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The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: - PowerPoint PPT Presentation

Dec 08, 2022 •150 likes •286 views

The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal & Stephen Sorokanich August 28, 2018 Mini Physics Intro Course Magnets and spins, magnetization Ferromagnetism Spins under an external magnetic field Ising

  1. The Ising Spin Lattice Model Alexander Brunmayr Graduate mentor: Nicholas Paskal & Stephen Sorokanich August 28, 2018

  2. Mini Physics Intro Course Magnets and spins, magnetization Ferromagnetism Spins under an external magnetic field

  3. Ising Nearest-Neighbor Lattice Model Models a ferromagnetic (or antiferromagnetic) material Only nearest-neighbor interactions Square Lattice with periodic boundary conditions

  4. 1D Ising Model We want to predict the magnetization of the system where

  5. 1D Ising Model

  6. 1D Ising Model Critical behavior at T=0

  7. 2D Ising Model Phase transition for temperatures below the Curie Temperature (Tc > 0) T > Curie Temperature: T = Curie Temperature no phase transition T < Curie Temperature: phase transition

  8. 2D Ising Model

  9. 3D and Beyond No analytical solution for dimensions 3 and higher Numerical solutions (see computer simulations)

  10. Simulations Step 1: select a location at random Step 2: ● Metropolis Algorithm: ○ if Energy decreases → flip spin ○ if Energy increases → flip spin with probability e – β ΔE ● Glauber Algorithm: ○ Flip spin with probability 1/(1+e β ΔE ) ● Voter Algorithm: ○ Set spin to be like a randomly selected neighbor

  11. Applications Magnets Neurology Computer science Solar magnetograms

  12. Texts and Sources R.J. Baxter : Exactly Solved Models in Statistical Mechanics R.A. Minlos : Introduction to Mathematical Statistical Physics (Lecture Series) Linda E. Reichl : A Modern Course in Statistical Physics (4th ed.) D.A. Levin : Glauber Dynamics for Ising Model I (AMS Short Course, UOregon) Raissa D’Souza : Simulating Glauber Dynamics for the Ising Model (UC Davis)

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