ising model 1.pdf

7/27/2019 ising model 1.pdf

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Lecture 8: The Ising model
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Introduction

Up to now: Toy systems with interesting properties(random walkers, cluster growth, percolation)

Common to them: No interactions

Add interactions now, with significant role

Immediate consequence: much richer structure in model,in particular: phase transitions

Simulate interactions with RNG (Monte Carlo method)

Include the impact of temperature: ideas from

thermodynamics and statistical mechanics important Simple system as example: coupled spins (see below),

will use the canonical ensemble for its description
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The Ising model

A very interesting model for understanding someproperties of magnetic materials, especially thephase transition ferromagnetic paramagnetic

Intrinsically, magnetism is a quantum effect,triggered by the spins of particles aligning with each other

Ising model a superb toy model to understand thisdynamics

Has been invented in the 1920s by E.Ising

Ever since treated as a first, paradigmatic model
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The model (in 2 dimensions)

Consider a square lattice with spins at each lattice site

Spins can have two values: si = 1 Take into account only nearest neighbour interactions

(good approximation, dipole strength falls off as 1/r3)

Energy of the system:

E = Jij sisj Here: exchange constant J> 0 (for ferromagnets),

and ij denotes pairs of nearest neighbours.

(Micro-)states characterised by the configuration ofeach spin, the more aligned the spins in a state , thesmaller the respective energy E.

Emergence of spontaneous magnetisation (withoutexternal field): sufficiently many spins parallel
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Adding temperature

Without temperature T: Story over.

With temperature: disordering effects, spins flip randomly. In the following assume system to be in contact with

external heat bath - can use canonical ensemble (look itup, if youre interested).

Effect: System will again explore all possibilities(like in the case of milk in tea, lecture 6)

In contrast here: No uniform probability, but

P exp

E

kBT (Boltzmann factor)

for the system to be in a micro-state .

Consequence: Increasing temperatures open micro-stateswith larger energies less alignment
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Observables

How to calculate macroscopic observables?Example: the magnetisation of the system M.

First step: calculate the observable for a micro-state:M =

i

si

Sample over all micro-states:M =

MP Therefore, name of the game:

How to calculate the P/sample over the micro-states. Will present two solutions in the following
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Analytic solution: Mean field theory (MFA)

Mean field theory an extremely powerful approximation

Will introduce it with the specific example of the Isingmodel at hand.

However nice: It is not precise!

Basic idea: Replace the individual spins si = 1 with anaverage value s [1, 1]. Then

M =i

si i

s = Ns Nsi In other words: Deduce an average alignment

(works for an infinitely large system - all spins equivalent).

With the equation above we used that we can pick everyspin as average spin. But how can we calculate withoutthe micro-states?

Answer: A trick (see below)
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The trick for the solution

Add a magnetic field (seems a detour, but wait & see!):E = Jij

sisj Hi

si

(Magnetic field H interacts with spins through their magnetic moment .)

Consider a system made of one single spin: E = H. Two micro-states with P = Cexp HkBT Normalisation C from P+ + P = 1

= C = 1exp

H

kBT

+exp

H

kBT

=1

2 cosh HkBT

Therefore thermal average of the single spin:si = P+ P = tanh HkBT
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Rounding it off

Having the solution for a single spin in a background field,

we replace the background field with the average spins!E =

i

ij

sj + H

si = Heff

i

si

The effective magnetic field is thus

Heff =J

ij sj + H

Mean field approximation then sets H = 0 and replacesthe actual sj with the average value:

Hmfa =jJ

s

(Here j = 4 is number of nearest neighbours) This implies the following implicit equation:

s = tanh jJskBT
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Results: Schematic

Notice two different regimes: either 1 or 3 solutions
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True results

Obtained from solving f(s) = s tanh 4sT

= 0
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Discussion of results

In plots use Mmax = N for normalisation and J = kB tokeep things simple.

Phase transition at Tc = 4 - second order(1st derivative of order parameter magnetisation jumps)

Around Tc: dM/dT. Exact form of singularity from Taylor expansion of tanh:

tanh x = x x33

+O(x4) Therefore, around T = Tc:

s = jJkBTs 1

3

jJ

kBT

3 s3
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Quantifying the phase transition

Non-trivial solution for:

s =

3T

kBT

jJ

3 jJ

kB T

1/2 (Tc T)

Critical temperature and exponent:

Tc = jJ/kB, = 1/2.

But exact results (analytically known):

Tc = 2/ ln(1 +

2)

2.27, = 1/8for a square lattice with j = 4 and J = kB.

Will now turn numerical/simulation
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Strategy of simulation

Strategy very similar to whats been done before: Use therandom number generator to evolve the system

This time: RNG to describe interactions

Necessary: Interactions in probabilistic language

Algorithm will look like: Go over the spins, check whetherthey flip (compare Pflip with number from RNG), repeatto equilibrate.

To calculate Pflip: Use energy of the two micro-states(before and after flip) and Boltzmann factors.

While running, evaluate observables directly and takethermal average (average over many steps).
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MetropolisA classical method for the program above

1. Initialise the lattice, i.e. fix all si(either at random, or si = 1i, or similar)

2. In each time step go through the entire lattice.For each spin decide whether it flips or not

Calculate E = J sisj for both configs Determine characteristic energy gain by flip

Eflip = Eafter Ebefore ifEflip < 0, the spin is flipped ifEflip > 0, the spin is flipped with probability

Pflip = expEflip

kBT

3. Repeat with a huge number of time steps

= allow the system to equilibrate
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Why Metropolis is correct: Detailed balance

Consider one spin flip, connecting micro-states 1 and 2.

Rate of transitions given by the transition probabilities W IfE1 > E2 then W12 = 1 and W21 = exp

E1E2

kBT

In thermal equilibrium, both transitions equally often:

P2

W21 =

P1

W12

This takes into account that the respective states areoccupied according to the Boltzmann factors.

In principle, all systems in thermal equilibrium can bestudied with Metropolis - just need to write transition

probabilities in accordance with detailed balance, asabove.

Metropolis algorithm simulates the canonical ensemble,summing over micro-states with a Monte Carlo method.
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Some code specifics

Initialise the L L lattice with spins si.Take the spins either fixed (all si = 1) or at random (all si = 1)

A single time-step: sweep through the latticeGo systematically through the lattice, line by line,spin by spin, and decide, whether the spin should flipor not. Important here: boundary conditionsBoundary conditions (finite-size lattices)

We use periodic boundary conditions. This meansthat the spins at the right edge of the lattice aretaken as neighbours of spins at the left edge. For 2-

D lattices this renders our lattice effectively a donut-shaped object. Such a treatment reduces finite-sizeeffects, but one should keep in mind that correlationswith a length larger than

2L cannot be simulated.

S i th h th l tti
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Sweeping though the lattice

Fix temperature, use a 10 10 lattice
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Discussion of the sweeps

At low temperatures (T = 1): system quite stable, only

small fluctuations, relative magnetisation around 1.

At larger temperatures, but below Tc (T=2): largerfluctuations due to more favourable Boltzmann factor(10% anti-aligned), average magnetisation around 0.9.

At large temperatures (T = 4): system disordered, noaverage magnetisation left.

Around the critical temperature (T = 2.25): Hugefluctuations, after periods of stability, system jumps from

sizable positive to negative magnetisation and vice versa. This is in agreement with fluctuation-dissipation theorem

(next lecture).

Phase transition the MC look at things
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Phase transition - the MC look at things

Still on a 10 10 lattice
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Some simulation nitty-gritty

Results above after equilibrating the system, but

critical slowdown around critical point:

The systems time to equilibrate diverges. Independent of this: Monte Carlo results in agreement

with exact calculation and in disagreement with theresults from mean field approximation.
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Summary

First simulation of a system with interactions

Used the Ising model as laboratory: well-defined,well-studied system, analytical results known, a favouriteof the simulators

A (simple) analytical approximation: mean field theorygives qualitatively correct results: existence of a phasetransition, estimate of critical temperature

Exact calculations (and simulation) agree and arequantitatively different from MFA.
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