3.3 From Micro to Macro: The Bottom-Up Approach
85
K =
1
2
cosh
−1 (e
2K ′
)
ξ(k) =
1
2
ln 2 +
1
2
K
′ +
1
2
ξ(K
′ )
(3.67)
The one-dimensional Ising chain is not the most interesting magnetic system to
study with renormalization theory, since it is known from the exact solution that
there is no phase transition, the chain is disordered at any finite temperature. The
two-dimensional Ising lattice does have an order/disorder phase transition, nicely
reproduced with the renormalization procedure as discussed in Refs. [15, 16]. Such
phase transition does not exist in a two-dimensional lattice described with the Heisenberg Hamiltonian. For this model, a non-zero interaction along the third dimension is
needed to have an ordered (anti-)ferromagnetic system at finite temperature as stated
by the Mermin-Wagner theorem.
Monte Carlo simulations: An alternative strategy to calculate thermodynamic properties is to explicitly follow the trajectory of a magnetic system by a computer simulation of the system. Along such trajectory, the system will adopt many conformations
with different energy, magnetization and other microscopic observables. If the sampling of the conformational space is done correctly, a good estimate of the partition
function can be made and with this all type of thermodynamic functions can be
calculated.
There are basically two types of simulations to sample the conformational space.
The first one is known as Molecular Dynamics and propagates a system in time by
integrating the Newton’s equations of motion. In its most rudimentary form the procedure can be described as follows. For a given set of atomic positions r (t = t 0 ),
one calculates the forces and from these the velocities v(t 0 ), accelerations a(t 0 ) and
usually some higher derivatives. The atoms are then moved from r (t 0 ) to r (t 0 + ∆t)
by the formula r (t 0 + ∆t) = r (t 0 ) + v(t 0 )∆t + (1/2)a(t 0 )∆t 2 + ...and the time is
updated from t 0 to t 0 + ∆t. Then the cycle is repeated as long as one wants to follow the trajectory. The second method, the so-called Monte Carlo method, does not
propagate the system in time but rather performs a random walk through the conformational space to calculate the partition function. Whereas numerical integration on
a regular grid is much more efficient for low-dimensional functions, such approach
is absolutely out of the question for extremely high dimensional functions, such as
the partition function for any interesting N -particle system. In these cases a smart
random walk is more effective and can be used to extract macroscopic properties as
function of microscopic interactions.
To illustrate the procedure, we come back to the Ising model, but now focusing on
the two-dimensional lattice with nearest neighbour interactions only. The sampling of
the conformational space is usually done with the Metropolis algorithm, which starts
by creating the initial spin conformation S 0 . This can be done in many ways, one of
them is assigning a random spin direction M S =±
1
2 to each lattice point as shown
in the left part of Fig. 3.11. After calculating the energy of this spin distribution, a
trial step in conformational space is taken by inverting the spin at one of the lattice
85
K =
1
2
cosh
−1 (e
2K ′
)
ξ(k) =
1
2
ln 2 +
1
2
K
′ +
1
2
ξ(K
′ )
(3.67)
The one-dimensional Ising chain is not the most interesting magnetic system to
study with renormalization theory, since it is known from the exact solution that
there is no phase transition, the chain is disordered at any finite temperature. The
two-dimensional Ising lattice does have an order/disorder phase transition, nicely
reproduced with the renormalization procedure as discussed in Refs. [15, 16]. Such
phase transition does not exist in a two-dimensional lattice described with the Heisenberg Hamiltonian. For this model, a non-zero interaction along the third dimension is
needed to have an ordered (anti-)ferromagnetic system at finite temperature as stated
by the Mermin-Wagner theorem.
Monte Carlo simulations: An alternative strategy to calculate thermodynamic properties is to explicitly follow the trajectory of a magnetic system by a computer simulation of the system. Along such trajectory, the system will adopt many conformations
with different energy, magnetization and other microscopic observables. If the sampling of the conformational space is done correctly, a good estimate of the partition
function can be made and with this all type of thermodynamic functions can be
calculated.
There are basically two types of simulations to sample the conformational space.
The first one is known as Molecular Dynamics and propagates a system in time by
integrating the Newton’s equations of motion. In its most rudimentary form the procedure can be described as follows. For a given set of atomic positions r (t = t 0 ),
one calculates the forces and from these the velocities v(t 0 ), accelerations a(t 0 ) and
usually some higher derivatives. The atoms are then moved from r (t 0 ) to r (t 0 + ∆t)
by the formula r (t 0 + ∆t) = r (t 0 ) + v(t 0 )∆t + (1/2)a(t 0 )∆t 2 + ...and the time is
updated from t 0 to t 0 + ∆t. Then the cycle is repeated as long as one wants to follow the trajectory. The second method, the so-called Monte Carlo method, does not
propagate the system in time but rather performs a random walk through the conformational space to calculate the partition function. Whereas numerical integration on
a regular grid is much more efficient for low-dimensional functions, such approach
is absolutely out of the question for extremely high dimensional functions, such as
the partition function for any interesting N -particle system. In these cases a smart
random walk is more effective and can be used to extract macroscopic properties as
function of microscopic interactions.
To illustrate the procedure, we come back to the Ising model, but now focusing on
the two-dimensional lattice with nearest neighbour interactions only. The sampling of
the conformational space is usually done with the Metropolis algorithm, which starts
by creating the initial spin conformation S 0 . This can be done in many ways, one of
them is assigning a random spin direction M S =±
1
2 to each lattice point as shown
in the left part of Fig. 3.11. After calculating the energy of this spin distribution, a
trial step in conformational space is taken by inverting the spin at one of the lattice