= e-G(H,T)/kT =

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JV j(Dm)

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(17.1)

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where E[m, H], the effective Hamiltonian divided by kT, is a functional of the order parameter density m(x) and its conjugate field H(x):

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E[m, H]

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j(dx)1/;(m(x), H(x))

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(17.2)

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where (dx) is short for d dX . The dependence on temperature is suppressed.

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*L. D. Landau, Phys. Z. Sowjetunion 11, 26 (1937), reprinted in Collected Papers of L. D. Landau, D. ter Haar, ed. (Pergamon, London, 1965), p. 193; V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).

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THE LANDAU APPROACH

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We refer to 1/; as the "Landau free energy." In the neighborhood of a critical point, where m (x) is small, we can expand it in powers of m (x) and its derivatives:

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1/;(m(x),H(x)) = tlV'm(x)1 - kTm(x)H(x) +tYom2(x)

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+ som 3 (x) + u om 4 (x) +

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where V'm(x) is a d-dimensional vector whose components are am(x)jax. Terms not shown may involve higher powers of m(x) and its derivatives. The external field H(x) enters only linearly, because we assume it is very weak. We have assumed for simplicity that m( x) is a single real field. More generally it could have many components, or be complex. The coefficient of 1V'm (x) 12 is chosen to be t, to fix the scale of m (x). All other coefficients are phenomenological parameters that may depend on the temperature and the cutoff. In practice we simply choose them to suit our purpose. However, there are constraints to be observed. For example, the term linear in m(x) must have H(x) as a coefficient for the order parameter to be correctly given by - aG j aH. The coefficient So for the cubic term must vanish for systems invariant under reversal of m(x) when H = 0, such as a ferromagnet. It is customary to assume that U o is independent of T, while

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(17.4) Yo = aot, t = (T - Tc)jTc where a o is a positive constant. The integration in (17.1) is a functional integration which extends over all possible functional forms of m(x) that do not vary over an atomic distance a. The volume element (Dm) is defined only up to a factor, which is absorbed into a normalization constant JV in (17.1). Since it always cancels in ensemble averages, we need not specify it. We shall spell out in greater detail how one might actually do a functional integral in the next section. We now give an argument to justify the Landau theory. A more rigorous derivation for simple models will be given later. Assume we know how to express m ( x) in terms of the microscopic state of the system. We calculate the partition function by first summing over microscopic states holding the form of m (x) fixed, and then integrating over all possible forms of m(x):

(17 .5) The trace in the last integral denotes a sum-over-states with the functional form of m (x) held fixed: (Tre- /kT)m = W[m] e-<[ml/kT (17.6) where [m] is the energy of the system when m(x) has a particular functional form, and W[m] is the number of microscopic states having that energy. Denoting the "entropy" by S[m] = log W[m], we have

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E[m] =

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[m]

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S[m]

(17.7)

SPECIAL TOPICS IN STATISTICAL MECHANICS

From the physical interpretation of m(x) as a magnetic moment density, it is reasonable that the above can be expressed as a spatial integral over some function of m(x). The integrand is then the Landau free energy. But why do we choose to hold fix m (x), instead of any other variable, while performing the microscopic average The underlying assumption is that the order parameter is slow to come to thermal equilibrium, long after all other degrees of freedom have become thermalized. In this sense, it is similar to the macroscopic variables of hydrodynamics. To have a completely macroscopic theory, the cutoff A must somehow disappear from final physical answers. Since 1/A is of microscopic scale, the obvious thing to do is to take the limit A ~ 00. That cannot be done in a straightforward manner, however, because it produces divergences in the free energy, (except in the mean-field approximation discussed later.) The correct procedure for taking such a limit involves "renormalization," a scheme originally invented to circumvent divergences of a similar nature in quantum electrodynamics. This will be the subject of discussion in 18. That the inspiration for renormalization came from quantum electrodynamics was not accidental. The Landau theory for d = 4 is equivalent to a relativistic quantum field theory. * More precisely, the partition function with a spatially varying external field H(x) corresponds to the generating functional for the" vacuum Green's functions" of a relativistic quantum field theory in the presence of an external source H( x). Thus the theory has practical application for d = 1,2,3,4:

d= 2: d= 3: