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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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