Download Topological Defects 18.354 L24 Order Parameters, Broken Symmetry, and Topology James P. Sethna

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

[email protected] 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 x Here is shown the hedgehog defect, M ˆ. You can’t 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]

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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. maˆ x + naˆ y. 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 maˆ x + naˆ y. 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 + aˆ x = ⃗u + maˆ x + naˆ y.

(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

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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 n ˆ 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.

Disclineations 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 either arou number of times, DEFECTS then en IV. CLASSIFY THE TOPOLOGICAL cannot bent When which I was in graduate school, be the big fashion or was tw topological defects. Everybody was studying homotopy can’t change an intege groups, and finding exotic systems toby write papers about. It was, in the end, a reasonable thing to do.[11] It is true that in a fashion. typical application you’ll be able to figure out what the defects are without homotopy theory. You’ll do we categorize spend forever How drawing pictures to convince anyone else, t though. Most important, homotopy theory helps you to tals? think about defects. Well, there are two in A defect is a tear in the order parameter field. A topogo isaround the central hol logical defect a tear that can’t be patched. Consider the piece of 2-D crystal shown in figure 10. Starting in pass through it.is anInextrathe tra the middle of the region shown, there row of atoms. (This is called a dislocation.) Away from the midsponds precisely to distorted, the num dle, the crystal locally looks fine: it’s a little but there is no problem seeing the square grid and definofparameter. atomsCanwe passtheby. ing an order we rearrange atomsThis in a small region around the start of the extra row, and patch in the old days, and nobod the defect? No. The problem is that we can tell there is an exunderstand it.to the Wecenter. now tra row without ever coming near The ca traditional way of doing this is to traverse a large loop ofthethe surrounding defect,torus: 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 1 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-

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Defects in nematics winding number

Defects in nematics winding number

Topological Defects

evicˇ,1,2* Miha Sˇkarabot,1 Urosˇ Tkalec,1 Miha Ravnik,2 Slobodan Zˇumer2,1

crystal display technology. When foreign particles are introduced into the nematic liquid crystal, the orientation of nematic molecules is locally disturbed because of their interaction with the surfaces of the inclusions. The disturbance spreads on a long (micrometer) scale and can be considered as an elastic deformation of the nematic liquid crystal. Because the 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 313, 954 (2006); DOI: 10.1126/science.1129660

y to generate regular spatial arrangements of particles is an important technological and tal aspect of colloidal science. We showed that colloidal particles confined to a fewer-thick layer of a nematic liquid crystal form two-dimensional crystal structures that are topological defects. Two basic crystalline structures were observed, depending on the of the liquid crystal around the particle. Colloids inducing quadrupolar order crystallize y bound two-dimensional ordered structure, where the particle interaction is mediated by g of localized topological defects. Colloids inducing dipolar order are strongly bound into lectric-like two-dimensional crystallites of dipolar colloidal chains. Self-assembly by al defects could be applied to other systems with similar symmetry.

persions of colloids or liquid droplets a nematic liquid crystal show a divery of self-assembled structures, such as ns (1, 2), anisotropic clusters (3), twoal (2D) hexagonal lattices at interfaces ays of defects (6), particle-stabilized and cellular soft-solid structures (8). y of liquid crystals to spontaneously reign particles into regular geometric therefore highly interesting for develw approaches to building artificial colctures, such as 3D photonic band-gap 9). Current approaches to fabrication e controlled sedimentation of colloids ions (10), growth on patterned and pretemplates on surfaces (11), externalted manipulation (12), and precision y combined with mechanical microion (13). ropic solvents, the spatial aggregalloids is controlled by a fine balance the attractive dispersion forces and mb, steric, and other repulsive forces. e of colloidal interactions in nematic stals is quite different. Nematic liquid re orientationally ordered complex

fluids, in which rodlike molecules are spontaneously and collectively aligned into a certain direction, called the director. Because of their

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wing resources related to this article are available online at encemag.org (this information is current as of May 7, 2014 ): Fig. 1. Dipolar and quadrupolar colloids in a thin layer of a nematic liquid crystal. (A) Micrograph of a

nstitute, Jamova 39, 1000 Ljubljana, Slovenia. Mathematics and Physics, University of Ljubljana, , 1000 Ljubljana, Slovenia.

d 0 2.32 mm silica sphere in an h 0 5 mm layer of 5CB with a hyperbolic hedgehog defect (black spot on top). (B) The nematic order around the colloid has the symmetry of an electric dipole. (C) Dipoles spontaneously form dipolar (ferroelectric) chains along the rubbing direction. (D) The same type of colloid in a thin (h 0 2.5 mm) 5CB layer. The two black spots on the right and left side of the colloid represent the

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distorted nematic liquid crystal that had a colloid is now quadrupolar (Fig. 1E), with a rubbing direction director field with a symmetry reminiscent of closed disclination line (Saturn ring) surround- growth of kinked that of an electric quadrupole (18–21). In thicker ing the colloid (24). The two black spots on the rection perpendic parts, the nematic liquid crystal around the right and left side of the colloid in Fig. 1D C). Comparison o colloids had a symmetry like that of an electric represent the top view of the Saturn ring, en- an additional col Two-Dimensional Nematic Colloidal Crystals Self-Assembled by dipole (1, 2, 18, 19). circling the colloid. Quadrupolar colloids spon- position that crea Topological Defects Figure 1A shows a micrograph of a silica taneously self-assemble into kinked chains mote the growth with diameter d 0 2.32 T 0.02 mm in a oriented perpendicular to the rubbing direction with respect to th Igorsphere Musevic et al. shows that the nematic layer with a thickness (h) of 5 mm. The (Fig. 1F). Science 313 , 954 (2006); structure of the director field around the colloid In the experiments, laser tweezers were used tracted laterally t DOI:is shown 10.1126/science.1129660 in Fig. 1B. It is distorted dipolarly, to position colloids (25) and assist their assem- promotes the gro colloidal crystals attracted to a spe aration of severa strates the long-r structural nematic This copy is for your personal, non-commercial use only. energy of an ad along the kinked 10–18 J (È800 k traction of an is side of a quadru (È120 kBT) than sh to distribute this article to others, you can order high-quality copies for your in a quadrupolar chains can rearr es, clients, or customers by clicking here. structure with a m on to republish or repurpose articles or portions of articles can be obtained by Saturn ring defec An example the guidelines here. quadrupolar collo 2E. A single col tweezers close to wing resources related to this article are available online at the optical trap. T encemag.org (this information is current as of May 7, 2014 ): strates the attract the unoccupied co information and services, including high-resolution figures, can be found in the online structural force b

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 services, including high-resolution figures, can be found in the online (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 http://www.sciencemag.org/content/333/6038/62.full.html 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 http://www.sciencemag.org/content/suppl/2011/06/29/333.6038.62.DC1.html 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 are knitted by the another distinct tangle (Fig. 2E). We laser-induced defect fu- in the following subject collections: This article appears exactly three tangles by reversibly tra sion. Physics, The defectApplied lines are http://www.sciencemag.org/cgi/collection/app_physics 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