is always non–negative, the integral exists for all values of t. As required, unlike eq.(VI.75), the exact result is valid at all temperatures. There is a singularity when the argument vanishes for 2tc + 2tc − 1 = 0, = ⇒ tc = −1 ±√2. (VI.81) The positive solution describes a ferromagnet, and leads to a value of Kc = ln √2 + 1 /2, in agreement with the duality arguments of sec.VI.D. Setting δt = t − tc, and expanding eq.(VI.79) in the vicinity of q 0 gives → A ∗ (t, q) ≈ [(−" 2tc − 2)δt]2 + t# c(1 − t2 c)q 2 + ··· 2 ≈2tc 2 q 2 + 4 δt . (VI.82) tc The important difference from eq.( VI.76) is that (δt/tc) appears at quadratic order. Fol-lowing the steps in eqs.(VI.77) and (VI.78), the singular part of the free energy is ln Z  =1 Z Λ 2πqdq ln q 2 + 4  δt 2 N  sing. 2 0 4π2 tc " ! #Λ =1 q 2 + 4  δt 2 ln  q2 + 4(δt/tc)2 (VI.83) 8π tc e 0  2   1 δt  δt  = ln + analytic terms.− π tc tc The heat capacity is obtained by taking two derivatives and diverges as C(δt)sing. = A± ln δt . The logarithmic singularity corresponds to the l imit α = 0; the peak is sym-| |metric, characterized by the amplitude ratio A+/A− = 1. The exact partition function of the Ising model on the square lat tice was originally calculated by Lars Onsager in 1944 (Phys. Rev. 65, page 117). Onsager used a 2L × 2L transfer matrix to study a lat tice of width L. He then identified various symmetries of this matrix which allowed him to diagonalize it and obtain the l argest eigenvalue as a function of L. For any finite L, this eigenvalue is non-degenerate as required by Frobenius’s theorem. In the limit L → ∞, the top two eigenvalues become degenerate at Kc. The result in this limit naturally coincides w ith eq.(VI.74). Onsager’s paper is quite long and complicated, and regarded as a tour de force of mathematical physics. A somewhat streamlined version of this solution was developed by B. Kaufman (Phys. Rev. 76, 1232 (1949)) and is 112more or less reproduced in chapter 15 of Huang. This exact solution, for the first time, demonstrated that critical behavior at a phase transition can in fact be quite different from predictions of Landau (mean–field) t heory. It took almost three decades to reconcile the two results by the renormalizati on group. The graphical method presented in this section was originally developed by Kac and Ward (Phys. Rev. 88, 1332 ( 1 952)). The main ingredient of the derivation is the result that the correct accounting of the paths can be achieved by including a factor of (-1) for each intersection. (This conjecture by Feynman is proved in S. Sherman, J. Math. Phys. 1, 202 (19 60).) The change of sign is reminiscent of the exchange factor between fermions. Indeed, Schultz, Mattis, and Lieb (Rev. Mod. Phys. 36, 8 56 (1964)) describe how the Onsager transfer matrix can be regarded as a Hamiltonian for the evolution of non-interacting fermions in one dimension. The Pauli exclusion principle prevents intersections of the world-lines of these fermions in two dimensional space-time. In addition to the partiti on function, the correlation functions hσiσji can a lso be calculated by summing over paths (see Itzykson and Drouffe, Statistical field theory: 1). Since the combination q2 + 4(δt/tc)2 in eq.(VI.82) describes the behavior of these random walks, we expect a correlation length ξ ∼ |tc/δt|, i.e. diverging with an exponent ν = 1 on both sides of the phase tra nsition with an amplitude ratio of unity. The exponents α and ν are relat ed by the hyperscaling identity α = 2 − 2ν. The critical correlati ons at tc are more subtle and decay as hσiσjic ∼ 1/|i − j|η, with η = 1/4. Integrating the