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The Ising model (or Lenz–Ising model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics.The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1).
In the Ising model, we have say N particles that can spin up (+1) or down (-1). Say the particles are on a 2D grid. We label each with an x and y coordinate. Glauber's algorithm becomes: [3] Choose a particle , at random. Sum its four neighboring spins.
The transverse field Ising model is a quantum version of the classical Ising model.It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the axis, as well as an external magnetic field perpendicular to the axis (without loss of generality, along the axis) which creates an energetic bias for one x-axis spin direction ...
In d=2, the two-dimensional critical Ising model's critical exponents can be computed exactly using the minimal model,. In d=4, it is the free massless scalar theory (also referred to as mean field theory). These two theories are exactly solved, and the exact solutions give values reported in the table.
The Wolff algorithm, [1] named after Ulli Wolff, is an algorithm for Monte Carlo simulation of the Ising model and Potts model in which the unit to be flipped is not a single spin (as in the heat bath or Metropolis algorithms) but a cluster of them.
The original algorithm was designed for the Ising and Potts models, and it was later generalized to other systems as well, such as the XY model by Wolff algorithm and particles of fluids. The key ingredient was the random cluster model , a representation of the Ising or Potts model through percolation models of connecting bonds, due to Fortuin ...
Apparently independent of the work of Young and Elcock, Bortz, Kalos and Lebowitz [1] developed a KMC algorithm for simulating the Ising model, which they called the n-fold way. The basics of their algorithm is the same as that of Young, [8] but they do provide much greater detail on the method.
A Boltzmann machine (also called Sherrington–Kirkpatrick model with external field or stochastic Ising model), named after Ludwig Boltzmann, is a spin-glass model with an external field, i.e., a Sherrington–Kirkpatrick model, [1] that is a stochastic Ising model. It is a statistical physics technique applied in the context of cognitive ...