1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 647

K

K

1

2

.P

Fig. 16.19 Example curves defined by u 1 (K 1 ,K 2 ). The critical curve defined by
u 1 (K 1 ,K 2 ) = 0 is shown in black and iterates to the critical fixed pointP.

16.13Problems

16.1. Consider a block spin transformation of the one-dimensional Ising model with

h 6 = 0, in which every other spin is summed over. Such a procedure is also

called adecimationprocedure.

a. Write down the transformation operatorTfor this transformation, and

show that the transformation leads to a value ofb= 2.

b. Derive the RG equation for this transformation, and find the fixed points.

c. Sketch the RG flow in the (x,y) plane. What is the nature of the fixed

points, and what do they imply about the existence of a critical point?

16.2. A general class of models for magnetic systems, called Potts models, allow,

in general,Ndiscrete states for each spin variable on the lattice. The one-

dimensional three-state Potts model is defined as follows: each siteiis a ‘spin’

which may take on three values, 1, 2, or 3. The Hamiltonian is given by

H=−J

∑

i

δsi,si+1.

Use the same decimation procedure as in problem 16.1, derive the RG equa-

tion, and find the fixed points. Does this model have a critical point?