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17.1 The classical Heisenberg model is the same as the XY model defined in (17.27), except that the vector 5, can rotate in three dimensions, instead of being confined to a plane. Put the partition function of the classical Heisenberg model in Landau form, by methods similar to that used for the XY model in the text. 17.2 The infinite range model is an Ising model in which each spin interacts with all other spins. Take the Hamiltonian to be

-.YfjkT = lS2 + hS,

S ==

where h = H/kT, H being the magnetic field. Show that for this model mean-field theory is exact in the limit N -> 00, by the following steps: (a) Write down the partition function, and use (17.14) to express it as an integral over an auxiliary field .p. The configuration sum can then be done. (b) Assume 1 = a/No Use the saddle-point method to evaluate the partition function in the limit N -> 00. Show that the saddle-point condition gives the mean-field theory as described by (14.41).

Consider mean-field theory with a cubic term in the Landau free energy:

saucer pattern in December, but it took several more months before the stock was actually ready to move higher.

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If your document requires an index, you can use Word to build one. Indexes can contain main entries and subentries as well as cross-references (entries that refer to other entries). Before you can build an index, you must mark any words or phrases in your document that should appear in the index. When you do, Word adds a special index field, called an XE field, to the document; this field includes the marked word or phrase, as well as any crossreference information you might have added.

1 7.4

Sketch the shape of l/J(rn) for various values of '0 to show how the lowest minimum m of '0' Show that there is a first-order phase transition at a certain value of '0' Find that value, and the discontinuity of m at the transition. In the Gaussian model the Fourier transform of the correlation function is f (k) = rex) in d-dimensional space. Show the following for d 2:

p=--

d-2 2 '

where K v is a Bessel function:

Kv(z)

{Xl dxe-zcoshXcosh(vx)

FIGURE 9.11 Building Materials (symbol BMHC) saw its volume explode and its price go into a steady channel after moving above its saucer breakout.

Hz/2)"r(v)

(ro *- 0)

The Current Page radio button is selected. To mark just this occurrence of the text, click Mark. To mark all occurrences of the text, click Mark All. Word adds an index entry field to your document.

The asymptotic behaviors of the correlation function are r(x)~C'lxl(3-d)/2 e-.fO x ,

17.5 Prove (17.47), which gives the correlation length for T > ~ and T < ~. 17.6 Derive the Ginsburg criterion for a tricritical point. What is the critical dimension in this case

We have learned that the critical properties of a system depend not on shortdistance details, but only on the nature of long-wavelength fluctuations. This suggests that one should do away with the irrelevant degrees of freedom by continuing the coarse-graining procedure (through which the details on an atomic scale had been averaged out) to ever larger distance scales, until one reaches the correlation length. Kadanoff* first introduced this idea in terms of "block spin" transformations in Ising models. The idea here is that near the critical point the spins should act in concert in large blocks. Thus the important degree of freedom are the average spins of the blocks, rather than the original individual spins. One should describe the system in terms of an effective Hamiltonian involving only the "block spins." Consider an Ising model in d dimensions, with spin configurations specified by {s} = {Sl' sz""}' and a dimensionless Hamiltonian (in units of kT) denoted by

(i,j)

SiS) -

(18.1)

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