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a magnet is a collection of ³little´ magnets which have the property they can point in any 3-D
direction. The interactions between them are short range, tend to favor neighbours with parallel
directions.

Each magnet, a ferromagnet, will tend to align to each neighbour. The energy function is at a
minimum when the directions are aligned. There is the opposite, an anti-ferromagnet, will align
opposite to its neighbour.

Magnets only align to neighbours. In a high energy state (high temperature) the alignment is
erratic. At low T, low E ± all the magnets will tend towards alignment; At ground state, they will
be aligned (lowest energy state) ± no particular direction, just all the same direction.

Mathematical Models (03:45)
We build mathematical models as solving the interactions of a collection of individual magnets is
too complicated, especially since the interactions will affect magnets beyond the immediate
neighbours.

x ‘

In our examples there is an energy function which favors neighbours being in the same state.

At zero temperature there are only 2 states , everything up or everything down. Assuming
up/down states equally likely, or equal energy, then we have one energy state between magnets
both up or both down and a higher energy state when one is up, the other down.

Note: for this discussion we will call up,down orientation ³spins spin up, spin down´ ± nothing
to do with spinning, just a name.

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ground state, no matter the size of the field, will be all spin up or all spin down,
but which one?
Answer will depend on a tiny, tiny stray magnetic field (degeneracy?) which might favour one
direction or another.
will all align in one direction until something causes the magnetic field to reverse ± called
spontaneous symmetry breaking.

The system has a symmetry.
Total energy (each magnet aligned up, down, «) = Total energy (with each magnet having
direction reversed)

8.1

o

ı(i) is spin of i

th

magnet; (+1 is up)

8.2

o o

Symmetry between spin for all magnets reversing sign

ám apply an external magnetic field;

ám all ı(i) align to that direction ± a bias to the external field;

A collection of (group) Z2 magnets.Not really a magnet, Z2 is a group
name for up-down, 0-1, ± interactions)

low energy states:, same direction, both up or both down

high energy, opposite directions

ám by definition of spontaneous symmetry breaking, the bias does not disappear when you

shrink the external field;

we let the external field become smaller and smaller ± eventually removing the external field
altogether.
The symmetry of { ı(i) ĺ -ı(i) } will be broken in favor of a (now permanent) bias of all up or
all down, depending upon the (now removed) external field

O ‘ ‘

We can model many systems as magnets, for example; a lattice Gas: a fluid or gas as a
collection of molecules

Map the mathematical problem of the distribution of molecules into the mathematical problem of
the magnets.
--- all spin up state corresponds to an extremely dense gas or liquid; (high density limit)
--- all spin down corresponds to an empty collection. (low density limit)

--- equal numbers of ups and downs:
for a magnet this would be an unmagnetized state;
for gas/fluid corresponds to half full

O

What happens to magnet spin with small external magnetic field, a low finite temperature (15:00

tends to make the magnets point down, on the average. a weak bias ± as the temperature
increases, the spin flucuates more and the bias goes away.

now we lessen the external magnetic field until we reach zero ± then turn on in the opposite
direction ± what happens?
If spontaneous symmetry breaking was active there should be a sudden jump in the
magnetization (more spins down to more spins up)

Q?:what does this mean for the fluid?
A: the density makes a sharp jump.

The Lattice Model: collection of molecules in fluid or gas
state

break up fluid into a collection of cells, each ~ size of a
molecule
assume molecules have ³hard cores´, so that we cannot stick
them on top of each other ± which insures we cannot get two+
particles into a (equal sized) cell.
Either a box is empty (-, down) or has a molecule(+, up)

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jump from less than half full to greater than half full

called a phase transition (from a gas to a liquid)

The magnetic transition, where sudden spontaneous magnetization happens, is also called the
gas-liquid phase transition

‘

The ı spin change (h-axis) occurs by application of an external magnetic field. This is
called

® ®®® ® ®® ®®

At low temperatures:

ám phase transition line as described above;

ám sudden change in magnetization orientation or density in a gas-fluid;

ám persists to some finite (low) Temperature

At higher temperatures:

ám the flucuations due to high Temperature cause destruction of the ³tendency to align when

weak ext. magnetic fieldĺ0´

ám you still have the ۦıۧ>0 and ۦıۧ®® ®®

®

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A meta-stable state occurs when a gas-liquid or magnetic orientation state persists beyond the
h=0 critical point.

ám A temporary state.

Solid blue line

, phase transition line ± a jump in magnetization or density in a gas-fluid

Very low temperature, persists to some finite temperature. The spontaneous symmetry
breaking region.

Red line

, continuous transition line. At higher temperatures the ± ı spin, magnetism or

gas-fluid transition occurs smooothly, no jumps.

Green line

, meta-stable state. Liquid in gas state or gas in liquid state. No change in the ±

ı spin orientation. An unstable state that deterioates with time.

6®®®®

ۦıۧ

®®

T

temperature axis

h

spin axis, count of ± states

h

T

ۦıۧ> 0
Liquid

!
ۦıۧ< 0

ám Slowly change the orientation of a weak external field. The spins will stay in previous

direction. Then, after some (long) time, the spins will orientate themselves to the external
magnetic field.

ám similar situation in lattice gas-liquid situation. you can move a liquid into gas region, but

after a time, will turn to gas.

.
with real magnets a wave is emitted (generated, maybe a sound wave, a magnetic wave?) across
the collection which gradually causes spin change

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ther approximations:

ám Low temperature expansions: You start at zero temperature and expand by powers of a

function of the temperature ;

ám igh temperature expansions: Initially a very high temperature, erratic behaviour, expand

by powers of a 1/(function of the temperature) ;

ám Renormalization group:

Mean field approximation investigates the effects adjoining neighbours have on a molecule in a
Lattice cell. The approximate works best on a large number of dimensions (d) but is sufficiently
accurate in three dimensions. The effect can be studied in two dimensions.

Define an energy function.

ám The energy only depends on the orientation of nearest neighbours;

ám low when parallel, high when anti-parallel

8.3

2

o

o

i,j range over nearest neighbours

If ı values are equal, energy is minus one, ground state
8.4

o

o

2

If ı values are unequal, energy is plus one
8.5

o

o

2

Define bonds between neighbours

ám A broken bond occurs between two neighbours when the spins are anti-parallel;

ám An unbroken bond occurs between two neighbours when the spins are parallel;

ám broken bonds cost energy 2 units of energy

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ocus on one molecule, call ı

0

(sigma zero

8.6

o

o

energy is ı

0

× ™ ıs of nearest neighbours

ۦoۧ

suppose the spins have an average value

³s´ is called the mean field. can be positive or negative or zero

8.7

o

energy (2× # of dimensions) ı has 2d neghbours

keep mean value, ³s´, constant

8.8

"#

probability function e

-ȕ E

(Boltzman weighting factor?)

partition function:
8.9

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8.10

' &

average energy of the spin

3.15

2

ۦۧ

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(

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average energy

8.11

ۦۧ

® & "

# #

8.7

o

spin ± energy differ by a factor of 2ds

8.12

ۦ

ۧ ® &

average value of spin

where:

ۦ

ۧ

®)

The Lattice Model: collection of molecules in fluid or gas state

Çm

represents ı(i) = +1

mm

represents ı(i) = - 1

(blue line)

unbrocken bond, neighbours with the same ı value E= -1

(red line)

brocken bond, neighbours with the unequal ı values E=+1

The brocken bond stores 2 units of energy (over ground state E=-1)

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e

³small e´ energy of the single molecule

½

ám negative ȕ actually refers to the term (2ȕds < 0);

ám the mean value of the spin ³s´ can be negative

ám neither ȕ , temperature or dimensions are negative

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A major difficulty is the ۦ

ۧ® expected, or average, value of the spin may consist of large

strings of +1 or -1. The equation (8.8) only makes sense if the average value of a spin is the same
as the average value of every other spin.

8.12

# ۦۧ ® &

solutions indicate consistency requirement

r

#ۦۧ

+

-

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ám at high temperatures spin orientation is random;

ám There may be a slight magnetic bias;

ۦ

ۧ

2ȕd < 1 small ȕ highT

Plot showing how different temperatures (T=1/ȕ) affect the transition point spin 0
The intersection of the tanh curve with the ۦsۧ = s line shows where the consistency
requirements are met for the mean value of spin.
Critical temperature is when slope of tanh = slope ۦsۧ = s line = 45°

2ȕd = 1

2ȕd > 1 large ȕ lowT

ĸ Negative spin.

Positive spin. ĺ

Average value of spin ۦ

ۧ = tanh (2ȕds)

min/max value spin takes is 1
min/max value beta takes is ’
temperature (1/ȕ) is 0

ȕ

ı

Plot of expected spin versus ȕ values in the equation: ۦ

ۧ ® &

The max positive beta corresponds to minimum temperature ~0 (T=1/ȕ )

ám The only point where the (expected spin) equals (mean value spin) occurs at s=0 (™ ı(i)

= 0)

ám The single molecule mean value approximation is not valid at high temperatures

O #"

ám at low temperatures there are two pointswhere the (expected spin) equals (mean value

spin)

ám The single molecule mean value approximation is handled differently depending upon

initial s= 1

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ám this is the area where spontaneous symmetry breaking occurs;

ám The magnetic spin will exhibit a sudden reversal when a small external magnetic field is

gently applied;

ám A similar reversal occurs in the gas-liquid when a small external energy field is gently

applied

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The only stable solutions are the s=ۦ

ۧ 1 points for (2ȕd • 1) . The solution at s=ۦۧ

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Accuracy of mean field approximation

ám greater with greater dimensions; degrees of freedom

ám Lattice duality - better approximation by studying SM of pairs of molecules

Tc critical point.

ám for magnet. point where spontaneous magnetization disappears, the magnet no longer has

a defined magnetization

ám for gas-liquid, point where the density gas = density liquid. beyond Tc there is no

differentiation.

ám critiical temperature always lower than mean field approximate (in space Tc=2.7

8.13

­

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&

Critical temperature is related to #dimensions

Note a units constant is required. The originating mean value energy equation 2

o

o

needs

units of energy besides 1 values

description of what mean field approximation is

ám large degrees of freedom;

ám can be isolated to neighbouring effects only;

ám solve for one degree; assume equal for the rest

ám used in many aspects of physics

In one dimension

è

riginally theory by Ising. Wrong conclusion (no spontaneous magnetization) but name Ising

magnetic system persisted as a one dimensional system.

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a flip from + to ± cost 4 units of energy (2 broken bonds)

any subsequent flip , if the bits are next to each other is same 4 units (only 2 broken bonds)

ám large amount of configurations where little energy required for magnetic flip

ám no magnetic or gas-liquid phase transition in one dimension

Two dimensions

O &'(

gas-liquid. the phase transition occurs when the molecules go from a homogeneous gas state to a
sharply order crystal lattice state

you cannot tell a phase transition from a small sample or limited number of degrees of freedom
- phase transitions occur in the infinite limit of number of particles

ow does outside energy, used to affect transition, affect the system itself?

8.14

o

o

o

'

use sigmas meaning QM sigma states.

Assume external energy from ı3 which has components that act on (i,j) directions

2

o

'

o

'

(

8.15

) o

'

o

'

(

consider as a amilltonian ± ı3 components will not change with

time

8.16

*+ ,*)-

(that is the hamiltonian commutes with ı3)

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Some type of change is required for the system to come to thermal equilibrium (A  0) ± so
add a small number;
8.17

) ) . o

o

(

small ı1 to make hamiltonian non-commutative and

give it time dependence

) ) . o

o

(

and ı2

a flip costs 8 units 4 broken bonds; 2 flips ± not next to each other
costs 16 units

2 flips, next to each other; 6 broken bonds, 12 units of energy

if there are many sign flips, the number of broken bonds is
proportional to the boundary

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A system with only ı3 is similar to a system with PE (potential energy) only. Needs some KE to
move towards equilibrium.
Phase transition however only depends upon PE, not KE so the addition will not affectPhase
transition