MonteIsing

Overview

Run

To run open root and run .x compileMacro.C. You can try the examples loading them with .L.

Lattice Class

The class can build a multi dimensional square lattice of spins.

$H = - \sum_{\langle i, j\rangle}^{}J_{ij} \sigma_i \sigma_j$ $J_{ij} = 1 \text{ and } \mu = 1$

DrawLattice Class

The class can draw 2D or 3D lattice from the class Lattice.

Cooling 2D 1 Cooling 2D 2 Cooling 2D 3 Cooling 2D 4

1000000 spin 2D model at $T \approx 0.5$ . Each frame is the flipping of 1000 spins. First video goes from 0 to 20000th iterations, then each gif continues with the next 20000. Notice that periodic boundary conditions during the islands formation.

Structure of the Lattice

The lattice is a multidimensional grid. We store it in a one dimensional array, which means that we have to choose a convention. The lattice is periodical, so we can imagine a torus for a 2D lattice…

Torus

…and so on…

Torus 4D

Imagine the vector of spins wrapped on itself as many times as the dimension of the problem. Here, a 3D representation:

3D Lattice

It’s necessary to remember these assumptions when measuring quantities such as energy on the lattice.

Multidimensional Neighbour Interactions and Counting

Since the lattice is a mono dimensional bool * we need a method to count the neighbours of each spin.

Here are examples of neighbour counting for mono dimensional and bi dimensional lattice.

1D Model

2D model

Reduntant terms were omitted: (i // N) * N is zero when in max dimension, also, (i + 1) % N (i - 1 -N) % N may be reduntant.

We can easily see a pattern. We use i as the index of the spin in the lattice, d is a particular dimension where we want to find the neighbours. For a simple notation we used // and ** in a python-ish style meaning respectively integer division and power (power comes before all the other operations).

(i // N**(d + 1))*N**(d + 1) + (i + N**d             ) % N**(d + 1)
(i // N**(d + 1))*N**(d + 1) + (i - N**d + N**(d + 1)) % N**(d + 1)

We can loop on i and d to obtain all the neighbours. The process can be executed in parallel and be reduced with if statements on the first and last iteration.

The first reduction is much more sensitive since let us to evaluate a single power for each dimension > 1.

monteising