Critical Phenomena: Ising Model

Kevin Han

Mentor: Ram Reddy

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Statistical Mechanics

• Parameters

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• Hamiltonian

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• Probabilities; Partition function

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• Observables

Probability

Probability of Each State

After Normalization:

Partition Function

Observables (General Case)

Phase Transition

A discontinuity in a derivative of an observable with respect to a parameter of a system.

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Results in the system going from an ordered state to a disordered state (Loss of symmetry)

The Ising Model

What is an Ising Model?

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• Solid lattice of spins

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• Phase transitions

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• Ferromagnetism

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Barry A. Cipra. (1987). An Introduction to the Ising Model. The American Mathematical Monthly, 94(10), 937–959. https://doi.org/10.2307/2322600

Energy Operator (Hamiltonian)

1-D: Exact Calculation

Start with One-Dimensional Ising Model

All Possible States Generation

One-dimensional lattice: 2^N possible states where N is the size of the row

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Two-dimensional lattice: 2^N*N possible states where N is the size of the square edge

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Three-dimensional lattice: 2^N*N*N possible states where N is the size of the cube edge

Average Magnetization vs Temperature

Higher Dimensional Ising Models

Problem with Exact Calculation

Monte Carlo Algorithms

Monte Carlo Approximation

• What is it?

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• Dependence on iteration

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• Example for pi

Monte Carlo Algorithm for approximating Pi

Result

Metropolis Algorithm

• Markov Chain Monte Carlo (MCMC)

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• Randomness

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• Approximate probability distributions

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The Process

Ising Model MC “Filter”

• Start with state

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• Random spin chosen and flipped. New state is

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• If , make the flip and repeat with

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Else make the flip with a probability of

T < Tc : Ordered Phase

T > Tc : Disordered Phase

“Golden Number” of Iterations

“Golden Number” of Iterations

Critical Temperature

Conclusions

• Computational approach

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• Monte carlo algorithms

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• Ising model

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