What are topological defects?

 (image source)

It's often said that topology is the branch of mathematics where they can't tell the difference between a donut and a coffee mug.  They each have a single hole (the mug's handle and the donut hole), and that's all that matters.  If I may overanalyze this joke, the point seems to be that topology is so disconnected from our everyday experience.  How is this useful?

I wish to explain one particular use of topology in physics: topological defects.

A topological defect is a sort of knot that exists in the microscopic structure of a material.*  You can move the knot around from atom to atom, but you can't untie it.  We'll get into how that works soon enough.

*Material is a vague term for "stuff".  Later I'll discuss a few different materials including magnets, liquid crystals, and superconductors

2D Magnets

One classical material that everyone is familiar with are permanent magnets (aka ferromagnets).  Permanent magnets are made of lots of little atoms, each of which generates a tiny magnetic field.  In fact, many materials contain atoms that produce magnetic fields..  The miracle of permanent magnets is that all the magnetic fields of the atoms align together in the same direction.

(All images in this post, unless otherwise creditted, are my creations.  You may use them, if you credit me.)

Of course, none of the atoms knows which direction is the "right" direction.  Rather, each atom interacts with its nearby neighbors.  It's energetically favorable for nearby atoms to align with each other.  But if its neighboring atoms happen to be in the "wrong" direction (perhaps because of random motion, or impurities), there's no way for the atom to know.

Suppose that the atoms were instead in a configuration like one of these:

Both of these configurations are "smooth", meaning that each atom is pointing in nearly the same direction as its neighbors.  However, there seems to be a single discontinuity at the center of each configuration.  This is a topological defect.  Specifically, this is called a point defect, because it exists at one point.  It is not possible to remove the point defect without temporarily sacrificing the overall smoothness. Therefore, even though the topological defects are not the lowest-energy states, they are still somewhat stable.

To demonstrate, let's draw a loop around the point defect.  As we cross each arrow with our loop, take note of the direction of that arrow.  You'll note that the arrows change direction very smoothly (as long as we didn't draw the loop through the point defect).  Once you get back to the beginning of the loop, the direction of the arrow should be the same as where you started.  However, in the meantime, the direction of the arrow has gone around completely in a circle!

There's no way to smoothly deform the arrows so they don't go around in a circle.  To steal an analogy from my professor, it's like I tied a string in a loop around my finger.  I can't very well take the string off, because my finger is in the way (ignoring the possibility of slipping it off the end of my finger).

Generalizing order

To better understand these topological defects, we have to think about the order parameter manifold.  The "order parameter", in this case, is just the direction of each arrow.  The "order parameter manifold" is the set of all possible directions that the arrow can point.  In this example, the arrow may point in any direction in a 360 degree circle.  Therefore, the order parameter manifold is a circle.  If we draw a path around the circle, there is no way to continuously deform the path so that it no longer goes around the circle.  That's why there exists a point defect.

In fact, there will be two kinds of point defects.  One defect corresponds to going around the circle clockwise.  The other corresponds to going around the circle counter-clockwise.  (I showed one of each kind in an image above.)  As it happens, if these two point defects meet, they cancel each other out.  The defects behave a bit like particles and antiparticles, annihilating when they encounter each other.

Now let's consider the case where the magnetic field of each atom may point in any direction in three dimensions.  In this case, the order parameter manifold is not a circle, but a sphere (specifically, the surface of the sphere).  That makes a big difference, because if you draw a loop around the sphere, now it is possible to continuously deform the path into a single point.  (In mathematical terms, every loop is "homotopic" to a point.)  This means that there is no point defect (in 2D).

(Image source)

On the other hand, if you had an order parameter manifold which was shaped like a donut, then there would be some loops which you could not shrink down to a point.  These loops would correspond to point defects.

Some order parameter manifolds may be even more complicated.  For example, liquid crystals are made of long molecules.  In the "nematic" phase of liquid crystals, the orientation of the molecules becomes an order parameter.  But the order parameter manifold isn't quite a sphere, because you only need to go around 180 degrees to get back where you started.  Instead, it is an exotic topology called the real projective plane.  The real projective plane looks like a sphere, except that opposite points of the sphere are identified as the same point.  Here is a point defect in a 2D liquid crystal:

Even though I drew all the orientations to be in the plane, they are still free to move out of the plane.  But still there is no way to smoothly get rid of the defect.

What's interesting about defects in a real projective plane is that there are no defects and anti-defects.  There is only one kind of defect.  If any two of these defects meet, they will annihilate.  This makes sense, because one defect corresponds to going around the sphere 180 degrees.  Two defects corresponds to going around the sphere 360 degrees.

Other kinds of defects

All the above examples were point defects in 2D materials (though the order parameter manifold may have more dimensions).  But there are other kinds of defects in higher-dimensions.  For example, there is such a thing as a line defect, also called a vortex.

(I'm drawing more liquid crystals because my version of Mathematica can't draw arrows in 3D)

Whenever there would be a point defect in 2D, there will be a vortex in 3D.  All it takes is a loop in the order parameter manifold that cannot be continuously deformed to a point.

On the other hand, if there is a bubble in the order parameter manifold that cannot be continuously deformed to a point, then we can have point defects in 3D.  One example of such a defect is the "pincushion" pattern.

There's also another class of defects which aren't defects at all: topologically nontrivial field configurations.  Basically, the order parameter is smooth everywhere, but there's a funny pattern in the order parameter that cannot be removed.  Here is an example in a 2D liquid crystal:

The region in the center can be moved, squeezed, spread out, but it cannot be removed smoothly.  Topologically nontrivial field configurations exist in 3D as well, but it would be hard to visualize.  Incidentally, if there is a nontrivial field configuration in 2D, then there must be a point defect in 3D.  If there is a nontrivial field configuration in 3D, then hypothetically there would be a point defect in 4D.

Superconductors

I admit, I think it's just cool that topology appears in the real world, even if it's not so useful.  But at the same time, I'm sure it is useful.  I study superconductors, which have topological defects of their own.  The superconductor's order parameter manifold is the same as my first example: a circle.  (The order parameter, however, is not magnetic field direction, it's the complex phase of the superconducting condensate's wavefunction.)  So that means that superconductors can have vortices.

And superconducting vortices are really important!  As I've explained in a previous post, there are type I and type II superconductors.  Type I superconductors block any magnetic fields, and a sufficiently strong external magnetic field will destroy the superconductivity.  Type II superconductors can typically handle a stronger magnetic field, because they form vortices which the magnetic field can penetrate

(Image from Hyperphysics)

If we want superconducting wires with high electric current, we need them to tolerate the magnetic fields produced by that current.  The superconducting magnets in MRI machines, therefore, require type-II superconductors.  Topological defects: saving lives!

Skepticism spiralling

PZ Myers wrote a post called "I am so over the skeptical movement", criticizing a couple leaders in skepticism for their negative comments on feminism.  I will reproduce a comment I left there:

I find this a disturbing turn for The Great Internet Atheist-Feminist Wars–the increasing identification of skeptics with anti-feminists. And I can hardly fault PZ for that, when skeptical leaders are being so obtuse and/or disingenuous. I’m especially disappointed with Harriet Hall, who I’ve long respected.

Can we not let this happen? Harriet Hall and Ben Radford don’t own the skeptical movement anymore than Dawkins owns the atheist movement. We can change the culture by naming and shaming the bad (such as in this post), and highlighting skeptical leaders doing good.

As an example of Harriet Hall's obtuseness/disingenuousness, she was trying to defend her shirt that said, "I am a skeptic.  Not a 'woman skeptic.'  Not a 'skep-chick.'  Just a skeptic."  She wore this at a major skeptical conference, and it was largely seen as a "take that!" towards the Skepchicks, because I guess she doesn't like the Skepchicks, despite all their hard work.

Here's what Harriet Hall says in her defense:

 I have never criticized others for identifying as women skeptics; I only said I personally prefer to be identified as a skeptic rather than as a woman skeptic.

And then she approvingly cites the following comment on her shirt:

Harriet Hall’s T-shirt was brilliant! It encompassed free speech and equality. (just think we are all equal…we are all skeptics, not female skeptics and male skeptics but simply skeptics)

Which is it?  Is it her personal preference to not identify as a woman skeptic, or is it a public statement that we are all "just" skeptics?  Amanda Marcotte catches her saying lots of other things that betray her intention.

She's also largely missing the point.  Skeptical women's organizations and skeptical women's discussion spaces are not about separating skeptics into the men and women.  It's about dedicating a part of our time discuss and respond to specialized issues (if any).  Women's organizations are not just needed in skepticism, they're ubiquitous in all sorts of institutions.  For example, at my university, there's a society of women in the physical sciences.  Not a big deal, you know?

In my comment I talk about taking the skeptical movement back, and I'm semi-serious about that.  There's no reason organized skepticism has to be identified with second-wave feminism.  I totally appreciate PZ Myers criticizing prominent skeptics, and it's okay to disidentify with organized skepticism.  But we also need skeptical leaders to stand up and say that's not okay.  Where are they now?

Unfortunately, I am not what you call an "skeptical leader".  I'm just this guy with a blog with "skeptic" in the title.

Religion at Creating Change

I went to Creating Change last month, a national conference held by the National Gay and Lesbian Task Force.  Overall I enjoyed the conference, but I also got this overall impression that it was dominated by religious perspectives.  I don't wish to make a big deal out of it, but it was a minor annoyance to me, and I will use my inconsequential corner of the internet to vent.

The first visible sign was the workshop track called "Practice Spirit, Do Justice".  The descriptions lead me to believe that these workshops are:
1) Directed at making religious communities more LGBT-friendly.
2) Coming from a religious perspective.

Since my perspective is not religious, and not sympathetic to religion, these workshops are basically shouting out that they're not for me. 

It's fine if the workshops are not for me; there are over 25 workshops offered each session, and maybe three or four of them are in the "Practice Spirit, Do Justice" track.  In fact, I can totally get behind the idea of helping LGBT folk get more support from every community they interact with, and that includes their religious communities.

But at the same time, I really do not think it such a tragedy if many LGBT people choose to abandon religion. You know what's a tragedy?  Lots of people have to deal with being queer in a heteronormative family on top of being an atheist in a religious family.  Where are the workshops on this double coming out experience?  None!  You gotta find an atheist conference instead, because atheists are way more willing to talk about queer stuff than queers are to talk about atheist stuff.

(To be fair, there was a single atheist/agnostic caucus in the program hosted by Zack Ford.  Unfortunately it occurred at the same time as the asexual caucus, which I was obligated to attend instead.)

Now I'm just speculating on the content of these workshops (since I didn't feel comfortable in attending any), but I imagine at least a few of them were advancing religious arguments for being LGBT-positive.  That also annoys me.  I can accept that religious arguments are a good political strategy, but I disagree with the arguments, and I worry a lot about the adverse consequences of such a strategy.  I agree with what Greta said:

When we make a religious case for same-sex marriage -- heck, when we make a religious case for any matter of public policy -- we're conceding that public policy should be based on religion. And that means we're conceding the idea that policy and law should be decided, not on the basis of solid evidence and sound reasoning and basic human compassion, but on personal faith.

I worry that the same personal faith will be used to justify the wrong side of the next culture war.  I think this is already happening when people say that loving marriages or sexual activity are endorsed by religion (never mind people who want non-traditional relationship structures or who are asexual).

The above quote, by the way, is reacting to religious arguments advanced by Bishop Gene Robinson, who was given an award at this same conference.

There was also one workshop where religion came up, and it illustrates some of the dynamics going on between LGBT activists and atheism.

The workshop was about inclusivity in campus groups, and was led by a few student leaders.  One of the leaders explained that though they were raised "agnatheist", they'd come to understand that we need to be inclusive of people with many religious views.  The sentiment is nice, but the devil is in the details.  Their advice was to avoid putting Christopher Hitchens stickers on your laptop, and to avoid bringing the subject of religion up.  I wasn't the only one who objected to this.

Of course, the bad advice was mixed with good as well.  The student leader related a story where someone aggressively questioned another queer student about their religious faith.  How could they be queer and still identify with religion?  I agree that's a bad thing to do.  I'm up there with those "militant" atheists who thinks it's great to persuade religious people that they're wrong, and I don't even think it's necessary to be polite when you do it.  But even I recognize that you don't push people in private conversation to talk about something they don't want to talk about.  Save it for the public conversations, where no one is forced to stay and listen.

As you can see, there's this image of the atheists being overly pushy to the religious LGBT people.  Some atheists can be pretty pushy, I appreciate that.  It's weird though, that "religious inclusivity" is all about being nice to religious people, and not at all about how it feels to be in the non-religious minority.

As a final note, my experience is that the asexual community is way better about this.  There are not just lots of asexual atheists, there are lots of asexual atheists of the sort that participate in the atheist community.  I'm not the only one.

A fallacy diagram

For some reason I was inspired to draw a diagram of fallacies as I understand them.

The main point of the diagram is that there are really two kinds of fallacies.  First, there are formal fallacies, which are inferences that are not deductively correct.  Second, there are fallacies of induction, which are inductive arguments that are faulty or especially weak.

Formal fallacies are a category that technically includes induction.  This can lead to errors when you hold an inductive argument up to deductive standards.  For example, it is correct to argue that since the sun regularly has come up every morning, it will probably come up tomorrow morning as well.  But it's not deductively true, so you could call it a fallacy if you wanted to be a smartass.

The boundaries between proper induction and fallacious induction is somewhat fuzzy, and I tried to represent this in the diagram.  For example, argument from authority is a bit of a fallacy, since authorities are frequently wrong.  However, if the authority is shown to represent expert opinion, and if our resources are limited enough that we cannot investigate very deeply for ourselves, the expert opinion might be acceptable.  It could be very difficult to determine when inductive reasoning crosses that line into fallacious reasoning.

I also put a third category in there, assertions.  Assertions are not fallacies or deductions, because they involve no inferences.  Assertions can be useful in saving resources, since there's no point in advancing evidence or arguments for points we all agree on.  They are also useful to understand the differences between our positions.  They are not substitutes for arguments.

Earlier I mentioned mounting a critique of "name that fallacy"-style arguments.  This is where my critique would begin.  If all you have is a list of fallacies, and you don't distinguish between fallacies of induction and formal fallacies, then you're going to spot fallacies everywhere, even in good inductive arguments.  Or rather, you're going to spot fallacies everywhere you cast a critical eye, which will mostly be on your opponents.  This is a good way to get entrenched in your own beliefs, regardless of whether those beliefs are true or not.

More Brillouin Zones with origami

These are the first and second brillouin zones of the bcc crystal structure.  This is a geometrical shape that is sometimes used in condensed matter physics, which is my field.

These models are much smaller and simpler than the model I created for the second brillouin zone of the fcc crystal structure:

That's great. Simpler is better.  Anyway, now I can cross this off my bucket list:

1. Be a physicist.
2. Get in the top 25 of the US Puzzle Championship.
3. Write a novel.
4. Learn to play "Pyramid Song".
5. Create the second brillouin zones of the fcc and bcc structures using origami.

I realize that I haven't written about physics in a few months (depending on what counts).  Writing about physics takes a lot more work than writing about other things.  Also, it is hard to be in grad school and not be all cynical and jaded.  But I wanted to write something about "coffee cups and donuts" topology in condensed matter, because that's fun.

Pope Benedict resigns

I'm sure you've heard the news that Pope Benedict resigned (see NY Times, The Onion), probably due to declining health.  Chris H and PZ said they didn't care either way, but I care because I have the interest of an ex-Catholic.

I was still Catholic when Benedict became pope in 2005.  I recall most people (students and teachers) at my high school didn't like him.  The school was full of liberal Catholics, and Pope John Paul II was known to be one of the most liberal Popes ever.  Benedict, on the other hand, entered the papacy with the goal of healing rifts between the Church and its ultra-conservative factions.

Of course, John Paul II was only liberal in the relative sense.  It's worth noting that the scandals of priests raping children occurred before John Paul died.  And it was John Paul who decided that an appropriate response was to block people with "deep-seated homosexual tendencies" from being ordained (Ref).  He essentially ended the "don't ask don't tell" policy of the Catholic holy orders.  Sometimes the Catholic Church really is marching backwards.

The next pope may be slightly better or slightly worse than Benedict.  I don't have very high expectations, but the Church leads millions of people, so I hope for the best that is realistically possible.  Then I hope that people leave the Catholic Church so it doesn't matter.

The psi gulf

Skeptoid recently did an episode on Ganzfeld Experiments.  These are parapsychology experiments looking for telepathy.  A sender concentrates for 30 minutes on a target object, while the receiver wears headphones that play white noise and can only see uniform red light.  Then the receiver tries to pick out the target object from a line-up based on their impressions.  The theory behind the experiments is that any psi abilities we have are washed out by everyday noise.  The receiver is placed under conditions of reduced noise so that psi abilities may emerge.

Ganzfeld Experiments were able to find some small effects, one review showing a success rate of 30% (which is significantly higher than the 25% success rate from chance alone).  But there are numerous biases compromising the results, such as sensory leakage, the file drawer effect, and poor statistical analysis.  Ultimately, researchers seem to have seem to lost interest, and there have not been further attempts to replicate.

In this discussion, I was struck by the gulf between parapsychology claims and psychic claims.

In the Ganzfeld experiments, there's an effect of 5% increased success rate, and that's if you believe the methodology is sound.  It's also under perfect conditions with as little noise as possible.  I suspect that few psychics make specific claims about their success rates, but I gather that they're supposed to be right most of the time, with just enough exceptions that it doesn't hurt them when you point out a few past errors.  They're obviously claiming something better than 5%.

If it were only a 5% difference, I don't think we'd even be able to tell.  Are you able to estimate the probabilities of everyday occurrences in your life with precision of 5%?  What's the probability that there are enough seats when I get on the bus every morning?  What's the probability that I'm the first person in the office?  What's the probability that it will be warm enough for me to unzip my jacket?

If the parapsychology researchers are honest, then they have to admit that the small effects they observe have nothing to do with the paranormal perceived by most people.  This also goes for experiments on the efficacy of prayer.

FYI: Statistics on belief in paranormal.

A realistic way to categorize atheists

It's common to talk about different "kinds" of atheists, and it's often a lot of bullshit.  People really like talking about two kinds, because then they can say there's the good kind and the bad kind, and you should really be more like the good kind because I say so.  And then there's the atheism/agnosticism divide, which I also think is stupid.

You may rightly wonder, if I dislike all those other atheist categorization schemes, what categorization scheme do I use, if any?

I think there are two kinds of atheists.  There are those who participate in the atheist community, and there are those who don't.

Participation means: attending meatspace groups or conferences, reading popular atheist books, reading blogs or periodicals, watching youtube videos, following reddit, etc. etc.

There are obviously different levels of participation.  For example, on a blog there are lurkers, commenters, and bloggers.  In a college student group, there are people who only join the associated facebook page, people who only go to the big events, people who go to meetings, and officers.  It's also possible for different groups to be more or less closely associated with the atheist community.  Many skeptical groups, for example, are only tangentially related (and may even officially disavow any association).

So there's a lot of gray area between the two kinds of atheists.  Which makes sense, because I'm trying to create a realistic categorization, not a rhetorical device.  In the real world there are no clear-cut divisions.

To make things messier, this is not just a categorization scheme for atheists.  It applies to nonreligious people, nontheists, agnostics, deists, skeptics, humanists, and all the rest.  You can participate in the atheist community and not be an atheist.  It's just a matter of whether you feel a part of the community, or if you feel like an outsider who is always visiting.  Likewise, you can be an atheist and have no connection to the atheist community.  Some people just don't like communities, go figure.

It is hard to generalize about atheists who do not participate in the community.  Atheism is just a lack of belief in gods.  Sure, maybe most people who don't believe in gods also disbelieve in witches or psi, but who is to say?  It is much easier to generalize about people who participate in the community.  Generally speaking, atheists who believe in psi do not participate, for several reasons:

1. They may feel like they don't have much in common with the community.
2. Other people in the community may not welcome them to the community.
3. In participating with the community, they may stop believing in psi.

The values of the atheist community are an emergent property of atheism.  Generally speaking, the views of the atheist community consists of the majority view of atheists, whatever that may be.  If you have views which are in a sizable minority, you may stick around, because there's nothing wrong with a little disagreement, right?  If you have views which are in a tiny minority, you may feel like you have nothing in common with the community and feel no need to participate.

And of course, we in the community have some say in the process as well.  White males seem to be in the majority in the community, which tends to make women and people of color feel less welcome.  But writers and groups can go out of their way to be more welcoming.  Liberalism and feminism also seem to be disproportionately common, which may make conservatives and anti-feminists feel less welcome.  But then, many of us wonder, is it so bad if they feel unwelcome?  Not sure about that one.

None of this is all that surprising or unusual.  It's just the way a lot of human communities work.

Ragequitting is ineffective

I had in mind a post whose central thesis was, "Ragequitting a blog (as a reader) isn't very effective at whatever it's supposed to accomplish."  Ragequitting is when you announce loudly that you're going to stop reading.  This hasn't happened to me recently or anything, I was just thinking about the general phenomenon.

Alas, this will not be a very substantial post, since my argument is too brief.  When you quit reading something for political reasons, the blogger knows all too well that you are vastly outnumbered by people who quit reading for much more mundane reasons.  For example, length.

For my part, I have stopped reading a number of blogs simply because there were too many fluff posts.  I stopped reading Skepchick because I didn't care about their various regular posts.  After many years I stopped reading Bad Astronomy, because of too many posts about... I don't remember exactly.  Nothing political about it, and nothing against those blogs.

Rhetorical strategies against untestable claims

PZ Myers has spent more time complaining about organized skepticism, this time in response to Steven Novella.  You don't need to read all that though, it's mostly people talking past each other.

Anyway, a major point of contention is the correct response to someone who makes an untestable claim.  Steven Novella says that we cannot counter claims that are purely based on faith.  They are outside the realm of scientific inquiry:

As skeptics we can now say – that belief is not science-based. It is faith. Now the rules of faith apply – which means, in a secular society (see above) you don’t get to teach such belief in the public school classroom, and you don’t get funding for scientific research, you can’t impose your beliefs on others without violating their religious freedom, you cannot claim that insurance companies should cover your therapy, etc. It becomes a matter of personal faith only.

PZ Myers is unsatisfied with this.  He'd like to go further and directly counter faith-based claims:

“Faith” is not a magic get-out-of-jail-free word; I don’t think Novella would be stopped cold in his tracks if a homeopath invoked faith and god as a mechanism behind succussed water. Faith-based claims are empirical claims! When someone claims a vast cosmic intelligence named Jesus created the universe, I’m going to ask for their evidence for that claim; it is an empirical claim not just about how the universe works, but about how they arrive at their conclusions and what the chain of evidence that led them to that assertion is.

If you only read PZ Myers, it might seem like Steven Novella was advocating that we stop questioning beliefs as soon as their believers invoke faith.  But I do not think this is accurate.  When properly understood, this is not a matter of attacking faith vs letting faith be.  Rather, Novella and PZ advocate different kinds of attacks on faith.

Novella says, if your belief is faith based, then you cannot get into classrooms or acquire scientific funding, and your belief is "not even wrong".  PZ says, you think your belief is faith-based, but it's still a belief about empirical reality, and you better come up with evidence for it.

Viewed in this way, the disagreement between PZ Myers and Steven Novella seems much more trivial and unimportant then they make it out to be.  So we all don't like faith-based claims.  What does it matter if we take different rhetorical strategies to counter faith-based claims?

Novella's strategy is essentially the more conservative one.  It's the kind of argument you can make in the courtroom or other professional settings, without getting bogged down with philosophical details.  "No matter what you think about faith as a concept, you now have to agree that other people may reasonably disagree with you, and that your idea does not deserve public funds."  It's going after the believer's political power, because faith-based claims deserve no power.

PZ Myer's is the more radical strategy.  It's not just about attacking the believer's political power, it's going after the belief itself.  This is a useful strategy, because when believers retreat to faith, they generally don't quite grok what it is they're doing.  So we have to drive it home.  "In order for your claim to be faith based, it has to be a claim that does not affect reality in any observable way.  Are you sure that you want to say that the resurrection of Jesus has no effect on reality?  You know that even homeopaths and alien abductees invoke faith when criticized, so what makes you so different?"

Both of these rhetorical strategies seem useful, albeit in different contexts.  So I just don't see what the problem is.