Chapter 5 - WebRing

CHAPTER 5. MAGNETIC SYSTEMS 255
and set H = 0. For simplicity, the initial orientation of the spins is all spins up.
(a) Choose T = 10 and run until equilibrium has been established. Is the orientation of the spins
random such that the mean magnetization is approximately equal to zero? What is a typical
size of a domain, a region of parallel spins?
(b) Choose a low temperature such as T = 0.5. Are the spins still random or do a majority
choose a preferred direction? You will notice that M ≈ 0 for sufficient high T and M = 0
for sufficiently low T. Hence, there is an intermediate value of T at which M first becomes
nonzero.
(c) Start at T = 4 and determine the temperature dependence of the magnetization per spin m,
the zero-field susceptibility χ, the mean energy E, and the specific heat C. (Note that we have
used the same notation for the specific heat and the heat capacity.) Decrease the temperature
in intervals of 0.2 until T ≈ 1.6, equilibrating for at least 1000mcs before collecting data at
each value of T. Describe the qualitative temperature dependence of these quantities. Note
that when the simulation is stopped, the mean magnetization and the mean of the absolute
value of the magnetization is returned. At low temperatures the magnetization can sometimes
flip for small systems so that the value of 〈|M|〉 is a more accurate representation of the
magnetization. For the same reason the susceptibility is given by
χ = 1 2 2
〈M 〉−〈|M|〉
kT
, (5.103)
rather than by (5.17). A method for estimating the critical exponents is discussed in Problem
5.41.
(d) Set T = Tc ≈ 2.269 and choose L ≥ 128. Obtain 〈M〉 for H = 0.01, 0.02, 0.04, 0.08, and 0.16.
Make sure you equilibrate the system at each value of H before collecting data. Make a log-log
plot of m versus H and estimate the critical exponent δ using (5.99).
(e) Choose L = 4 and T = 2.0. Does the sign of the magnetization change during the simulation?
Choose a larger value of L and observe if the sign of the magnetization changes. Will the sign
of M change for L ≫ 1? Should a theoretical calculation of 〈M〉 yield 〈M〉 = 0 or 〈M〉 = 0
for T < Tc?
∗ Problem 5.14. Ising antiferromagnet
Sofarwehaveconsideredthe ferromagneticIsingmodelforwhichtheenergyofinteractionbetween
two nearest neighbor spins is J > 0. Hence, all spins are parallel in the ground state of the
ferromagnetic Ising model. In contrast, if J < 0, nearest neighbor spins must be antiparallel to
minimize their energy of interaction.
(a) Sketch the ground state of the one-dimensional antiferromagnetic Ising model. Then do the
same for the antiferromagnetic Ising model on a square lattice. What is the value of M for
the ground state of an Ising antiferromagnet?