Energy technology and research opportunities - DOE quadrennial review

If you are interested in the US Department of Energy's take on the current status and trends in energy technology and related research, I strongly encourage you to watch this talk, by Franklin "Lynn" Orr, current US undersecretary of energy for science and energy.  It's a 40 minute talk, full of a lot of information.

If you want to see the actual, detailed, referenced document with graphs and bibliography, see here.

Colloquium: Pluto and New Horizons

We had an excellent colloquium here today from John Spencer, one of the investigators on the New Horizons mission to Pluto.   Amazing stuff - if you ever get the chance to hear a talk by one of the mission members, don't pass it by.  A few facts that were striking:

• The ambient surface temperature on Pluto is something like 40 K, basically because of the slow release of energy from residual radioactive material in there.  I guess that makes it too warm for the Outsiders, so we'll just have to wait longer to purchase a hyperdrive.
• The surface of much of Pluto is geologically young - there seems to be something like a "nitrogen cycle" analogous to the earth's water cycle, whereby nitrogen ice sublimes, precipitates out on km-tall water ice mountains, and eventually flows in glacial form back down to nitrogen ice seas.
• The New Horizons spacecraft was the fastest thing ever launched directly from the earth's surface, and it passed the moon within 9 hours after launch, having already been boosted to solar escape velocity.  (It's slower in the end than Voyager 1 and 2 because those spacecraft got close gravity assists from both Jupiter and Saturn.)
• Pluto's moons other than Charon are, well, complicated.  Their rotational axes are nearly in the plane of their orbit about the Pluto system barycenter, and they're not all round, so they rotate and interact in complicated ways.
• Space is big.  Really big.  You just won't believe how vastly hugely mindbogglingly bit it is.  I mean, you may think it's a long way down to the road to the chemist's, but that's just peanuts to space.

APS March Meeting, day 4

I spent a big chunk of my last day at the meeting having conversations with a couple of my collaborators, but I did get to see a couple of impressive talks.

Prof. Martin Aeschlimann of Kaiserslautern presented the remarkable work by his group using time-resolved 2-photon photoemission microscopy (PEEM) to drive and monitor plasmons on the nanoscale and femtosecond timescale.  The technique is a mouthful.  It's like electron microscopy, only instead of shooting an electron beam at the sample and looking at the secondary electrons that come out, you illuminate the sample with ultrafast, intense pulses of 800 nm light.  If these excite a plasmon mode, then the very intense local electromagnetic field leads to nonlinear two-photon processes that cause photoemission of electrons from the sample, and those photoelectrons are collected by a high resolution electrostatic column similar to that in an electron microscope.  The result is, you can "see" plasmons with ~ 10 nm or better spatial resolution, and by varying the time delay between pump and probe optical pulses, you can watch plasmons decay, or transport energy coherently, or interfere with each other.  Amazing stuff.

After watching some talks about spin Hall physics (hugely growing activity there, and definitely worth multiple blog posts down the line), I watched a fascinating talk by Scott Kemp of MIT about the Iran nuclear deal - he was one of the US negotiators.  It was great to get a sense of the scientific and political reasoning behind the negotiations and their outcome, and there was information in the talk that I hadn't seen anywhere else.

Final thoughts on the meeting:

• The variety of topics and the level of activity in condensed matter physics these days is great to see.  It's an active, thriving field, with deep ideas, open questions, and some topics that could well have major technological impact.  More than ever, I feel like there is an untapped potential here for informing the public about this stuff.
• The meeting is almost too big at this point.  It's unwieldy, and often there are multiple great talks on similar topics scheduled simultaneously.   I'm curious to learn what the long-term plans are in terms of meeting (re)organization and abstract sorting.  It feels like there has to be a better way to do some of these things, but if there were easy answers they would have been implemented already.
• Finally there was coffee and tea available without making everyone pay through the nose.  Whoo-hoo!

APS March Meeting, day 3

(Note that I'm leaving out the parts of the meeting where I did things like chat with friends and colleagues, and visit the trade show - I doubt anyone wants to read that stuff.)

I started day 3 with some plasmonics talks.  A particularly remarkable piece of work was presented by Teri Odom, discussing her group's plasmonic lasing efforts.   Metal nanoparticles excited at their local plasmon resonance can support very large local enhancements of the electromagnetic field, effectively confining light to incredibly small, sub-wavelength volumes.  However, usually the plasmon modes are relatively broad, so that a photon doesn't "live" very long in those tiny volumes.  By combining many nanoparticles in a regular array, the interparticle coupling can lead to a collective, coherent narrowing of those resonances.  When combined with a gain medium (in this case IR-140, an infrared dye with emission commensurate with the resonance of the metal nanoparticle array), the result is an optically pumped laser, with emission that can be tuned across the dye's bandwidth by changing the index of refraction of the surrounding medium.

I then tried to learn about Weyl fermions. This is another example of a particle originally proposed in the high energy physics context, with some peculiar relationships between energy, momentum, and angular momentum, and then seen in the emergent properties of a condensed matter system.  Truth be told, the talks I saw focused much more on the photoemission techniques, materials, and the steady stream of high profile publications than on providing a pedagogical approach to these funky (quasi)particles.

Eli Yablonovitch gave a fun, informative Buckley Prize talk, on the history of photonic band gap systems and their use to engineer spontaneous emission, optical antennas, and lastly structural color in nature.  Regarding optical antennas, he argues strongly that it's useful to think of these things in the context of classical antenna theory (basically modeling the antenna as an equivalent circuit made from discrete inductors, capacitors, and resistors) rather than other approaches involving quantum optics concepts.  I'm sure he's right in many cases, but fundamentally it seems to me that lumped element models can't really work well when worrying about a number of problems.

Nadya Mason gave a compelling talk about the nature of superconductivity in islands of granular Nb, as a test case to better understand the low-T metallic state of many thin systems in which superconductivity can be suppressed.  It's elegant work gaining new insights into a classic problem.  Many aspects can be explained with a simple model involving the distribution of grain sizes (and hence local superconducting transition temperatures), though mysteries remain, such as how nearby islands coupled by a normal metal film really talk to each other.

After a fun lunch with blogger extraordinaire Chad Orzel, I heard Yong Chen from Purdue present his group's work on transport in small devices made from 3d topological insulators of sufficiently high quality that the bulk is actually insulating, like it's supposed to be.  The favorite materials are apparently BiSbTeSe₂, which can be exfoliated from bulk or grown in film form, and vapor-grown nanowires of Bi₂Te₃.  That work is here and here, respectively.

After some talks on VO₂ (it's still complicated), I rounded out the day by going to the end of the Kavli Frontiers symposium.  My colleague Naomi Halas gave an extremely impressive talk about plasmonic particles for heat transfer and steam generation, and this was followed by an exhuberent lecture from Duncan Brown, who presented the LIGO gravitational wave detection experiment.  His excitement and joy about the result were infectious.

Next:  my last half-day of the meeting, + final thoughts.

APS March Meeting, day 2

Another eclectic bunch of talks today:

• There was a very interesting session this morning about coupling superconductors to semiconductors - this is a topic that has a long history and has enjoyed a huge resurgence as people have figured out ways to create composite systems with wild properties, like Majorana fermions.  Amir Yacoby gave a talk about what happens when a superconductor (Al) is coupled to a strong spin-orbit semiconductor, a HgCdTe quantum well.  The superconducting order parameter leaks into the semiconductor (the proximity effect), and more interestingly, it oscillates in space between \(s\)-wave pairing (the electrons in each Cooper pair form an antisymmetric spin configuration,  \( (1/\sqrt{2})(| \uparrow \downarrow\rangle - |\downarrow \uparrow \rangle) \), that flips sign if you swap the electrons ) and \(p\)-wave pairing (the electrons forming a symmetric spin configuration, like \((1/\sqrt{2})(|\uparrow \uparrow\rangle + |\downarrow \downarrow \rangle)\).  From current data as a function of in-plane magnetic field and out-of-plane magnetic field, plus some disorder in the contact region, you can explain almost everything.  The next talk, by Dale van Harlingen, discussed superconductors coupled to the 2d surface of a 3d topological insulator, Bi₂Se₃, making Josephson junctions.  These things end up playing host to Majorana fermions, and can be used to push them around in interesting ways.
• Later, after seeing some contributed talks, chatting with folks, and visiting the trade show to get literature from a bunch of vendors, I stood through a talk about trying to detect evidence of dark energy with a (comparatively) "tabletop" atomic physics experiment.   A very cool topic, but the room was so claustrophobic I couldn't stay for the talk about the gravity-decoherence paper I'd mentioned here.
• After learning about "Advanced undergraduate labs:  why bother?", I went to the extremely dense session about spintronic devices beyond spin-transfer torque.   The metal spin device toolkit is now very extensive, and it will be interesting to see if the materials issues can be worked out well enough to produce devices that will really revolutionize information storage and processing.  Power dissipation remains a big issue.  Here is a recent review article on this stuff (sorry - I didn't want to direct-link to someone's private copy of the pdf).  I should write a separate post on this stuff.

More tomorrow....

APS March Meeting, day 1

First, hat tip to Chad Orzel for this article, and ZapperZ for his.  While condensed matter physics is harder to describe to a general audience, it's shaped your everyday life far more than string theory or neutrino oscillations.  We as a community need to do a better job getting that across, as well as the wonder that some of these topics inspires.  Interesting talks that I saw today (aside from those of my group members, of course):

• There is a lot of interest in trying to capture optical energy (e.g., from the sun) and not waste so much of it.  Plasmons in metals provide one way of converting a photon into electron-hole excitations in a metal - the trick is to then do something useful with those "hot" electrons and holes.  Lisa Krayer spoke about a clever approach of putting a metal film grating on the back of a Si photovoltaic system, to grab photons too low in energy for the Si itself into plasmons, and then kick "hot" electrons back into the Si.  As an added bonus, the optical properties of the Si (high index of refraction) end up implying that the grating can capture light over a much larger range of incident angles than if the grating was on the front side.  Similar in spirit, Prinaha Narang spoke about theoretical modeling of the electrons in these and similar plasmonic structures, with an eye toward manipulating (through geometry) the momentum and energy distributions of the hot electrons and holes.
• Hsin-Zon Tsai gave an interesting talk about using an underlying gate electrode to change not just the charge density in a layer of graphene, but also to manipulate the amount of charge on a molecule (called F4TCNQ) tethered to the graphene.  Measuring by scanning tunneling microscope, Tsai and coworkers showed that the highest occupied molecular level always sat lower in energy than the Dirac point of the graphene, and made a nice argument in support of this involving image charges.
• In his talk in honor of receiving the Adler Prize, Harry Atwater gave a nice overview of his group's plasmonics efforts, including a discussion of their concept of the plasmoelectric effect:  Illuminating a plasmonic object in an environment where it can gain or lose charge can drive charge transfer, as explained here.  
• We are used to employing ferromagnets in electronic devices.  Maxim Tsoi gave a very clear talk about some remarkable work using antiferromagnets, both for magnetoresistive devices and for the manipulation of and by spin currents.  The next talk in that session, by Wei Zhang, described recent work where antiferromagnetic alloys were used as sources of spin currents.  Very pretty stuff.
• I also caught part of the session where various historians of science (and a noted blogger) critiqued/commented on Steven Weinberg's latest book, with Weinberg in the room to offer rebuttal.  

More tomorrow....

APS March Meeting 2016

It's that time of year again, when a bit under 10,000 condensed matter/materials/polymer physicists gather in a meeting that is now 1.7 times as large as it was when I first started going to these things.   This year the festivities are in Baltimore, and as I've done in the past I will try to give some snapshot of bits that caught my interest (though my session attendance is of course partly driven by my group's talks).  If there are particular things my readers think I should see, hopefully they will point them out in the comments.  If you are at the meeting, I encourage you to stop by the Cambridge booth at the exhibition and pick up some copies of my book as gifts for your friends.

Unidentified Superconducting Objects

The search for new superconductors has been going on for decades, because the potential promise of room temperature superconductors (with useful properties, like high critical fields, high critical currents, chemical stability, the ability to be integrated in some way into wires, ribbons, or tapes) is so enormous.  Littering the metaphorical laboratory floor are various claims over the years of "unidentified superconducting objects" - a term attributed to Paul Chu to describe one-off, irreproducible hints of 200-300 K superconductivity, often features in resistivity or magnetization that look like they could originate in some unknown impurity phase of an already complex material.  I was reminded of this by a paper that showed up on the arxiv last night.  Most likely this will fade away, but these things are always intriguing.  Extraordinary claims require extraordinary evidence, of course.

Google scholar question

Readers:  I suspect many of you are familiar with Google Scholar, Google's free approximation of what Thomson-Reuters offer for a fee.   Scholar is a nice tool for searching references, though as a Google product it uses something similar to their pagerank algorithm, meaning that it can heavily bias searches in favor of papers that have been highly cited (though this can be tweaked or avoided in many ways).

Google Scholar gives you the opportunity to create a public profile as well, so that people can see at a glance your publications and their citations, keywords that you choose to describe your research area, instantly calculated metrics such as citation counts and the h-index, etc.  (Some people like the Google Scholar h-index because it is systematically higher than the one from Thomson-Reuters, since it does a better job of catching bibliographic references in books and online resources.  That, and our culture of encapsulating complex things in single numbers biases us toward preferring higher numbers.)  I do have a profile, though I have mixed feelings about the score-keeping aspects of these things.

One reason I do have a profile is that Google Scholar has a feature that I've found interesting (if not necessarily useful) in the past:  Based on your papers, where they're being cited, your research interests, etc., every few days Google Scholar comes up with suggested literature that it lists under a "My Updates" tab on your profile.  These are new papers that either cite your work or Google's algorithm computes that you would likely be interested in the subject matter.

A month ago, I stopped receiving new "updates".  The most recent one that shows up in my queue is from January 31.  Moreover, when I look at my profile now, the "Co-authors" list, which previously had been populated automatically by Google Scholar based on my publications, is now completely empty.  As far as I know, I have made no changes to my profile or settings.  I have looked extensively and not found any reason why this should have changed.  I used the feedback link to ask Google Scholar support about this, to no avail so far.  I received an automated response with pieces of their FAQ list, and was told to reply to the email if that was not sufficient.  I did so several days ago, with no response yet.

Has anyone else had these issues?  Anyone have any suggestions for resolving this?  I don't really care about the "Co-author" bit as I don't use that for anything, but I actually liked the article updates.

Update:  The issue seems to have been fixed by Google!  Woo-hoo!

Patience and scientific research

The Washington Post ran this nice column by Alan Lightman the other day, where Lightman points out that the recent LIGO discovery is the result of a long, sustained research program going back decades.  In it, Lightman points out that the modern go-go instant-gratification culture is the antithesis of this, and he hints (but does not say explicitly) that we need to worry about whether we are rewarding the right things in terms of research.

There is no question that we have arrived at a scientific research culture that, at least at the individual investigator level rather than large collaborations, tends to reward rapid progress and nimbleness.  One of the more damning comments that can appear in a proposal or paper review is that some piece of work or proposed research idea is "incremental" or "just the next logical step".   It is hard for me to see in the present environment how an individual investigator could get sustained federal funding to work on a single extremely ambitious, long timescale project unless there were many high-impact milestones along the way.  Of course the right answer is probably that we should support a mixed portfolio of research with inherently different timescales for payoff, but the continual trend of short-termism (how many "research products" came out in the last reporting period?  how fast is some promotion candidate's h-index growing year-over-year?) is not comforting.

Light-induced superconductivity

The physics of systems driven out of equilibrium remains a frontier topic, and as new techniques are enabled involving ultrafast lasers, exciting developments have been coming along.  Light-induced (possible) superconductivity is one example that has gotten a lot of attention lately.  Several years ago, the group of Andrea Cavalleri started with a non-superconducting copper oxide material closely related to the high-T_c superconductors.  By smacking this system with a light pulse intended to disrupt an intervening phase ("stripe order", a kind of spontaneous modulation of the charge density in the material into stripes), they were able to get the material to have an optical response that looked just like that of a superconducting cuprate, at least on the picosecond timescale.

Around this same time, Cavalleri and others pointed out that carefully tailored light pulses could also be created that would couple to particular vibrational modes of crystals.  That way, again on the picosecond timescale, one could imagine reaching into a crystal and distorting it transiently.   The Max Planck group made use of this approach in a very deliberate way.  There is a trend toward higher superconducting transition temperatures in the cuprate superconductors as particular bonds within the lattice are distorted due to the overall crystal structure.  What these folks did was hit a cuprate, YBa₂Cu₃O_6.5, with a pulse designed to transiently distort the bonds even more in the favorable-for-superconductivity direction.  Again, they found an optical response that indicated, below T_c, strengthened superconductivity, and above T_c, optical signatures similar to that of superconductivity all the way up to room temperature!  Subsequent ultrafast x-ray diffraction measurements indicated that the lattice really was distorting as desired in those experiments.  

The very recent attention has resulted from this paper, where this group has again optically driven some lattice modes, inducing signatures in the optical response that look very much like superconductivity well above T_c, this time in K₃C₆₀.  Interestingly, in this case when the material is already superconducting, it doesn't seem like the optical pulse enhances the superconductivity.

All of this is very cool, though it's important to remember that these are transient effects, and on the timescales so far it is extremely difficult to perform any other measurements (e.g., non-optical ones, like magnetometry) that would independently test for superconductivity.  Still, this is an impressive strategy, and beyond nonequilibrium physics it strongly suggests that greater control over material structure (than what we have so far been able to achieve) could pay enormous dividends.

The end of Moore's Law

There is a nice article at Nature this past week talking about the possible impending end of Moore's Law.  The quick summary:  It's really looking like we are approaching the end of one of Moore's laws (that the number of transistors on a chip doubles roughly every 18 months).  Bear in mind that the endurance of this form of Moore's law is not an accident - the growth of transistor density transformed from an empirical observation made by Moore to a growth target adopted by the semiconductor industry decades ago.

There are many reasons why continued aggressive transistor scaling is difficult.  I write about these at some length in my book.  Clearly we are starting to approach the limit where devices are so small that atomic-scale differences in geometry and composition can start to affect performance.  Power density, even when transistors are nominally "off", is becoming a major problem.  This is one reason mentioned in the article why clock speeds on processors have basically stopped climbing.  (Oddly, I never hear anyone mention the other major reason that clock speeds have plateaued at a few GHz:  Going much higher in frequency makes layout and circuit design a much more difficult microwave engineering task.)  

The article discusses possible radical shifts in strategy to extend the trend of increasing processor performance.  These include major changes in materials (obligatory mention of graphene and two-dimensional semiconductors) and architecture (going 3d in circuit design; quantum computing; the increasingly trendy neuromorphic computing).  There are also major efforts to think about computing at lower powers.  While it's cool to talk about these, I have to say that the enormous economic advantage of silicon (an individual Si transistor costs an infinitesimal fraction of a cent, and we know how to make a billion of them at a time and have them all work for a decade) makes it very difficult to see how any competing material gains significant ground for a long time.  

Brief news items

As I go on some travel, here are some news items that looked interesting to me:

• Rumors are really heating up that LIGO has spotted gravity waves.  The details are similar to some things I'd heard, for what that's worth, though that may just mean that everyone is hearing the same rumors.  update:  Press conference coming (though they may just say that the expt is running well....)
• The starship Enterprise is undergoing a refit.
• This paper reports a photocatalytic approach involving asymmetric, oblong, core-shell semiconductor nanoparticles, plus a single Pt nanoparticle catalyst, that (under the right solution conditions) can give essentially 100% efficient hydrogen reduction - every photon goes toward producing hydrogen gas.   If the insights here can be combined with improved solution stability of appropriate nanoparticles, maybe there are ways forward for highly efficient water splitting or photo production of liquid fuels.
• Quantum materials are like obscenity - hard to define, but you know it when you see it.

What is density functional theory? part 3 - pitfalls and perils

As I've said, DFT proves that the electron density as a function of position contains basically all the information about the ground state (very cool and very non-obvious).  DFT has become of enormous practical use because one can use simple noninteracting electronic states plus the right functional (which unfortunately we can't write down in simple, easy-to-compute closed form, but we can choose various approximations) to find (a very good approximation to) the true, interacting density.

So, what's the problem, beyond the obvious issues of computing efficiency and the fact that we don't know how to write down an exact form for the exchange-correlation part of the functional (basically where all the bodies are buried)?  

Well, the noninteracting states that people like to use, the so-called Kohn-Sham orbitals, are seductive.  It's easy to think of them as if they are "real", meaning that it's very tempting to start using them to think about excited states and where the electrons "really" live in those states, even though technically there is no a priori reason that they should be valid except as a tool to find the ground state density.  This is discussed a bit in the comments here.  This isn't a completely crazy idea, in the sense that the Kohn-Sham states usually have the right symmetries and in molecules tend to agree well with chemistry ideas about where reactions tend to occur, etc.  However, there are no guarantees.

There are many approaches to do better (e.g., some statements that can be made about the lowest unoccupied orbital that let you determine not just the ground state energy but get a quantitative estimate of the gap to the lowest electronic excited state, and that has enabled very good computations of energy gaps in molecules and solids; time-dependent DFT, which looks at the general time-dependent electron density).  However, you have to be very careful.  Perhaps commenters will have some insights here.  

The bottom line:  DFT is intellectually deep, a boon to many practical calculations when implemented correctly, and so good at many things that the temptation is to treat it like a black box (especially as there are more and more simple-to-use commercial implementations) and assume it's good at everything.  It remains an impressive achievement with huge scientific impact, and unless there are major advances in other computational approaches, DFT and its relatives are likely the best bet for achieving the long-desired ability to do "materials by design".  

What is density functional theory? part 2 - approximations

So, DFT contains a deep truth:   Somehow just the electronic density as a function of position within a system in its lowest energy state contains, latent within it, basically all of the information about that ground state.  This is the case even though you usually think that you should need to know the actual complex electronic wavefunction \(\Psi(\mathbf{r})\), and the density (\(\Psi^{*}\Psi\)) seems to throw away a bunch of information.

Moreover, thanks to Kohn and Sham, there is actually a procedure that lets you calculate things using a formalism where you can ignore electron-electron interactions and, in principle, get arbitrarily close to the real (including interaction corrections) density.  In practice, life is not so easy.  We don't actually know how to write down a readily computable form of the complete Kohn-Sham functional.  Some people have very clever ideas about trying to finesse this, but it's hard, especially since the true functional is actually nonlocal - it somehow depends on correlations between the density (and its spatial derivatives) at different positions.   In our seating chart analogy, we know that there's a procedure for finding the true optimal seating even without worrying about the interactions between people, but we don't know how to write it down nicely.  The correct procedure involves looking at whether each seat is empty or full, whether its neighboring seats are occupied, and even potentially the coincident occupation of groups of seats - this is what I mean by nonlocal.

Fig. from here.

We could try a simplifying local approximation, where we only care about whether a given chair is empty or full.  (If you try to approximate using a functional that depends only on the local density, you are doing LDA (the local density approximation)).  We could try to be a bit more sophisticated, and worry about whether a chair is occupied and how much the occupancy varies in different directions.  (If you try to incorporate the local density and its gradient, you are doing GGA (the generalized gradient approximation)).   There are other, more complicated procedures that add in additional nonlocal bits - if done properly, this is rigorous.  The real art in this business is understanding which approximations are best in which regimes, and how to compute things efficiently.

So how good can this be?  An example is shown in the figure (from a summer school talk by my friend Leeor Kronik).  The yellow points indicate (on both axes) the experimental values of the ionization energies for the various organic molecules shown.  The other symbols show different calculated ionization energies plotted vs. the experimental values.  A particular mathematical procedure with a clear theoretical justification (read the talk for details) that mixes in long-range and short-range contributions gives the points labeled with asterisks, which show very good agreement with the experiments.

Next time:  The conclusion, with pitfalls, perils, and general abuses of DFT.

What is density functional theory? part 1.

In previous posts, I've tried to introduce the idea that there can be "holistic" approaches to solving physics problems, and I've attempted to give a lay explanation of what a functional is (short version: a functional is a function of a function - it chews on a whole function and spits out a number.).  Now I want to talk about density functional theory, an incredibly valuable and useful scientific advance ("easily the most heavily cited concept in the physical sciences"), yet one that is basically invisible to the general public.

Let me try an analogy.  You're trying to arrange the seating for a big banquet, and there are a bunch of constraints:  Alice wants very much to be close to the kitchen.  Bob also wants to be close to the kitchen.  However, Alice and Bob both want to be as far from all other people as possible.  Etc. Chairs can't be on top of each other, but you still need to accommodate the full guest list.  In the end you are going to care about the answers to certain questions:  How hard would it be to push two chairs closer to each other? If one person left, how much would all the chairs need to be rearranged to keep everyone maximally comfortable?    You could imagine solving this problem by brute force - write down all the constraints and try satisfying them one person at a time, though every person you add might mean rearranging all the previously seated people.  You could also imagine solving this by some trial-and-error method, where you guess an initial arrangement, and make adjustments to check and see if you've improved how well you satisfy everyone.  However, it doesn't look like there's any clear, immediate strategy for figuring this out and answering the relevant questions.

The analogy of DFT here would be three statements.  First, you'd probably be pretty surprised if I told you that if I gave you the final seating positions of the people in the room, that would completely specify and nail down the answer to any of those questions up there that you could ask about the room.¹  Second, there is a math procedure (a functional that depends on the positions of all of the people in the room that can be minimized) to find that unique seating chart.²  Third, even more amazingly, there is some mock-up of the situation where we don't have to worry about the people-people interactions directly, yet (minimizing a functional of the positions of the non-interacting people) would still give us the full seating chart, and therefore let us answer all the questions.³

For a more physicsy example:  Suppose you want to figure out the electronic properties of some system.  In something like hydrogen gas, H₂, maybe we want to know where the electrons are, how far apart the atoms like to sit, and how much energy it takes to kick out an electron - these are important things to know if you are a chemist and want to understand chemical reactions, for example.  Conceptually, this is easy:  In principle we know the mathematical rules that describe electrons, so we should be able to write down the relevant equations, solve them (perhaps with a computer if we can't find nice analytical solutions), and we're done.  In this case, the equation of interest is the time-independent form of the Schroedinger equation.  There are two electrons in there, one coming from each hydrogen atom.  One tricky wrinkle is that the two electrons don't just feel an attraction to the protons, but they also repel each other - that makes this an "interacting electron" problem.  A second tricky wrinkle is that the electrons are fermions.  If we imagine swapping (the quantum numbers associated with) two electrons, we have to pick up a minus sign in the math representation of their quantum state.  We do know how to solve this problem (two interacting electrons plus two much heavier protons) numerically to a high degree of accuracy.  Doing this kind of direct solution gets prohibitively difficult, however, as the number of electrons increases.

So what do we do?  DFT tells us:
¹If you actually knew the total electron density as a function of position, \(n(\mathbf{r})\), that would completely determine the properties of the electronic ground state.  This is the first Hohenberg-Kohn theorem.

²There is a unique functional \(E[n(\mathbf{r})]\) for a given system that, when minimized, will give you the correct density \(n(\mathbf{r})\).  This is the second Hohenberg-Kohn theorem.

³You can set up a system where, with the right functional, you can solve a problem involving noninteracting electrons that will give you the true density \(n(\mathbf{r})\).  That's the Kohn-Sham approach, which has actually made this kind of problem solving practical.

The observations by Kohn and Hohenberg are very deep.  Somehow just the electronic density encodes a whole lot more information than you might think, especially if you've had homework experience trying to solve many-body quantum mechanics problems.  The electronic density somehow contains complete information about all the properties of the lowest energy many-electron state.  (In quantum language, knowing the density everywhere in principle specifies the expectation value of any operator you could apply to the ground state.)

The advance by Kohn and Sham is truly great - it describes an actual procedure that you can carry out to really calculate those ground state properties.  The Kohn-Sham approach and its refinements have created the modern field of "quantum chemistry".

More soon....

CalTech wins the whole internet - public outreach for quantum.

This makes my public outreach efforts look lame by comparison.  Well done!

What is a functional? Ex: the Action Principle

Working our way toward the biggest theory most people have never heard of, let's talk about functionals, using the non-rigorous language that physicists like and which annoys mathematicians.

Here's an analogy.   You want to drive from your house to the store.  There are many possible routes, and for each route we could come up with a single number that depends on the route - it could be the total distance traveled, or the total time it took to get from the house to the store, or it could be the total fuel consumed, or it could be the number of times you turned left minus the number of times you turned right.  We could take all your possible routes, and we could somehow process each possible route into a number.  The operation that chews on your route information and converts it to a number is a functional of your path from the house to the store.  (Why would you want to do this?  Well, perhaps you value your time, and you want to pick the route that has the least accumulated time.  Perhaps you value fuel costs, and you want to pick the route that has the least fuel consumption.  The point is, depending on what you care about, a functional can let you pick between alternatives, here the routes, that are described by a huge, effectively infinite number of variables.)

In the spirit of MTW, a function of a single variable is a machine that takes a number, chews on it, and spits out a number.   This could be \(y(x) = x^{2}\), for example.  A function of multiple variables is a machine that takes more than one number, chews on them, and spits out a number -- like \(y(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + 3x_{2} - x_{3}\).  For this example, for any set of three numbers \( \{x_{1}, x_{2}, x_{3}\} \), you can compute a value of \(y\).  

A functional is the "continuum limit" of a function of multiple variables - it's a machine that takes an infinite number of numbers (!), chews on it, and spits out a single number.  We can cast our example of Fermat's principle of least time this way.  Suppose light starts out at point P, and we let it take some wild path like the one shown in the figure.  We're eventually going to have the light wind up at point Q.  How long does it take the light to get from P to the interface?  Well, that depends on how you think it goes.  If you knew all the intervening points \((x_{i},y_{i})\), you could compute the distance between successive points, and add up all the times.  The transit time \(t_{\mathrm{tot}}\) depends on the whole trajectory that the light takes from P to Q.  Instead of writing \(t_{\mathrm{tot}}(x_{1}, y_{1}, x_{2}, y_{2}, .....)\), we write \(t_{\mathrm{tot}}[x,y]\), where the square brackets indicate that this is a functional.  For any goofy trajectory we could draw from P to Q, we could compute \(t_{\mathrm{tot}}\).  Fermat's principle of least time says that the one actually taken by light is the one that gives the smallest value of \(t_{\mathrm{tot}}\).  Why does this work?  That's actually a very deep question, and I won't try to answer it now.

The Action Principle is the most famous example of showing that functionals can be incredibly useful in physics.  I'm going to do a simple 1d example involving mechanical motion of a particle, but everything I will say generalizes to much more complicated cases.  Suppose we have a particle that starts at some initial position position \(x_{\mathrm{i}}\) at some initial time \(t_{\mathrm{i}}\), and ends up at some final position \(x_{\mathrm{f}}\) at some final time \(t_{\mathrm{f}}\).  We want to know, how does the particle get there?  Which of the essentially infinite number of possible trajectories \(x(t)\) did the particle take?  (Note that by allowing any arbitrary path \(x(t)\), we're also basically permitting any arbitrary velocity as a function of time in there.)

The local way to answer this problem is to start with the particle at the initial location and time, and apply Newton's laws.  From its position find the force acting on the particle, use that force to find the acceleration, and take a little timestep forward, updating the particle's position and velocity.  Now repeat this.

The Action Principle is a global approach.  It says that there is some functional called the action, \(S[x(t)]\).  For any trajectory \(x(t)\), you can compute a number \(S\).  The trajectory that a classical particle takes is the one that starts and ends in the right places and times, and produces the minimum* value of \(S\).  The form of \(S\) contains all the physics.  (For a 1d particle obeying Newton's laws, the correct form for \(S\) is the integral as a function of time over the whole trajectory of (the kinetic energy minus the potential energy).)  This is one of the stranger things to learn when studying physics - with the right procedure for writing down and expression for \(S\), and the right procedure for minimizing it (techniques called variational calculus), it seems like the (global) Action Principle is nearly magical, giving you ways to solve problems that would seem hopelessly complex in traditional (local) approaches.   Why does this actually work?   Again, this is a deep question, and I'll revisit it some other time.  The fact that you can actually come up with a functional-based formalism does indicate that there is "hidden" structure to nature beyond what you might guess just from, e.g., Newton's laws.

To revisit the analogy:  If I told you that there was a way to predict how you would drive from home to the store based on a single number related to each possible route, you would realize:  (1) you don't necessarily have to know all the detailed rules of driving to find the preferred route, just how to calculate that number; and (2) there clearly is some deeper principle at work than just the rules of driving that picks out the route you take.

Next time, I'll finally get to the point about density functional theory.

*Technically, a maximum could also work here, but for many many cases, there is no maximum possible value of \(S\).

"Local" vs "global" ways to solve physics problems

Inspired by a recent post of Ross McKenzie, I thought it would be fun to try to write a popularly accessible piece about the enormously successful, wholly remarkable  theory that most people have never heard of, density functional theory.

To get there will require a couple of steps.  First, it's important to appreciate that sometimes, thanks to the mathematical structure of the universe, it is possible to think about and solve physics problems with two seemingly very different approaches - call them "local" and "global".  In the local approach, we write down equations that describe the underlying problem in great detail, and by carefully working out their solution, we arrive at an answer.  In the global approach, we come at the problem from an overview perspective of considering possible solutions and figuring out which one is correct.

For example, let's think about a light ray propagating from point P (in air) to point Q (in water), as shown in the figure (courtesy wikipedia).  It turns out that light travels at a speed \(c/n\) in a medium, where \(c\) is the speed of light in vacuum, and \(n\) is the "index of refraction" that depends on the material and the frequency of the light.  (This is already short-hand for solving the complicated problem of electromagnetic radiation and its interactions with a material containing charges, something that Feynman wrote about elegantly in this book, based on these lectures.)  The "local" approach would be to write down the equations describing the electromagnetic light waves, and solve these, including the description of the air, the water, and their interface.  The result we would find is so simple and compact that we teach it to freshmen, Snell's Law:  \(n_{1}\sin(\theta_{1}) = n_{2}\sin(\theta_{2})\), where the angles are defined in the figure.

The "global" way to solve this problem (and again arrive at Snell's Law) was found by Fermat (yes, the one with the "last" theorem).  He didn't have the option of solving the microscopic equations governing the radiation, since he died two hundred years before Maxwell published them.  Instead, Fermat knew that light seems to travel in straight lines within a given medium.  Therefore, he considered all the possible paths that a light ray could take from P to Q (such as the blue and green alternatives shown in the modified figure), trying to figure out which combination of straight segments (and hence which angles) were picked out by nature.  The answer he posited was that the correct path for the light is the one that minimizes the overall time taken by the light in going from P to Q.   This does give Snell's Law as a consequence, and seems to hint at a deeper organizing principle or structure at work than just "we solved complex equations with tricky boundary conditions, and Snell's Law fell out".  (These days, if a student is asked to derive the Snell's Law from Fermat's Principle of Least Time, they would use calculus to do so, since that plus coordinate geometry provides a clear way to right down an expression for the transit time and a way to minimize that function.  Fermat couldn't do that, as modern calculus didn't exist at the time, though he was among the people thinking along those lines.  He was pretty sharp.)

Next up:  another example of a "global" approach, the Action Principle.

APS elections - reminder

Sorry for the year-end lull in posting.  Work-related writing is taking a lot of my time right now, though I will be posting a few things soon.

In the meantime, a reminder to my APS colleagues:  The APS divisional elections are going on right now, ending on January 4.  The Division of Condensed Matter Physics and the Division of Materials Physics are both holding elections, and unfortunately there were some problems with the distributions of the electronic ballots, particularly to people with ".edu" email addresses.  These issues have been resolved and reminders sent, but if you are a member of DCMP or DMP and have not received your ballots electronically, I urge you to contact the respective secretary/treasurers (linked from the governance sections of the division webpages).   (Full disclosure:  I'm a candidate for a DCMP "member-at-large" position.)