Correlation function in 1D Ising model
Consider the Ising model in one dimension with periodic boundary condition and
with zero external field. Evaluate <i> by using the transfer matrix method. What
is <i> at T=0? Find the correlation function G(r)=<1r+1> and show that in the
limit of infinite sample (N∞) it has the form G(r)~ exp(r/At what
temperature  diverges and what is its significance?
Ising with long range interaction:
Consider the Ising model of magnetism with long range interaction: the energy of a
spin configuration is given by E = (J/2N)i,j sisj  hi si where J>0, and the sum
is on all i and j, not restricted to nearest neighbors. The energy E in terms of
m=isi/N can be written as E(m,h)= (1/2)JNm2  hNm. Explain why N is
included in the definition of the coupling J/N. Evaluate the free energy F(m;T,h)
assuming that it is dominated by a single m. Hence m becomes a variational
parameter. By minimizing F find m(h,T) and a critical temperature Tc. Plot
qualitatively m(h) above and below the transition. Plot qualitatively F(m) for T>Tc
and T<Tc with both h=0 and h≠0. Explain the meanings of the various extrema.
Expand F(m; T, h=0) up to order m4. What is the meaning of the m2 coefficient?
Ising model with disorder
 1, where i=1,2,3,...,N and
Consider a one dimensional Ising model of spins i=+
N+1=1. Between each two spins there is a site for an additional atom, which if
present changes the coupling J to J(1). The Hamiltonian is then
H = J SUM ii+1(1ni), where ni=0 or 1. There are N' = SUM ni atoms, so
that N' < N. Evaluate the partition sum by allowing all configurations of spins and
of atoms. If the atoms are stationary impurities one needs to evaluate the free
energy F for some given random configuration of the atoms: Then one can average
F over all configurations. Evaluate the averaged F. Find the entropy difference
between the two results and explain its origin.
Mean field: ferromagnetism with classical spins
Apply the mean field approximation to the classical spin-vector model
H = J SUM si · sj  h · SUM si where si is a unit vector and i,j are neighboring
sites on a lattice. The lattice has N sites and each site has z neighbors. Find the
magnetization M=<cosi> in the mean field approximation, where i is the angle
relative to an assumed orientation of M. Find the transition temperature Tc by
solving for M at h=0. Find M(T) for T<Tc to lowest order in TcT and identify the
exponent  in M~(TcT). Of what order is the transition? Find the susceptibility
(T) at T>Tc and identify the exponent in ~(TcT).
Mean Field: antiferomagnetism
Antiferromagnetism is a phenomenon akin to ferromagnetism. The simplest kind
of an antiferromagnet consists of two equivalent antiparallel sublattices A and B
such that memebers of A have only nearest neighbors in B and vice versa. Show
that the mean field theory of this type of (Ising) antiferromagnetism yields a
formula like the Curie-Weiss law for the susceptibility ~(TTc)1, except that
TTc is replaced by T+Tc where Tc is the transition temperature into
antiferromagnetism (Neel's temperature). Below Tc the susceptibility of an
antiferromagnet drops again. Show that in the above mean field theory the rate of
increase of immediately below Tc is twice the rate of decrease immediately
above. Assume that the applied field is parallel to the antiferromagnetic
orientation.
Mean Field: ferroelectricity
Consider electric dipoles on sites of a simple cubic lattice which point along one of
the crystal axes ±p<100>. The interaction between dipoles is
p1.p2  3(p1.r)(p2.r)/r2
U=
40r3
where r is the distance between the dipoles, r=|r|, and 0 is the dielectric constant.
Assume nearest neighbor interactions and find the ground state configuration.
Consider either ferroelectric (parallel dipoles) or anti-ferroelectric alignment (antiparallel) between neighbors in various directions. Develop a mean field theory for
the ordering and for the average polarization P(T) at a given site at temperature T.
Find the critical temperature Tc and the susceptibility at T>Tcassuming an
electric field in the <100> direction.
Mean Field with field h(r)
Consider a ferromagnet with magnetic moments m(r) on a simple cubic lattice
interacting with their nearest neighbors. The ferromagnetic coupling is J and the
lattice constant is a. Extend the mean field theory to the situation that the
magnetization is not uniform but is slowly varying: Find the mean field equation in
terms of m(r) and its gradients (to lowest order). Assume an external magnetic
field h(r), that in general can be a function of r. Consider T>Tc where Tc is the
critical temperature, so that only lowest order in m(r) is needed. For a small h(r)
find the response m(r), and evaluate it explicitly in two limits: (I) The response is
characterized by the susceptibility for uniform h, and (II) the response is
characterized by the correlation function (why?) for h(r)~3(r). In the latter case
identify the correlation length.
Ferromagnetism for cubic crystal
A cubic crystal which exhibits ferromagnetism at low temperatures, can be
described near the critical temperature Tc by an expansion of a Gibbs free energy
1
3
G(H,T) = G0 + 2 rM2 + uM4 + v Mi4  H.M
i=1
where H=(H1,H2,H3) is the external field, and M=(M1,M2,M3) is the total
magnetization, and r=a(TTc). The other parameters, namely G0 and a>0 and u>0
and v, are independent of H and T. The constant v is called the cubic anisotropy,
and can be either positive or negative. At H=0 find the possible solutions of M
which minimize G, and the corresponding expressions for G(0,T). These solutions
are characterized by the magnitude and direction of M. Show that the region of
stability is u+v>0. Determine the stable equilibrium phases when T<Tc for the
different cases v>0 and u<v<0. Show that there is a second order phase transition
at T=Tc, and determine the critical indices ,  and  for this transition. These are
defined by the expressions CV,H=0 ~ |TTc| for both T>Tc and T<Tc, and
|M|H=0 ~ (TcT) for T<Tc and ij = (∂Mi/∂Hj) ~ ij |TTc| for T>Tc.
Mechanical model for symmetry breaking
The following mechanical model illustrates the symmetry breaking aspect of
second order phase transitions. An air tight piston of mass M is inside a tube of
cross sectional area a. The tube is bent into a semicircular shape of radius R. On
each side of the piston there is an ideal gas of N atoms at a temperature T. The


volume to the right of the piston is aR(2  ) while to the left is aR( 2 + ). The
free energy of the system has the form
F(T, = MgRcos  NkBT [2+ ln

aR(2  )
N3

+ ln
aR(2  )
N3
]
Explain the terms in F. Interpret the minimum condition for F() in terms of the
pressures in the two chambers.
Expand F up to 4th order in . Show that there is a symmetry breaking transition
and find the critical temperature Tc. Describe what happens to the phase transition
if the number of atoms on the left and right of the piston is N(1) and N(1)
respectively. Note that it is sufficient to consider ||<<1 and include a term ~ in
the expansion.
At a certain temperature the left chamber (containing N(1+) atoms) is found to
contain a droplet of liquid coexisting with its vapor. Which of the following
statements may be true at equilibrium: (a) The right chamber contains a liquid
coexisting with its vapor; (b) The right chamber contains only vapor; (c) The right
chamber contains only liquid.
M

R
a