(dx)lV'm(x)1 ex

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(Volume of wall)

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(16.64)

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wall

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We used the estimate lV'ml2 ex I/L 2 on the grounds that L is the only relevant length in the problem. By the same token, the thickness of the wall is also proportional to L. Thus, the volume of the wall is proportional to L d, and we conclude that (16.65) Reasoning as in the previous case, we conclude that entropy favors domain creation for d :$ 2. Therefore the lower critical dimension is d = 2 for continuous systems, in agreement with an earlier result. None of the arguments presented above are rigorous. But they give us physical insight into the importance of dimensionality. Rigorous arguments in special models support these conclusions. Hohenberg has shown that the superfluid order parameter must vanish in d::s; 2. Using a similar argument, Mermin

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*F. Bloch, Z. Phis. 61,206 (1930); C. H. Herring and C. Kittel, Phys. Rev. 81, 869 (1950).

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and Wagner proved that in d:s: 2 the Heisenberg model can exhibit neither ferromagnetic nor antiferromagnetic order. *

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Dimensional Reductiont

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Experimentally it is possible to create metals containing randomly distributed magnetic impurities that are out of thermal equilibrium with the host system, with a very long relaxation time. This is called "quenched" randomness (as opposed to "annealed" randomness, which obtains when the impurities are in thermal equilibrium with the system). The host system sees the impurities as sources of external fields, which vary from point to point at random. Experimentally the nonequilibrium status can be shown by checking the concentration of the impurities against known equilibrium values. The randomness of the impurities can be demonstrated by X-ray diffraction. Suppose a system in the presence of a quenched random external field magnetizes spontaneously. Since the random field varies from point to point, there will be regions in which it would pay for the magnetization to reorient itself, so as to lower the interaction energy in the local external field. Thus a domain will be formed, provided the domain wall does not cost too much energy. To find the energy balance, assume that the random field has Gaussian fluctuations. The value of the external field averaged over a domain will then be of order {ii, where N is the number of impurities in the domain. For a domain of linear size L, we have N a: L d. Hence the volume energy is proportional to - L d / 2 Adding this to the energy of the domain wall, which is proportional to L d - 2 as before, we obtain the energy cost of one domain:

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(16.66)

where c1 ' c2 are positive constants. The lower critical dimension is obtained by equating the powers of the two terms:

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d-2=

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. .d = 4

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(16.67)

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For d < 4 the energy is negative for sufficiently large L, and hence domain formation is energetically favored. For d = 4, the negative term also dominates for sufficiently large L, because it is a volume energy instead of surface energy. The result suggests that a system in a quenched random magnetic field in d dimensions behaves like a system without magnetic field in d - 2 dimensions. In other words, the presence of a quenched random field reduces the effective

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*P. C. Hohenberg, Phys. Rev. 158,383 (1967); N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17,1133 (1966). tY. Imry and S. K. Ma, Phys. Rev. Lett. 35,1399 (1975).

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dimensionality by 2.* However, experiments indicate that this behavior is not universal, and we have yet to understand the phenomenon more fully.t

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16.1 Consider a fluid near its critical point, with isotherms illustrated qualitatively in the sketch. For T ~ T" assume that the Gibbs free energy of the fluid has a singular part satisfying the scaling form

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