13.9 Exercises 453
the initial configuration consist of all spins pointing up, i.e.,sk= 1. Compute the mean energy
and magnetization for each cycle and find the number of cyclesneeded where the fluctuation
of these variables is negligible. What kind of criterium would you use in order to determine
when the fluctuations are negligible?
Change thereafter the initial condition by letting the spins take random values, either− 1 or
1. Compute again the mean energy and magnetization for each cycle and find the number of
cycles needed where the fluctuation of these variables is negligible.
Explain your results.
- Letmcs≥ 1000 and compute〈E〉,〈E^2 〉andCVas functions ofTfor 0. 1 ≤T≤ 5. Plot the results
and compare with the exact ones for periodic boundary conditions.
- Using the Metropolis sampling method you should now find the number of accepted config-
urations as function of the total number of Monte Carlo samplings. How does the number of
accepted configurations behave as function of temperatureT? Explain the results.
- Compute thereafter the probabilityP(E)for a system withN= 50 atT= 1. Choosemcs≥ 1000
and plotP(E)as function ofE. Count the number of times a specific energy appears and build
thereafter up a histogram. What does the histogram mean?
13.4.Here we will simulate the two-dimensional Ising model.
- Assume that the number of spins in thexandydirections are two, vizL= 2. Find the closed-
form expression for the partition function and the corresponding mean values forE,M, the
capacityCVand the suceptibilityχas function ofTusing periodic boundary conditions.
- Write your own code for the two-dimensional Ising model with periodic boundary conditions
and zero external fieldB. SetL= 2 and compare your numerical results with the closed-form
ones from the previous exercise. usingT= 0. 5 andT= 2. 5. How many Monte Carlo cycles
do you need before you reach the exact values with an unceertainty less than1%? What are
most likely starting configurations for the spins. Try both an ordered arrangement of the
spins and a randomly assigned orientations for both temperature. Analyse the mean energy
and magnetisation as functions of the number of Monte Carlo cycles and estimate how many
thermalization cycles are needed.
- We will now study the behavior of the Ising model in two dimensions close to the critical
temperature as a function of the lattice sizeL×L, withLthe number of spins in thexand
ydirections. Calculate the expectation values for〈E〉and〈M〉, the specific heatCVand the
susceptibilityχas functions ofTforL= 10 ,L= 20 ,L= 40 andL= 80 forT∈[ 2. 0 , 2. 4 ]with
a step in temperature∆T= 0. 05. Plot〈E〉,〈M〉,CVandχas functions ofT. Can you see an
indication of a phase transition?
- Use Eq. (13.5) and the exact resultν= 1 in order to estimateTCin the thermodynamic limit
L→∞using your simulations withL= 10 ,L= 20 ,L= 40 andL= 80.
- In the remaining part we will use the exact resultkTC/J= 2 /ln( 1 +
√
2 )≈ 2. 269 andν= 1.
Determine the numerical values ofCV,χandMat the exact valueT=TCforL= 10 ,L= 20 ,
L= 40 andL= 80. Plotlog 10 Mandχsom funksjon avlog 10 Land use the scaling relations in
order to determine the constantsβandγ. Are your log-log plots close to straight lines? The
exact values areβ= 1 / 8 andγ= 7 / 4.
- Make a log-log plot using the results forCVas function ofLfor your computations at the
exact critical temperature. The specific heat exhibits a logarithmic divergence withα= 0 ,
see Eqs. (13.1) and (13.3). Do your results agree with this behavior? Make also a plot of the
specific heat computed at the critical temperature for the given lattice.
The exact specific heats behaves as
CV≈−^2
π
(
2 J
kBTC
) 2
ln
∣∣
∣∣ 1 −T
TC
∣∣
∣∣+const.
Comment your results.
