Hjorth-Jensen M. Computational Physics
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11.3. SELECTED RESULTS FOR THE ISING MODEL 199
n_spins ;
o f i l e < < s e t i o s f l a g s ( ios : : showpoint | ios : : uppercase ) ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < temp ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < Eaverage / n_spins / n_spins ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < Evariance / temp / temp ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < Maverage / n_spins / n_spins ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < Mvariance / temp ;
o f i l e < < setw (15) < < se t p r e c i s ion (8) < < Mabsaverage / n_spins / n_spins
< < endl ;
} / / end output f unct io n
11.3 Selected results for the Ising model
11.3.1 Phase transitions
The Ising model in two dimensions and with undergoes a phase transition of second
order. What it actually means is that below a given critical temperature , the Ising model ex-
hibits a spontaneous magnetization with . Above the average magnetization is zero.
The one-dimensional Ising model does not predict any spontaneous magnetization at any finite
temperature. The physical reason for this can be understood from the following simple consid-
eration. Assume that the ground state for an
-spin system in one dimension is characterized by
the following configuration
which has a total energy and magnetization . If we flip half of the spins we arrive at a
configuration
with energy and net magnetization zero. This state is an example of a disordered
state. The change in energy is however too small to stabilize the disordered state. In two di-
mensions however the excitation energy to a disordered state is much higher, and this difference
can be sufficient to stabilize the system. In fact, the Ising model exhibits a phase transition to a
disordered phase both in two and three dimensions.
For the two-dimensional case, we move from a phase with finite magnetization
to
a paramagnetic phase with at a critical temperature . At the critical temperature,
quantities like the heat capacity
and the susceptibility diverge in the thermodynamic limit,
i.e., with an infinitely large lattice. This means that the variance in energy and magnetization
diverge. For a finite lattice however, the variance will always scale as
, being e.g.,
the number of configurations which in our case is proportional with . Since our lattices will
always be of a finite dimensions, the calculated or will not exhibit a diverging behavior.
200 CHAPTER 11. MONTE CARLO METHODS IN STATISTICAL PHYSICS
We will however notice a broad maximum in e.g., near . This maximum, as discussed
below, becomes sharper and sharper as
is increased.
Near
we can characterize the behavior of many physical quantities by a power law behav-
ior. As an example, the mean magnetization is given by
(11.27)
where
is a so-called critical exponent. A similar relation applies to the heat capacity
(11.28)
and the susceptibility
(11.29)
Another important quantity is the correlation length, which is expected to be of the order of
the lattice spacing for
. Because the spins become more and more correlated as
approaches , the correlation length increases as we get closer to the critical temperature. The
divergent behavior of near is
(11.30)
A second-order phase transition is characterized by a correlation length which spans the whole
system. Since we are always limited to a finite lattice,
will be proportional with the size of the
lattice.
Through finite size scaling relations it is possible to relate the behavior at finite lattices with
the results for an infinitely large lattice. The critical temperature scales then as
(11.31)
with
a constant and is defined in Eq. (11.30). The correlation length is given by
(11.32)
and if we set
one obtains
(11.33)
(11.34)
and
(11.35)
11.3.2 Heat capacity and susceptibility as functions of number of spins
in preparation
11.4. OTHER SPIN MODELS 201
11.3.3 Thermalization
in preparation
11.4 Other spin models
11.4.1 Potts model
11.4.2 XY-model
11.5 Physics project: simulation of the Ising model
In this project we will use the Metropolis algorithm to generate states according to the Boltzmann
distribution. Each new configuration is given by the change of only one spin at the time, that is
. Use periodic boundary conditions and set the magnetic field .
a) Write a program which simulates the one-dimensional Ising model. Choose
, the
number of spins
, temperature and the number of Monte Carlo samples
. Let the initial configuration consist of all spins pointing up, i.e., .
Compute the mean energy and magnetization for each cycle and find the number of cycles
needed 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
or
. 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.
b) Let
and compute , and as functions of for . Plot
the results and compare with the exact ones for periodic boundary conditions.
c) Using the Metropolis sampling method you should now find the number of accepted con-
figurations as function of the total number of Monte Carlo samplings. How does the num-
ber of accepted configurations behave as function of temperature ? Explain the results.
d) Compute thereafter the probability
for a system with at . Choose
and plot as function of . Count the number of times a specific energy
appears and build thereafter up a histogram. What does the histogram mean?
Chapter 12
Quantum Monte Carlo methods
12.1 Introduction
The aim of this chapter is to present examples of applications of Monte Carlo methods in studies
of quantum mechanical systems. We study systems such as the harmonic oscillator, the hydrogen
atom, the hydrogen molecule, the helium atom and the nucleus He.
The first section deals with variational methods, or what is commonly denoted as variational
Monte Carlo (VMC). The required Monte Carlo techniques for VMC are conceptually simple,
but the practical application may turn out to be rather tedious and complex, relying on a good
starting point for the variational wave functions. These wave functions should include as much as
possible of the inherent physics to the problem, since they form the starting point for a variational
calculation of the expectationvalueof the hamiltonian
. Givena hamiltonian and a trial wave
function , the variational principle states that the expectation value of , defined through
(12.1)
is an upper bound to the ground state energy of the hamiltonian , that is
(12.2)
To show this, we note first that the trial wave function can be expanded in the eigenstates of the
hamiltonian since they form a complete set, viz.,
(12.3)
and assuming the set of eigenfunctions to be normalized, insertion of the latter equation in
Eq. (12.1) results in
(12.4)
203
204 CHAPTER 12. QUANTUM MONTE CARLO METHODS
which can be rewritten as
(12.5)
In general, the integrals involved in the calculation of various expectation values are multi-
dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be ad-
equate for say the computation of the energy of a many-body system. The fact that we need to
sample over a multi-dimensional density and that the probability density is to be normalized by
the division of the norm of the wave function, suggests that e.g., the Metropolis algorithm may
be appropriate.
We could briefly summarize the above variational procedure in the following three steps.
1. Construct first a trial wave function
, for say a many-body system consisting of
particles located at positions . The trial wave function depends on
variational parameters .
2. Then we evaluate the expectation value of the hamiltonian
3. Thereafter we vary according to some minimization algorithm and return to the first step.
The above loop stops when we reach the minimum of the energy according to some specified
criterion. In most cases, a wave function has only small values in large parts of configuration
space, and a straightforward procedure which uses homogenously distributed random points in
configuration space will most likely lead to poor results. This may suggest that some kind of
importance sampling combined with e.g., the Metropolis algorithm may be a more efficient way
of obtaining the ground state energy. The hope is then that those regions of configurations space
where the wave function assumes appreciable values are sampled more efficiently.
The tedious part in a VMC calculation is the search for the variational minimum. A good
knowledge of the system is required in order to carry out reasonable VMC calculations. This is
not always the case, and often VMC calculations serve rather as the starting point for so-called
diffusion Monte Carlo calculations (DMC). DMC is a way of solving exactly the many-body
Schrödinger equation by means of a stochastic procedure. A good guess on the binding energy
and its wave function is however necessary. A carefully performed VMC calculation can aid in
this context.
12.2 Variational Monte Carlo for quantum mechanical sys-
tems
The variational quantum Monte Carlo (VMC) has been widely applied to studies of quantal
systems. Here we expose its philosophy and present applications and critical discussions.
12.2. VARIATIONAL MONTE CARLO FOR QUANTUM MECHANICAL SYSTEMS205
The recipe, as discussed in chapter 4 as well, consists in choosing a trial wave function
which we assume to be as realistic as possible. The variable stands for the spatial coordinates,
in total if we have particles present. The trial wave function serves then, following closely
the discussion on importance sampling in section 9.5, as a mean to define the quantal probability
distribution
(12.6)
This is our new probability distribution function (PDF).
The expectation value of the energy
is given by
(12.7)
where
is the exact eigenfunction. Using our trial wave function we define a new operator, the
so-called local energy,
(12.8)
which, together with our trial PDF allows us to rewrite the expression for the energy as
(12.9)
This equation expresses the variational Monte Carlo approach. For most hamiltonians,
is a
sum of kinetic energy, involving a second derivative, and a momentum independent potential.
The contribution from the potential term is hence just the numerical value of the potential.
At this stage, we should note the similarity between Eq. (12.9) and the concept of importance
sampling introduced in chapter 4, in connection with numerical integration and Monte Carlo
methods.
In our discussion below, we base our numerical Monte Carlo solution on the Metropolis
algorithm. The implementation is rather similar to the one discussed in connection with the Ising
model, the main difference residing in the form of the PDF. The main test to be performed is
a ratio of probabilities. Suppose we are attempting to move from position to . Then we
perform the following two tests.
1. If
where is the new position, the new step is accepted, or
2.
where is random number generated with uniform PDF such that , the step is
also accepted.
206 CHAPTER 12. QUANTUM MONTE CARLO METHODS
In the Ising model we were flipping one spin at the time. Here we change the position of say a
given particle to a trial position
, and then evaluate the ratio between two probabilities. We
note again that we do not need to evaluate the norm
1
(an in general impossible
task), since we are only computing ratios.
When writing a VMC program, one should always prepare in advance the required formulae
for the local energy
in Eq. (12.9) and the wave function needed in order to compute the ratios
of probabilities in the Metropolis algorithm. These two functions are almost called as often as a
random number generator, and care should therefore be exercised in order to prepare an efficient
code.
If we now focus on the Metropolis algorithm and the Monte Carlo evaluation of Eq. (12.9), a
more detailed algorithm is as follows
Initialisation: Fix the number of Monte Carlo steps and thermalization steps. Choose an
initial
and variational parameters and calculate . Define also the value of the
stepsize to be used when moving from one value of to a new one.
Initialise the energy and the variance.
Start the Monte Carlo calculation
1. Thermalise first.
2. Thereafter start your Monte carlo sampling.
3. Calculate a trial position
where is a random variable .
4. Use then the Metropolis algorithm to accept or reject this move by calculating the
ratio
If , where is a random number , the new position is accepted, else
we stay at the same place.
5. If the step is accepted, then we set .
6. Update the local energy and the variance.
When the Monte Carlo sampling is finished, we calculate the mean energy and the standard
deviation. Finally, we may print our results to a specified file.
The best way however to understand a specific method is however to study selected examples.
12.2.1 First illustration of VMC methods, the one-dimensional harmonic
oscillator
The harmonic oscillator in one dimension lends itself nicely for illustrative purposes. The hamil-
tonian is
(12.10)
1
This corresponds to the partition function in statistical physics.
12.2. VARIATIONAL MONTE CARLO FOR QUANTUM MECHANICAL SYSTEMS207
where is the mass of the particle and is the force constant, e.g., the spring tension for a
classical oscillator. In this example we will make life simple and choose
. We
can rewrite the above equation as
(12.11)
The energy of the ground state is then . The exact wave function for the ground state is
(12.12)
but since we wish to illustrate the use of Monte Carlo methods, we choose the trial function
(12.13)
Inserting this function in the expression for the local energy in Eq. (12.8), we obtain the following
expression for the local energy
(12.14)
with the expectation value for the hamiltonian of Eq. (12.9) given by
(12.15)
which reads with the above trial wave function
(12.16)
Using the fact that
we obtain
(12.17)
and the variance
(12.18)
In solving this problem we can choose whether we wish to use the Metropolis algorithm
and sample over relevant configurations, or just use random numbers generated from a normal
distribution, since the harmonic oscillator wave functions follow closely such a distribution. The
latter approach is easily implemented in few lines, namely
208 CHAPTER 12. QUANTUM MONTE CARLO METHODS
. . . i n i t i a l i s a t i o n s , decl a rati o n s of v ar i abl e s
. . . mcs = number of Monte Carlo samplings
/ / loop over Monte Carlo samples
for ( i =0; i < mcs ; i ++) {
/ / generate random variable s from gaussian d i s t r i b u t i o n
x = normal_random (&idum ) / sq rt 2 / alpha ;
local_energy = alpha alpha + x x (1 pow( alpha ,4 ) ) ;
energy += local_energy ;
energy2 += local_energy local_energy ;
/ / end of sampling
}
/ / write out the mean energy and the standard d ev ia ti on
cout < < energy / mcs < < s q rt (( energy2 /mcs (energy / mcs ) 2) / mcs ) ) ;
This VMC calculation is rather simple, we just generate a large number of random numbers
corresponding to the gaussian PDF and for each random number we compute the local
energy according to the approximation
(12.19)
and the energy squared through
(12.20)
In a certain sense, this is nothing but the importance Monte Carlo sampling discussed in chapter
4. Before we proceed however, there is an important aside which is worth keeping in mind when
computing the local energy. We could think of splitting the computation of the expectation value
of the local energy into a kinetic energy part and a potential energy part. If we are dealing with
a three-dimensional system, the expectation value of the kinetic energy is
(12.21)
and we could be tempted to compute, if the wave function obeys spherical symmetry, just the
second derivative with respect to one coordinate axis and then multiply by three. This will
most likely increase the variance, and should be avoided, even if the final expectation values are
similar. Recall that one of the subgoals of a Monte Carlo computation is to decrease the variance.
Another shortcut we could think of is to transform the numerator in the latter equation to
(12.22)