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.� � � � � VII. Continuous Spins at Low Temperatures VII.A The non-linear σ-model Previously we considered low temperature expansions for discrete spins (Ising, Potts, etc.), in which the low energy excitations are droplets of incorrect spin in a uniform back-ground selected by broken symmetry. These excitations occur at small scales, and are easily described by graphs on the lattice. By contrast, for continuous spins, the lowest energy excitations are long-wavelength Goldstone modes, as discussed in section II.C. The thermal excitation of these modes destroys the long-range order in dimensions d ≤ 2. For d close to 2, the critical temperature must be small, making low temperature expansions a viable tool for the study of critical phenomena. As we shall demonstrate next, such an approach requires keeping track of the interactions between Goldstone modes. Consider unit n-component spins on the sites of a lattice, i.e. ~s (i) = (s1, s2, ··· , sn), with |~s (i)| 2 = s 21 + ··· + s 2 n = 1. (VII.1) The usual nearest neighbor Hamiltonian can be written as −βH = K � ~s (i) · ~s (j) = K � 1 − (~s (i) −2 ~s (j))2 . (VII.2) hiji hiji At low t emperat ures, the fluctuations between neighboring spins are small and the differ-ence in eq.(VII.2) can be replaced by a gradient. Assuming a unit latti ce spacing, −βH = −βE0 − Kdd x (∇~s (x))2 , (VII.3) 2 where the discrete index i has been replaced by a conti nuous vector x d . A cutoff ∈ ℜof Λ ≈ π is thus implicit in eq.(VII.3). Ignoring the ground stat e energy, the partition function is K Z = D � ~s (x)δ � s(x)2 − 1 �� e−2 dd x(∇�s )2 . (VII.4) A possible ground state configuration is ~s (x) = (0, , 1). There are n −1 Goldstone ··· modes describing the transverse fluctuations. To examine the effects of these fluctuations close to zero temperature, set ~s (x) = (π1(x), , πn−1(x), σ(x)) ≡ (~π (x), σ(x)) , (VII.5) ··· 114� � � � � � � � � � � � � � where ~π (x) is an n −1 component vector. The unit length of the spin fixes σ(x) in terms of ~π (x). For each degree of freedom d~s δ(s 2 − 1) = ∞ d~π dσδ � π2 + σ2 − 1 � � −∞ � �� �� �� ∞∞ d~π = d~π dσδ σ − 1 − π2 σ +1 − π2 =2√1 − π2 , −∞ −∞ (VII.6) where we have used the identity δ(ax) = δ(x)/ a . Using this result, the partition function | |in eq.(VII. 4 ) can be written as K Z ∝ � 1D−~π (πx() x)2 e−2 dd x (∇�π )2+(∇√1−π2 )2 � ��� �� (VII.7) KK �� �2 ρ = D~π (x) exp − dd x 2(∇~π )2 +2 ∇ 1 − π2 +2ln(1 − π2) . In going from the lattice to the continuum, we have introduced a density ρ = N/V = 1/ad of lat tice points. For unit lattice spacing ρ = 1, but for the purpose of renormalization we shall keep an arbitrary ρ. Whereas the original Hamiltonian was quite simple, the one describing the Goldstone modes ~π (x), is rather complicated. In selecting a particular ground sta te, the rotational symmetry was broken. The nonlinear terms in eq.(VII.7) ensure that this symmetry is properly reflected when considering only ~π. We can expand the nonlinear terms for the effective Ha miltonian in powers of ~π (x), resulting in a series βH[~π (x)] = βH0 + U1 + U2 + , (VII.8) ··· where βH0 = Kdd x(∇~π )2 , (VII.9) 2 describes independent Goldstone modes, while U1 = dd x K 2(~π · ∇~π )2 − ρ 2 π2 , (VII.10) is the first order perturbation when the terms in the series are organized according to powers of T = 1/K. Since we expect fluctuations π2 ∝ T , βH0 is order of one, the two 115� � � � � � � � � �> terms in U1 are o rder of T ; remaining terms are order o f T2 and higher. In the language of Fourier modes, K dd q 2 2βH0 =2 (2π)d q |~π (q)| , K ddq1 ddq2 ddq3 U1 = − 2 (2π)3d πα(q1) πα(q2) πβ(q3) πβ(−q1 − q2 − q3) (q1 · q3) − ρ 2 (2dπdq )d |~π (q)| 2 . (VII.11) For the non-interacting (quadratic) theory, the correlation functions of the Goldstone modes are δα,β(2π)dδd(q + q ′) hπα(q)πβ(q′)i0 = Kq2 . (VII.12) The resulting fluctuations in real space behave as � π( x)2� = ddq � ~π (q)2� =(n − 1) 1/a ddq 1 =(n − 1) Kd � a2−d − L2−d � .0 (2π)d | | 0 K 1/L (2π)d q2 K (d − 2) (VII.13) For d > 2 the fluctuations are indeed proportional to T . However, for d ≤ 2 they diverge as L → ∞. This is a consequence of the Mermin–Wagner theorem on the absence of long range order in d ≤ 2. Polyakov (1975) argued that this implies a critical temperature Tc ∼ O(d − 2) for such systems, and that an RG expansion in powers of T may provide a systematic way to explore critical behavior close to two dimensions. To construct a perturbative RG, consider a spherical Brillouin zone of radius Λ, and divide the modes as ~π (q) = ~π<(q) + ~π>(q). The modes ~π< involve momenta 0 < |q| < Λ/b, while we shall integrate over the short wavelength fluctuations ~π> with moment a in the shell Λ/b < q < Λ. To order of T , t he coarse–grained Hamiltonian is given by | |� � � � � � ��>< < < >β˜H ~π = V δfb 0 + βH0 ~π + U1 ~π + ~π0 + O(T2), (VII.14) where h i> indicates averagi ng over ~π> . The term proportional to ρ in eq.( VII.11) results 0 in two contributions, one is a constant addition to free energy (from (π>)2 ), a nd the other is simply ρ(π<)2 . (The cross terms proportional to ~π< ~π> vanish by symmetry.) · The quartic part of U1 generates 1 6 terms. Nontrivial contributions arise from products of two ~π< and two ~π> . There are