Selected Topics in Solid State Physics – Exercise 2
1. Consider a one-dimensional ring, made of a lattice with N sites. A random walker
(which moves with equal probabilities one step to the right or left) is released at an
arbitrary site, which we denote by i=0. The probability to find the walker at site n after t
time steps is denoted by P(n,t,N), and we use P(n,t,N)=P(n+N,t,N)=P(n,t,N).
a) Show that P(n,t+1,N)-P(n,t,N)=[P(n-1,t,N)+P(n+1,t,N)]/2.
b) Show that the solutions can be written in the form P(n, t , N )  exp[t ] cos[kn] , and
find the possible values of k and  (k ) . Starting with P(n,0, N )   (n) , find the full
solution for P(n,t,N).
c) Formulate a scaling Ansatz for P(n,t,N), in terms of two scaled variables, and discuss
the limits of short times (short with respect to what?) and long times (long with respect to
what?).
d) Specifically, find the “equilibrium” solution P(n, , N ) , and show that it is approached
with a correction of order exp[ t /  ] . How does  depend on N? Does this behavior
obey your scaling Ansatz?
2. The spin-spin correlation function for the Ising model for a sample of linear size L at
temperature T is defined as G(r , T , L)   S 0 S r    S 0  S r  .
a) Replacing the variable T by the correlation length of the infinite sample at that
 (T )   0 | T  Tc | 
temperature,
,
assume
the
scaling
Ansatz
G(r ,  , L)  b  R(r / b,  / b, L / b) , and find the exponent  . What can you deduce from
this by choosing b=r? b=L? b   ? When is each of these choices justified?
b) Use the above Ansatz to find similar scaling expressions for the Fourier transform of
G, and also for the susceptibility.
3. (a) Formulate the mean field theory for the Ising model with a general spin S
(meaning that the spin on each site can have the (2S+1) values -S, -S+1, …, S-1, S). In
particular, obtain the self-consistent equation which determines the average
magnetization, M.
(b) Expand the above equation in powers of M, and find the critical temperature Tc ,
M(T) at zero field and M(H) at T  Tc . Deduce the relevant critical exponents. Do they
depend on S?
(c) Repeat the above analysis in the limit of infinite S, when each spin can have any
value between 1 and -1, with equal probabilities.
4. The free energy within the Landau theory for an Ising ferromagnet has the form
F
1
a (T  Tc ) M 2  BM 4  CM 6  HM .
2
(a) For C=0, Find the susceptibilities at H=0 for T  Tc and for T  Tc . Show that both
diverge as 1 / | T  Tc | , and find the ratio of the amplitudes in front of this factor. For
the same case, show that for small M, H and t  T  Tc one can write a scaling form,
M ( H , t )  b u M (b y H , b x t ).
What are the values of the exponents x, y and u? Express them in terms of  ,  ,  and  .
(b) For B<0, show that the model yields a first order transition, in which the
magnetization jumps from 0 to a finite value. Express the line of temperatures at which
this transition happens in terms of B and C. Hint: compare the free energies of the two
solutions.
(c) At B=0, the line of critical points (2nd order transition) meets the line of first order
transitions. This point is called the tricritical point. Find M(T) at this special point. Find
also the susceptibility at this point, and deduce the exponents  and  . Assuming that
the exponent  is still equal to ½ (explain?), use the Ginzburg criterion for deducing the
upper critical dimension for this special point.