1D Ising model Simple model of interacting spins in a lattice - - PowerPoint PPT Presentation

1D Ising model

Simple model of interacting spins in a lattice

• Nearest-neighbor only
• No external magnetic field

Here, we look at a 1D chain of spins

• Future: higher dimensions,

higher coordination numbers

Numerical Challenge

• Implement Metropolis algorithm for Ising model

– Pick random spin, flip according to algo. – Repeat for N random spins → 1 mcs (Monte Carlo

step)

• Compute system parameters, in particular

fluctuations after MC has settled into distr.:

– Mean energy, mean-squred energy – Mean magnetization; mean-squared magnetization – Heat capacity – Magnetic susceptibility

Generic but efficient implementation

• Metropolis algorithm taken as granted
• Nearest-neighbors implemented as an

array-of-arrays which point into the lattice array

– Neighbors computed once, in beginning (good);

interesting process for > 1D lattices

– Memory scales with coordination number (bad)

• How large a system can be simulated?

– mem = (1 bit/site)•(N sites) + N•ceil[log2(N)]•(c.n.) – If c.n.=12, N=1e8 yields mem = 3.8 GiB – If c.n.= 2, N=1e9 yields mem = 7.1 GiB

System energy vs. temperature: Simulation vs. analytic solution

No ferromagnetic phase transition: Heat capacity doesn't diverge! Magnetic Susceptibility