It is allowed to work in groups, but each student must hand it in separately (without copying)
and include information about collaborators. A maximum of three students are allowed per
group.
You may use basic programming libraries where meaningful (e.g. general numerical routines),
as long as they do not implement the Ising model for you (everything we use in the exercise
solutions is OK).
For the hand-in, please include your code in a ffle separate from your report. We will not
grade the quality of your code (only what is written in the report), but you must submit it.
In total, 10 points can be achieved by solving the problems 1 and 2 below (see annotation),
which map directly onto your grade (in percent of the total course grade). The remaining 90
percent can be obtained from a second midterm exam (10 percent) and the oral exam (80
percent). Note: 2 extra credits can optionally be achieved by also solving problem 3.
Figure 1: (a) Schematic showing the basic lattice. Any given site i has a connection to sites i − 1
and i + 1, modulo the lattice size N. Only three sites i − 1, i, i + 1 are highlighted in the schematic
for presentation clarity. (b) An example where each node has connectivity four, that is, each node
has the links shown in (a) and two additional random links (exempliffed here only for one site i,
other random links are not shown for clarity of presentation, but exist for each site).
Ising Model
Consider the two different lattice geometries in Fig. 1a and b. The schematic in Fig. 1a
displays a one-dimensional periodic closed ring consisting of N sites, where each site has two
nearest neighbors. The schematic in Fig. 1b shows a modiffcation of the closed ring, where
each of the N sites on average has two additional links to random sites so that each site now
has an average of four nearest neighbors. Notice that these neighbors have to be kept ffxed
during the simulation of a given system size. Also, be aware that if site i is neighbor to site
j then site j is also neighbor to site i.
In each of these two cases place an electron of spin1
2
on
each site — modeled as our usual
Ising model with si = ±1. Let the energy of any conffguration of spins be described by the
familiar ferromagnetic zero-ffeld Ising model (J > 0, h = 0), i.e.
H = −J
X
⟨ij⟩
sisj
, (1)
where ⟨ij⟩ denotes any bond that exists between sites i and j in the geometries shown.
1. Implement the Ising model for both geometries. Using a Metropolis algorithm, plot
the order parameter ⟨si⟩ vs. T. How does the order parameter depend on T in each
case? You will need to simulate various values of T and you will need to ensure that
you have simulated for a sufffciently long time. 5pt
2. Explore possible phase transition behavior of each system. In particular, try to assess
if a critical temperature Tc can be found for either of these two cases. Keep in mind
that Tc relates to the behavior in the thermodynamic limit, so, besides choosing a
sufffciently long simulation time, you will need to extrapolate to inffnite system size
(i.e., 1/N → 0). Hint: Try plotting the absolute value of average magnetization |m|
vs. T for various N and visually inspect the plots. Optionally, you can even try to
automatize the identiffcation of Tc for each ffnite system and then plot Tc vs. 1/N, to
obtain a ffnite-size scaling. Try to interpret your ffndings w.r.t. the value of Tc in the
thermodynamic limit. 5pt
3. Extra credit: In the case of Fig. 1a, try to support your reasoning about Tc by discussing
the introduction of a domain wall and the associated free energy change. How would
this reasoning change if you modiffed Fig. 1a so that each sites i has two further links
to i − 2 and i + 2? 2pt
