Topological Defects
18.354 L24
Order Parameters, Broken Symmetry, and Topology
James P. Sethna
Laboratory of Applied Physics, Technical University of Denmark,
DK-2800 Lyngby, DENMARK, and NORDITA, DK-2100 Copenhagen Ø,
DENMARK and Laboratory of Atomic and Solid State Physics (LASSP),
Clark Hall, Cornell University, Ithaca, NY 14853-2501, USA
(Dated: May 27, 2003, 10:27 pm)
We introduce the theoretical framework we use to study the bewildering variety of phases in
condensed–matter physics. We emphasize the importance of the breaking of symmetries, and develop
the idea of an order parameter through several examples. We discuss elementary excitations and
the topological theory of defects.
1991 Lectures in Complex Systems, Eds. L. Nagel and D. Stein, Santa Fe Institute Studies in
the Sciences of Complexity, Proc. Vol. XV, Addison-Wesley, 1992.
PACS numbers:
Keywords:
As a kid in elementary school, I was taught that there
dunkel@mit.edu
In important and precise ways, magnets are a distinct
•
optical effects
•
work hardening, etc
Π2 (S 2 ) = Z.
(7)
Here, the 2 subscript says that we’re studying the second
Homotopy group. It represents the fact that we are surrounding the defect with a 2-D spherical surface, rather
number:
FIG. 12: (a) Hedgehog defect. Magnets have no line defects (you can’t lasso a basketball), but do have point defects.
⃗ (x) = M0 x̂. You can’t
Here is shown the hedgehog defect, M
surround a point defect in three dimensions with a loop, but
you can enclose it in a sphere. The order parameter space, remember, is also a sphere. The order parameter field takes the
enclosing sphere and maps it onto the order parameter space,
wrapping it exactly once. The point defects in magnets are
categorized by this wrapping number: the second Homotopy
group of the sphere is Z, the integers.
(b) Defect line in a nematic liquid crystal. You can’t
lasso the sphere, but you can lasso a hemisphere! Here is the
defect corresponding to the path shown in figure 5. As you
pass clockwise around the defect line, the order parameter rotates counterclockwise by 180◦ .
This path on figure 5 would actually have wrapped around
the right–hand side of the hemisphere. Wrapping around the
left–hand side would have produced a defect which rotated
clockwise by 180◦ . (Imagine that!) The path in figure 5 is
halfway in between, and illustrates that these two defects are
really not different topologically.
8
(8)
Finally, why are these defect categories a group? A
group is a set with a multiplication law, not necessar-
FIG. 13: Multiplying two loops. The product of two loops
is given by starting from their intersection, traversing the first
loop, and then traversing the second. The inverse of a loop
is clearly the same loop travelled backward: compose the two
and one can shrink them continuously back to nothing. This
definition makes the homotopy classes into a group.
This multiplication law has a physical interpretation. If two
defect lines coalesce, their homotopy class must of course be
given by the loop enclosing both. This large loop can be
deformed into two little loops, so the homotopy class of the
coalesced line defect is the product of the homotopy classes
of the individual defects.
Two parallel defects can coalesce and heal, even though
each one individually is stable: each goes halfway around
the sphere, and the whole loop can be shrunk to zero.
Π1 (RP 2 ) = Z2 .
than the 1-D curve we used in the crystal.[13]
You might get the impression that a strength 7 defect
is really just seven strength 1 defects, stuffed together.
You’d be quite right: occasionally, they do bunch up,
but usually big ones decompose into small ones. This
doesn’t mean, though, that adding two defects always
gives a bigger one. In nematic liquid crystals, two line
defects are as good as none! Magnets didn’t have any
line defects: a loop in real space never surrounds something it can’t smooth out. Formally, the first homotopy
group of the sphere is zero: you can’t loop a basketball.
For a nematic liquid crystal, though, the order parameter space was a hemisphere (figure 5). There is a loop
on the hemisphere in figure 5 that you can’t get rid of
by twisting and stretching. It doesn’t look like a loop,
but you have to remember that the two opposing points
on the equater really represent the same nematic orientation. The corresponding defect has a director field n
which rotates 180◦ as the defect is orbited: figure 12b
shows one typical configuration (called an s = −1/2 defect). Now, if you put two of these defects together, they
cancel. (I can’t draw the pictures, but consider it a challenging exercise in geometric visualization.) Nematic line
defects add modulo 2, like clock arithmetic in elementary
school:
Topological defects
are discontinuities in
order-parameter fields
"umbilic defects" in a nematic liquid crystal
order = symmetry = invariance
!
(under certain group actions )
symmetry groups can be discrete,
continous, Lie-groups, ….
What is it which distinguishes the hundreds of different states of matter? Why do we say that water and
olive oil are in the same state (the liquid phase), while
we say aluminum and (magnetized) iron are in different
states? Through long experience, we’ve discovered that
most phases differ in their symmetry.[5]
ance: it c
the transl
by certain
The pictu
jumbled t
though, is
combinati
rotational
the same
More or less symmetric ?
FIG. 2: Which is more symmetric? The cube has many
symmetries. It can be rotated by 90◦ , 180◦ , or 270◦ about any
can rotat
a broken
picture is
number o
shown in
has comp
One of
fer by a s
smoothly
alcohol d
adding m
alcohol,
More or less symmetric ?
Mg2Al4Si5O18
http://www.doitpoms.ac.uk/tlplib/crystallography3/printall.php
More or less symmetric ?
2
heories,
s there
st, you
u must
to exclassify
ke from
heartily
ce. We
RY
FIG. 3:
Whichcontinuous
is more symmetric? At first glance, wabroken
ter seems to have much less symmetry than ice. The picture
translation/rotation
of “two–dimensional” ice clearly breaks the rotational invarisymmetry
(invariance)
ance:
it can be rotated
only by 120◦ or 240◦ . It also breaks
the translational invariance: the crystal can only be shifted
doughnut, bagel, or inner tube.)
u(x) as the local translation needed to bring the ideal lattice
their i
Finally, let’s mention that guessing the order
into registry with atoms in the local neighborhood of x.
try? T
(or the broken
isn’t
always so s
Also shown is the ambiguity in the definition of u. FIG.
Which
8: One eter
dimensional
crystal: symmetry)
phonons. The
order
parameter
for a one–dimensional
crystal
is themany
local dis“ideal” atom should we identify with a given “real” one?
This field
forward.
For example,
it took
years before
Long–wavelength waves in u(x) have low
ambiguity makes the order parameter u equivalent placement
to u + u(x).
figured
out that the order parameter for superc
frequencies,
and
cause
sound.
max̂ + naŷ. Instead of a vector in two dimensional space,
torsbecause
and superfluid
4 is symmea complex num
Crystals are rigid
of the brokenHelium
translational
the order parameter space is a square with periodic boundary
try. Because they
rigid,parameter
they fight displacements.
Because
Theare
order
field ψ(x)
represents
the “c
4
conditions.
Order parameters:
2D crystal
there is an underlying
translational
symmetry,
a uniform dis-loosely) is
sate wave
function”,
which (extremely
a headless vector ⃗n ≡ −⃗n. The order parameter space
placement costs no energy. A nearly uniform displacement,
quantum
stateand
occupied
byhave
a large
fraction
of the
is a hemisphere, with opposing points along the equator
thus,
will
cost
little
energy,
thus
will
a
low
freFor a identified
crystal,(figure
the important
degrees
of
freedom
are
as5). This space is called RP 2 by the
or helium
atoms inexcitations
the material.
quency. Thesepairs
low–frequency
elementary
are the The corr
sociatedmathematicians
with the broken
translational
order.
(the projective
plane), for obscure
rea- Consider
sound waves in
crystals.
ing
broken symmetry is closely4 related to the num
sons.
a two-dimensional
crystal which has lowest energy when
particles. In “symmetric”, normal liquid helium,
headlesswhich
vector ⃗n is
≡ −⃗
n. The order parameter
space
in a square lattice, a but
deformed
away A
from
good example
is given
sound
We won’t
cal number
of by
atoms
is waves.
conserved:
in superfluid
is a hemisphere, with opposing points along the equator
2 is dethat configuration (figure
6). 5).This
deformation
about sound
wavesnumber
in air: air
doesn’t becomes
have any indeterminate
broidentified (figure
This space
is called RP talk
by the
the
local
of
atoms
projective
plane), for obscure
reascribed by an arrow mathematicians
connecting(the
the
undeformed
ideal
latken
symmetries, so it doesn’t belong in this lecture.[9]
sons.
is because many of the atoms are condensed into t
tice points with the actual positions of the atoms.Consider
If we instead sound in the one-dimensional crystal
localized
wave
function.)
Anyhow,
the
magnitud
shown
in
figure
8.
We
describe
the
material
with
an
orare a bit more careful, we say that ⃗u(x) is that displacecomplex
number
is a fixed
function
of tempera
parameter
field u(x),
whereψ here
x is the
position
ment needed to align the ideal lattice in the localder
region
the order parameter space is the set of complex n
within
onto the real one. By saying it this way, ⃗u is also
de-the material and x − u(x) is the position of the
of within
magnitude
|ψ|.crystal.
Thus the order parameter sp
reference atom
the ideal
fined between the lattice positions: there still is a best
and
superfluids
is athe
circle S1 .
Now, theresuperconductors
must be an energy
cost
for deforming
displacement which locally lines up the two lattices.
ideal crystal. Now
There won’t
be any
cost,deformations
though, for aaway from
examine
small
FIG. 7: Order parameter we
space
for a two-dimensional
The order parameter ⃗u isn’t really a vector: there
is a translation:
uniform
≡
same energy as
crystal. Here
weform
see that
a u(x)
square
withuperiodic
boundary
0 has the
order
parameter
field.
conditions
a torus. (A (Shoving
torus is a surface
of a doughnut,
subtlety. In general, which ideal atom you associate
with
the
idealis crystal.
all the
atoms to the right
inner tube, or bagel, depending on your background.)
a given FIG.
real6: one
is ambiguous.
shown
inatoms
figuredoesn’t
6, thecost
any
energy.)
So,
theforenergy
will depend only
Two dimensional
crystal. As
A crystal
consists
FIG. 7:
Order
parameter
space
a two-dimensional
arranged vector
in regular,⃗
repeating
rowsby
and
columns.
At of
hightheon
crystal. of
Here
we see
that a square
with periodic
boundary
FIG. 9
derivatives
the
function
u(x).
The simplest
energy
displacement
u
changes
a
multiple
lattice
is a torus.
(A torus
is a surface of a doughnut,
temperatures, or when the crystal is deformed or defective,
III.
EXAMINE
THE ELEMENTARY
squareone
withconditions
periodic
boundary
conditions
has the same
rotatio
that
can
write
looks
like
inner
tube,
or
bagel,
depending
on
your background.)
will we
be displaced
from
lattice positions.
The atom:
constanttheaatoms
when
choose
a their
different
reference
2
topology as a torus, T . (The torus is the EXCITATIONS
surface of a
the ma
FIG.
6: TwoEven
dimensional
crystal
!
displacements ⃗
u are
shown.
better, crystal.
one can Athink
ofconsists atoms
arranged
in
regular,
repeating
rows
and
columns.
At
high
doughnut,
bagel,
or
inner
tube.)
u(x) as the local translation needed to bring the ideal lattice
2
temperatures, or when the crystal is deformed or defective,
E periodic
=that dx
(κ/2)(du/dx)
. the same
(2)
Finally, let’s
guessing
the
order has
paramsquaremention
with
boundary
conditions
into registry with atoms
in
the
local
neighborhood
of
x.
the atoms will be displaced from their lattice positions. The
2
topology as a torus, T . (The torus is the surface of a
Also shown is the displacements
ambiguity in⃗uthe
of u.
Which
are definition
shown. Even
better,
one can eter
think(or
of the broken symmetry) isn’t always so straightdoughnut,
bagel,
or inner
tube.)
“ideal” atom shouldu(x)
we identify
with
a given “real”
This
forward.
example,
it took
many
years before anyone
as the local
translation
needed one?
to bring
the ideal
lattice For
(Higher
derivatives
won’t
be
important
forparamthe low freFinally,
mention
that guessing
the order
ambiguity makes the
to u +
into order
registryparameter
with atomsuinequivalent
the local neighborhood
offigured
x.
out that
thelet’s
order
parameter
for superconduceter
(orhumans
the broken can
symmetry)
isn’tNow,
always you
so straightAlsoofshown
is theinambiguity
in the definition
Which
max̂ + naŷ. Instead
a vector
two dimensional
space, of u.quencies
that
hear.)
tors
and
superfluid
Helium
4
is
a
complex
number
ψ.may rememshould we
identify
with boundary
a given “real” one? This
forward. For example, it took many years before anyone
the order parameter“ideal”
spaceatom
is a square
with
periodic
The
order
parameter
field
ψ(x)
represents
thefor“condenber
F =the
ma.
force
here
is given by the
ambiguity makes the order parameter u equivalent
to uNewton’s
+
figured law
out that
orderThe
parameter
superconducconditions.
⃗u ≡ ⃗u + ax̂ = ⃗u + max̂ + naŷ.
(1)
The set of distinct order parameters forms a square
with periodic boundary conditions. As figure 7 shows, a
they ha
(b) Ne
are.liquid
Thec
Its amazing how slow human beings
inside your eyelash collide with one another a
million times during each time you blink your
Lone
netization (an arrow pointing to the “north” end of the
local magnet). The local magnetization comes from complicated interactions between the electrons, and is partly
due to the little magnets attached to each electron and
partly due to the way the electrons dance around in the
material: these details are for many purposes unimportant.
Order parameters:
magnets
⃗ as the orFIG. 4: Magnet. We take the magnetization M
ready ther
pound the
in this lec
can study
space.
The ord
alized in t
think of a
On the oth
real space
function w
the surfac
sphere S2 ,
don’t care
s partly
ron and
d in the
nimpor-
s the ora given
0 will be
ndepenn. (You
d of each
rections
why not
ent state
n arrow
in this lecture that most of the interesting behavior we
can study involves the way the order parameter varies in
space.
⃗ (x) can be usefully visuThe order parameter field M
alized in two different ways. On the one hand, one can
think of a little vector attached to each point in space.
On the other hand, we can think of it as a mapping from
⃗ is a
real space into order parameter space. That is, M
function which takes different points in the magnet onto
the surface of a sphere (figure 4). Mathematicians call the
sphere S2 , because it locally has two dimensions. (They
don’t care what dimension the sphere is embedded in.)
Order parameters:
nematic liquid crystals
“projective plane” =
half-sphere
with opposite points on
equator identified
FIG. 5: Nematic liquid crystal. Nematic liquid crystals are
made up of long, thin molecules that prefer to align with one
another. (Liquid crystal watches are made of nematics.) Since
they don’t care much which end is up, their order parameter
isn’t precisely the vector n̂ along the axis of the molecules.
Topological defects
6
and rotational waves (figure 9b).
In superfluids, the broken gauge symmetry leads to a
stiffness which results in the superfluidity. Superfluidity
and superconductivity really aren’t any more amazing
than the rigidity of solids. Isn’t it amazing that chairs
are rigid? Push on a few atoms on one side, and 109
atoms away atoms will move in lock–step. In the same
way, decreasing the flow in a superfluid must involve a
cooperative change in a macroscopic number of atoms,
and thus never happens spontaneously any more than
two parts of the chair ever drift apart.
The low–frequency Goldstone modes in superfluids are
heat waves! (Don’t be jealous: liquid helium has rather
cold heat waves.) This is often called second sound, but
is really a periodic modulation of the temperature which
passes through the material like sound does through a
metal.
O.K., now we’re getting the idea. Just to round things
out, what about superconductors? They’ve got a broken
gauge symmetry, and have a stiffness to decays in the
superconducting current. What is the low energy excita-
FIG. 10: Dislocation in a crystal. Here is a topological
defect in a crystal. We can see that one of the rows of atoms on
the right disappears halfway through our sample. The place
where it disappears is a defect, because it doesn’t locally look
like a piece of the perfect crystal. It is a topological defect
because it can’t be fixed by any local rearrangement. No
reshuffling of atoms in the middle of the sample can change
the fact that five rows enter from the right, and only four
leave from the left!
The Burger’s vector of a dislocation is the net number of extra
rows and columns, combined into a vector (columns, rows).
6
and rotational waves (figure 9b).
In superfluids, the broken gauge symmetry leads to a
stiffness which results in the superfluidity. Superfluidity
and superconductivity really aren’t any more amazing
than the rigidity of solids. Isn’t it amazing that chairs
are rigid? Push on a few atoms on one side, and 109
atoms away atoms will move in lock–step. In the same
way, decreasing the flow in a superfluid must involve a
cooperative change in a macroscopic number of atoms,
and thus never happens spontaneously any more than
two parts of the chair ever drift apart.
Work hardening
The low–frequency Goldstone modes in superfluids are
heat waves! (Don’t be jealous: liquid helium has rather
cold heat waves.) This is often called second sound, but
is really a periodic modulation of the temperature which
passes through the material like sound does through a
metal.
O.K., now we’re getting the idea. Just to round things
out, what about superconductors? They’ve got a broken
gauge symmetry, and have a stiffness to decays in the
superconducting current. What is the low energy excitation? It doesn’t have one. But what about Goldstone’s
theorem? Well, you know about physicists and theorems
...
That’s actually quite unfair: Goldstone surely had conditions on his theorem which excluded superconductors.
Actually, I believe Goldstone was studying superconductors when he came up with his theorem. It’s just that
everybody forgot the extra conditions, and just remembered that you always got a low frequency mode when
you broke a continuous symmetry. We of course understood all along why there isn’t a Goldstone mode for
superconductors: it’s related to the Meissner effect. The
high energy physicists forgot, though, and had to rediscover it for themselves. Now we all call the loophole in
Goldstone’s theorem the Higgs mechanism, because (to
be truthful) Higgs and his high–energy friends found a
much simpler and more elegant explanation than we had.
We’ll discuss Meissner effects and the Higgs mechanism
in the next lecture.
I’d like to end this section, though, by bringing up
another exception to Goldstone’s theorem: one we’ve
known about even longer, but which we don’t have a
nice explanation for. What about the orientational order
in crystals? Crystals break both the continuous translational order and the continuous orientational order. The
phonons are the Goldstone modes for the translations,
but there are no orientational Goldstone modes.[10] We’ll
discuss this further in the next lecture, but I think this
is one of the most interesting unsolved basic questions in
the subject.
FIG. 10: Dislocation in a crystal. Here is a topological
defect in a crystal. We can see that one of the rows of atoms on
the right disappears halfway through our sample. The place
where it disappears is a defect, because it doesn’t locally look
like a piece of the perfect crystal. It is a topological defect
because it can’t be fixed by any local rearrangement. No
reshuffling of atoms in the middle of the sample can change
the fact that five rows enter from the right, and only four
leave from the left!
The Burger’s vector of a dislocation is the net number of extra
rows and columns, combined into a vector (columns, rows).
IV.
CLASSIFY THE TOPOLOGICAL DEFECTS
When I was in graduate school, the big fashion was
topological defects. Everybody was studying homotopy
groups, and finding exotic systems to write papers about.
It was, in the end, a reasonable thing to do.[11] It is true
that in a typical application you’ll be able to figure out
what the defects are without homotopy theory. You’ll
spend forever drawing pictures to convince anyone else,
though. Most important, homotopy theory helps you to
think about defects.
A defect is a tear in the order parameter field. A topological defect is a tear that can’t be patched. Consider
the piece of 2-D crystal shown in figure 10. Starting in
the middle of the region shown, there is an extra row of
atoms. (This is called a dislocation.) Away from the middle, the crystal locally looks fine: it’s a little distorted,
but there is no problem seeing the square grid and defining an order parameter. Can we rearrange the atoms in a
small region around the start of the extra row, and patch
the defect?
No. The problem is that we can tell there is an extra row without ever coming near to the center. The
traditional way of doing this is to traverse a large loop
surrounding the defect, and count the net number of rows
crossed on the path. In the path shown, there are two
rows going up and three going down: no matter how far
we stay from the center, there will naturally always be
an extra row on the right.
How can we generalize this basic idea to a general problem with a broken symmetry? Remember that the order
parameter space for the 2-D square crystal is a torus (see
6
and rotational waves (figure 9b).
In superfluids, the broken gauge symmetry leads to a
stiffness which results in the superfluidity. Superfluidity
and superconductivity really aren’t any more amazing
than the rigidity of solids. Isn’t it amazing that chairs
are rigid? Push on a few atoms on one side, and 109
atoms away atoms will move in lock–step. In the same
way, decreasing the flow in a superfluid must involve a
cooperative change in a macroscopic number of atoms,
and thus never happens spontaneously any more than
two parts of the chair ever drift apart.
Disclinations
edge
The low–frequency Goldstone modes in superfluids are
heat waves! (Don’t be jealous: liquid helium has rather
cold heat waves.) This is often called second sound, but
is really a periodic modulation of the temperature which
passes through the material like sound does through a
metal.
O.K., now we’re getting the idea. Just to round things
out, what about superconductors? They’ve got a broken
gauge symmetry, and have a stiffness to decays in the
superconducting current. What is the low energy excitation? It doesn’t have one. But what about Goldstone’s
theorem? Well, you know about physicists and theorems
...
That’s actually quite unfair: Goldstone surely had conditions on his theorem which excluded superconductors.
Actually, I believe Goldstone was studying superconductors when he came up with his theorem. It’s just that
everybody forgot the extra conditions, and just remembered that you always got a low frequency mode when
you broke a continuous symmetry. We of course understood all along why there isn’t a Goldstone mode for
superconductors: it’s related to the Meissner effect. The
high energy physicists forgot, though, and had to rediscover it for themselves. Now we all call the loophole in
Goldstone’s theorem the Higgs mechanism, because (to
be truthful) Higgs and his high–energy friends found a
much simpler and more elegant explanation than we had.
We’ll discuss Meissner effects and the Higgs mechanism
in the next lecture.
screw
I’d like to end this section, though, by bringing up
another exception to Goldstone’s theorem: one we’ve
known about even longer, but which we don’t have a
nice explanation for. What about the orientational order
in crystals? Crystals break both the continuous translational order and the continuous orientational order. The
phonons are the Goldstone modes for the translations,
but there are no orientational Goldstone modes.[10] We’ll
discuss this further in the next lecture, but I think this
is one of the most interesting unsolved basic questions in
the subject.
FIG. 10: Dislocation in a crystal. Here is a topological
defect in a crystal. We can see that one of the rows of atoms on
the right disappears halfway through our sample. The place
where it disappears is a defect, because it doesn’t locally look
like a piece of the perfect crystal. It is a topological defect
because it can’t be fixed by any local rearrangement. No
reshuffling of atoms in the middle of the sample can change
the fact that five rows enter from the right, and only four
leave from the left!
The Burger’s vector of a dislocation is the net number of extra
rows and columns, combined into a vector (columns, rows).
IV.
CLASSIFY THE TOPOLOGICAL DEFECTS
When I was in graduate school, the big fashion was
topological defects. Everybody was studying homotopy
groups, and finding exotic systems to write papers about.
It was, in the end, a reasonable thing to do.[11] It is true
that in a typical application you’ll be able to figure out
what the defects are without homotopy theory. You’ll
spend forever drawing pictures to convince anyone else,
though. Most important, homotopy theory helps you to
think about defects.
A defect is a tear in the order parameter field. A topological defect is a tear that can’t be patched. Consider
the piece of 2-D crystal shown in figure 10. Starting in
the middle of the region shown, there is an extra row of
atoms. (This is called a dislocation.) Away from the middle, the crystal locally looks fine: it’s a little distorted,
but there is no problem seeing the square grid and defining an order parameter. Can we rearrange the atoms in a
small region around the start of the extra row, and patch
the defect?
No. The problem is that we can tell there is an extra row without ever coming near to the center. The
traditional way of doing this is to traverse a large loop
surrounding the defect, and count the net number of rows
crossed on the path. In the path shown, there are two
rows going up and three going down: no matter how far
we stay from the center, there will naturally always be
an extra row on the right.
How can we generalize this basic idea to a general problem with a broken symmetry? Remember that the order
parameter space for the 2-D square crystal is a torus (see
6
and rotational waves (figure 9b).
In superfluids, the broken gauge symmetry leads to a
stiffness which results in the superfluidity. Superfluidity
and superconductivity really aren’t any more amazing
than the rigidity of solids. Isn’t it amazing that chairs
are rigid? Push on a few atoms on one side, and 109
atoms away atoms will move in lock–step. In the same
way, decreasing the flow in a superfluid must involve a
cooperative change in a macroscopic number of atoms,
and thus never happens spontaneously any more than
two parts of the chair ever drift apart.
Disclineations
The low–frequency Goldstone modes in superfluids are
heat waves! (Don’t be jealous: liquid helium has rather
cold heat waves.) This is often called second sound, but
is really a periodic modulation of the temperature which
passes through the material like sound does through a
metal.
O.K., now we’re getting the idea. Just to round things
out, what about superconductors? They’ve got a broken
gauge symmetry, and have a stiffness to decays in the
superconducting current. What is the low energy excitation? It doesn’t have one. But what about Goldstone’s
theorem? Well, you know about physicists and theorems
...
That’s actually quite unfair: Goldstone surely had conditions on his theorem which excluded superconductors.
Actually, I believe Goldstone was studying superconductors when he came up with his theorem. It’s just that
everybody forgot the extra conditions, and just remembered that you always got a low frequency mode when
you broke a continuous symmetry. We of course understood all along why there isn’t a Goldstone mode for
superconductors: it’s related to the Meissner effect. The
high energy physicists forgot, though, and had to rediscover it for themselves. Now we all call the loophole in
Goldstone’s theorem the Higgs mechanism, because (to
be truthful) Higgs and his high–energy friends found a
much simpler and more elegant explanation than we had.
We’ll discuss Meissner effects and the Higgs mechanism
in the next lecture.
I’d like to end this section, though, by bringing up
another exception to Goldstone’s theorem: one we’ve
known about even longer, but which we don’t have a
nice explanation for. What about the orientational order
in crystals? Crystals break both the continuous translational order and the continuous orientational order. The
phonons are the Goldstone modes for the translations,
but there are no orientational Goldstone modes.[10] We’ll
discuss this further in the next lecture, but I think this
is one of the most interesting unsolved basic questions in
the subject.
FIG. 10: Dislocation in a crystal. Here is a topological
defect in a crystal. We can see that one of the rows of atoms on
the right disappears halfway through our sample. The place
where it disappears is a defect, because it doesn’t locally look
like a piece of the perfect crystal. It is a topological defect
because it can’t be fixed by any local rearrangement. No
reshuffling of atoms in the middle of the sample can change
the fact that five rows enter from the right, and only four
leave from the left!
The Burger’s vector of a dislocation is the net number of extra
rows and columns, combined into a vector (columns, rows).
rameter winds eith
number of times,
which cannot be be
can’t change by a
When I was in graduate
school, the big fashion was
fashion.
topological defects. Everybody was studying homotopy
groups, and finding exotic systems to write papers about.
How do we cate
It was, in the end, a reasonable thing to do.[11] It is true
that in a typical application you’ll be able to figure out
tals?
Well,theory.
there
what the defects are
without homotopy
You’ll a
spend forever drawing pictures to convince anyone else,
though. Most important,
homotopy theory the
helps youcen
to
go around
think about defects.
A defect is a tearpass
in the order
parameter field. A
topo- In
through
it.
logical defect is a tear that can’t be patched. Consider
the piece of 2-D crystal shown in figure 10. Starting in
sponds precisely to
the middle of the region shown, there is an extra row of
atoms. (This is called a dislocation.) Away from the midof looks
atoms
pass b
dle, the crystal locally
fine: it’s awe
little distorted,
but there is no problem seeing the square grid and defining an order parameter.
we rearrange
the atoms in an
a
in Can
the
old days,
small region around the start of the extra row, and patch
the defect?
understand it. We
No. The problem is that we can tell there is an extra row without ever coming near to the center. The
traditional way of of
doing the
this is totorus:
traverse a large loop
IV.
CLASSIFY THE TOPOLOGICAL DEFECTS
surrounding the defect, and count the net number of rows
crossed on the path. In the path shown, there are two
rows going up and three going down: no matter how far
we stay from the center, there will naturally always be
an extra row on the right.
How can we generalize this basic idea to a general problem with a broken symmetry? Remember that the order
parameter space for the 2-D square crystal is a torus (see
FIG. 11: Loop around the dislocation mapped onto or-
where Z represen
labeled by two int
Bacterial vortices
PIV
+1
Dunkel et al PRL 2013
-1
-1
+1
Active nematics
Dogic lab (Brandeis) Nature 2012
Active nematics
REVIEW LETTERS
week ending
31 MAY 2013
FIG. 2 (color online). Defect pair production in an active
suspension of microtubules and kinesin (top) and the same
phenomenon observed in our numerical simulation of an extenet100
al PRL
sile nematic fluid Giomi
with " ¼
and !2012
¼ "0:5. The experimental picture is reprinted with permission from T. Sanchez et al.,
Defects in nematics
winding
number
Defects in nematics
winding
number
l science. We showed that colloidal particles confined to a fewmatic liquid crystal form two-dimensional crystal structures that are
Two basic crystalline structures were observed, depending on the
round the particle. Colloids inducing quadrupolar order crystallize
onal ordered structure, where the particle interaction is mediated by
gical defects. Colloids inducing dipolar order are strongly bound into
nsional crystallites of dipolar colloidal chains. Self-assembly by
pplied to other systems with similar symmetry.
elastic energy of deformation depends on the
separation between inclusions, structural forces
between inclusions are generated. The structural forces in liquid crystals are long-range (on
the order of micrometers) and spatially highly
anisotropic, thus reflecting the nature of the
order in liquid crystals (14–17).
In our experiments, a dispersion of micrometersized silica spheres in the nematic liquid crystal pentylcyanobiphenyl (5CB) was introduced
into a rubbed thin glass cell with thickness
varying along the direction of rubbing from
Two-Dimensional Nematic Colloidal Crystals Self-Assembled by
Topological Defects
Igor Musevic et al.
Science
954molecules
(2006);
fluids, in313
which, rodlike
are spontaor liquid droplets
and collectively aligned into a certain
tal show a diverDOI:neously
10.1126/science.1129660
tructures, such as direction, called the director. Because of their
clusters (3), twotices at interfaces
particle-stabilized
d structures (8).
This copy is for your personal, non-commercial use only.
to spontaneously
egular geometric
resting for develing artificial colhotonic band-gap
es to fabrication
tation of colloids
patterned
presh to and
distribute
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es (11),
external-or customers by clicking here.
es,
clients,
2), and precision
echanical micro-
copies for your
on to republish or repurpose articles or portions of articles can be obtained by
the guidelines
here.
spatial
aggrega-
y a fine balance
sion
forces
and
wing
resources
related to this article are available online at
repulsive forces.
encemag.org (this information is current as of May 7, 2014 ):
ctions in nematic
t. Nematic liquid
rdered
complex Fig.
information
and1. services,
including
high-resolution
figures,
can (A)
beMicrograph
found in
Dipolar and quadrupolar
colloids
in a thin layer of a nematic
liquid crystal.
of athe
d 0 2.32 mm silica sphere in an h 0 5 mm layer of 5CB with a hyperbolic hedgehog defect (black spot
online
director field with a symmetry reminiscent of closed disclination line (Saturn ring) surround- growth of kin
that of an electric quadrupole (18–21). In thicker ing the colloid (24). The two black spots on the rection perpen
parts, the nematic liquid crystal around the right and left side of the colloid in Fig. 1D C). Compariso
colloids had a symmetry like that of an electric represent the top view of the Saturn ring, en- an additional
dipole (1, 2, 18, 19).
circling the colloid. Quadrupolar colloids spon- position that c
Two-Dimensional
Nematic
Colloidal
the grow
Figure 1A shows a micrograph
of a silica
taneously Crystals
self-assemble Self-Assembled
into kinked chains moteby
Topological
Defects
sphere with diameter
d 0 2.32 T 0.02 mm in a oriented perpendicular to the rubbing direction with respect to
shows that th
nematic
layer with et
a thickness
Igor
Musevic
al. (h) of 5 mm. The (Fig. 1F).
structure of the director field around the colloid
In the experiments, laser tweezers were used tracted lateral
Science
313
,
954
(2006);
is shown in Fig. 1B. It is distorted dipolarly, to position colloids (25) and assist their assem- promotes the
DOI: 10.1126/science.1129660
colloidal crys
attracted to a
aration of sev
strates the lon
structural nem
This copy is for your personal, non-commercial use only.
energy of an
along the kink
10–18 J (È80
traction of an
side of a qua
(È120 kBT) th
in a quadrupo
sh to distribute this article to others, you can order high-quality copies for your
chains can re
es, clients, or customers by clicking here.
structure with
Saturn ring de
on to republish or repurpose articles or portions of articles can be obtained by
An examp
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quadrupolar c
2E. A single
tweezers close
wing resources related to this article are available online at
the optical trap
encemag.org (this information is current as of May 7, 2014 ):
strates the att
the unoccupie
information and services, including high-resolution figures, can be found in the online
structural forc
ded from www.sciencemag.org on May 7, 2014
the entangled loops, we performed a
Fig. 1, B to E, all the loop conformations are calculated structure.
However, the true richness of the knots and of topology-preserving Reidemeister m
likewise topologically equivalent to the unknot.
Reconfigurable
Knots
Linkswhen
in Chiral
Nematic
linksand
is revealed
the colloidal
clustersColloids
are which virtually transform the real phy
The simplest nontrivial
topological configUros
Tkalecofetlocal,
al. extended to arrays of p × q particles (Fig. 1G). formation of the loops into its planar
uration that is created by
a sequence
Scienceand
333optically
, 62 (2011);
The laser-assisted knitting technique was applied with the minimum number of crossin
isotropic-to-nematic, temperature,
10.1126/science.1205705
induced micro-quenches isDOI:
the Hopf
link (Fig. 1F). at multiple knitting sites so as to connect the tive or left-handed crossings (1) are fa
left-twisted nematic profile because o
metric constraint of the cell. The relaxa
Fig. 1. Topological defect
pings, illustrated in Fig. 1, G to J (righ
lines tie links and knots
surprising result. There is a series of
in chiral nematic colloids.This copy is for your personal, non-commercial use only.
torus knots and links (1): the trefoil
(A) A twisted defect ring
Solomon link, the pentafoil knot, and
is topologically equivalent
David. This generically knotted serie
to the unknot and appears
and links shows that the confining latt
spontaneously around a
loidal particles allows for the productio
single microsphere. The
If youorientation
wish to distribute
this article to others, you can order high-quality copies for your links and knots of arbitrary complexi
molecular
on
colleagues, clients, or customers by clicking here.
by adding and interweaving addition
the top and bottom of the
particles—that is, by increasing q.
Permission
toorirepublish or repurpose articles or portions of articles can be obtained by
cell coincides
with the
The knots and links can also be revers
following
the guidelines
here.
entation
of the crossed
poTopologically, this corresponds to locally
larizers. (B to E) Defect
following
resources related to this article are available online at
the mutual contact—the unit tangle (1)
loopsThe
of colloidal
dimer,
www.sciencemag.org
(this information is current as of May 7, 2014 ):
the two segments of the knotted line,
trimer,
and tetramers are
either cross or bypass one another in
equivalent to the unknot.
Updated
information
and
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including
high-resolution
figures,
can
be
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(F) The Hopf link is the
pendicular directions. We were able to
version of this article at:
first nontrivial
topologidisclination lines in the region of th
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cal object, knitted from two
tangle by applying the laser-induc
Supporting
Online
Material
can
be
found
at:
interlinked defect loops.
quench, as shown in Fig. 2, thus tra
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In (A) to (F), the correspondthe unit tangles one into another and co
ing loop
A listconformations
of selected additional articles on the Science Web sites related to this article can bechanging the topology of the present
were found
calculated
at: numericalmations. Starting from a tangle inside th
http://www.sciencemag.org/content/333/6038/62.full.html#related
ly by using
the Landau-de
region in Fig. 2A, the laser beam initia
Gennes free-energy model
tangle, and then by using precise posit
This article cites 31 articles, 7 of which can be accessed free:
(13). (G
to
J)
A
series
of
alhttp://www.sciencemag.org/content/333/6038/62.full.html#ref-list-1
intensity tuning of the beam, the line
ternating torus knots and
were reknotted into a distinct tangle
has arbeen cited by 9 articles hosted by HighWire Press; see:
links This
on 3 ×article
q particle
Further, we reknotted a tangle (Fig
rays http://www.sciencemag.org/content/333/6038/62.full.html#related-urls
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laser-induced
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fu- in the following subject collections:
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sion. Physics,
The defectApplied
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schematically
redrawn by using a program for representing knots (33) to show the relaxation them one into another. These local tr
tions change the topology and the h