Interacting Quantum Systems Driven Out of Equilibrium - day 2

Continuing into day 2 of our workshop:

• Bryce Gadway of the University of Illinois spoke about using cold atoms in an optical lattice to simulate topological and disordered systems.  His group has implemented an optical lattice constructed not by interference of retroreflected lasers, but by interference between a laser and counterpropagating beams frequency shifted by precise, controlled amounts.  As a non-atomic physics person I'm a bit fuzzy on the details, but the point is that this allows his group to put in place precise control of the on-site potential of each lattice site and to dial in designer phase shifts associated with tunneling between adjacent sites, on demand.  This means it is possible to study transport problems (like Bloch oscillations) as well as introducing designer, time-varying, site-specific disorder if desired.  
• I spoke about my group's work on heating and dissipation in atomic- and molecular-scale junctions driven out of equilibrium (and into a steady state) by electronic bias.  I framed the discussion in terms of how hard it is to obtain truly local information about vibrational and electronic distributions in such driven systems.  On the vibration side, if you're interested, I suggest looking here, here, and here, with a recent related result here.  On the electronic front, I talked about published (here and here) and some unpublished data looking at electronic shot noise at high biases in atomic-scale metal junctions.  
• Eugene Demler from Harvard (my grad school classmate) gave a nice talk that addressed nonequilibrium aspects of both cold atoms and electrons.   For example, he and collaborators have developed some theoretical machinery for looking at a cold atom version of the orthogonality catastrophe - what happens if you suddenly "turn on" interactions between a cold Fermi gas and a single impurity, and watch the dynamics.  These same theoretical techniques can be applied to solid state systems as well.  (This is just a subset of what was presented.)
• Ryo Shimano from Tokyo University gave a very pretty talk about optical manipulation and driving of the Higgs mode inside superconductors.  You can hit a superconductor with THz radiation as a pump, and then probe at some delay with additional THz radiation.  If the pump is at the right frequency (energy half the superconducting gap, in the s-wave case), you can excite collective sloshing of the condensate (see here and scroll down to the first example).  As you might imagine, things get more rich and complicated with more exotic superconductors (multiband or unconventional).
• Emil Yuzbashyan from Rutgers presented a look at the fundamental issues involved in non-thermal steady states of ensembles of quantum particles at long times after a quench (a sudden change in some parameter).  As I wrote in the first-day discussion, the interesting question here is when does the system evolve seemingly coherently (i.e., the particles slosh around in recurring patterns, just as a Newton's cradle ticks back and forth), and when does the system instead tend toward a long-time state that looks like a randomized, thermalized condition?   To see how this relates to classical mechanics, see these articles (here and here) that I need to find time to read.  
• Lastly, my colleague Matt Foster from Rice spoke about quenched BCS superfluids, topology, spectral probes, and gapless (topological) superconductivity under intense THz pumping.  This was a neat pedagogical talk about this work.  It touches some of the same issues as the Shimano talk above.  One aspect that I found interesting to consider:  You can have a system where a quench drives some collective oscillations, and those collective oscillations act as a Floquet perturbation, changing the effective band structure and giving rise to nonlinearities that continue the oscillations.  Wild stuff - here are the slides.  

We then had a lunch that segued smoothly into a relaxed poster session for students and postdocs.  Overall, it was a great workshop and a good chance to get people from diverse areas of CM and AMO physics together over common interests in nonequilibrium response of quantum systems.  Thanks to everyone who was able to come, to our staff organizer who made everything actually come together, and of course to our sponsors (NSF DMR, ICAM-I2CAM, the Gordon and Betty Moore Foundation, Lakeshore, Advantest, and Coherent)!

Interacting Quantum Systems Driven Out of Equilibrium - day 1 (updated - complete)

Our workshop was fun and interesting.   There are multiple ways to drive physical systems out of equilibrium - you can take some system and push on it with some force, for example.  In the case of a condensed matter system (whether solid state or trapped cold atoms), you can apply a bias - some difference in population (or chemical potential or pressure) that drives the system, either by adding kinetic energy to it or encouraging the flow of matter and/or charge.  You can apply a temperature difference across the system, driving some average flow of energy through the system's degrees of freedom.  You can shine light on the system, adding energy and momentum either at a steady rate or in a sudden pulse.  One favorite piece of vocabulary these days is a quench - suddenly (compared with relaxation rates of the system) changing some condition like the potential energy of the particles, and then watching the response of the system's degrees of freedom.  Does the system "thermalize"?  That is, do the microscopic pieces of the system interact with each other and redistribute energy so that there seems to be some effective temperature?  Or does the system fail to thermalize, and instead slosh around in some non-thermal configuration for a long time?  There are many open issues.

We had 13 talks on the first day, and I don't want to write exhaustive summaries of all of them.  We will eventually be posting pdf files of the relevant slides.  That being said, I will give a super-brief description of each, and link to a relevant paper or two so that you can see what was discussed.  Here are the 13 talks we had on the first day.

• Nadya Mason from UIUC spoke about her group's work on engineered superconducting/normal metal structures in magnetic fields.  These devices allow studies of current-driven motion of trapped magnetic flux.  In some sense this is an old, established problem, but traditional models actually do a poor job of reproducing the experimental data.  The experiments are here, and it looks like it's important to include some "delayed friction" to understand vortex motion.
• Jonathan Bird from Buffalo spoke about his group's studies of quantum point contacts in semiconductors, where it's long been known how to measure electronic conduction down to the limit of discrete quantum channels, where the devices act like waveguides for the electrons.   His group has developed some high speed techniques for making sub-ns electronic measurements, and what really gets interesting is when systems are driven hard, so that the electronic bias is the largest energy scale in the problem - you have to worry quite a bit about exciting phonons and what they do.  A key result is the apparent formation of a specific, somewhat heating-immune transport mode when such a point contact is driven really hard.
• David Goldhaber-Gordon from Stanford spoke about his group's recent experiments looking at quantum dots, some building on work looking at the so-called two-channel Kondo effect.  An unpaired electron is placed in the position of trying to couple to two (carefully tuned to be) independent baths of electrons.  Some of the not-yet-published results look at interesting scaling as one tunes through the accessible regimes, and involved some stunningly pretty device fabrication done at the Weizmann Institute.  Other experiments looked at the apparent emergence of symmetry in systems comprising two quantum dots.
• Tilman Esslinger of ETH presented his group's great work on using cold atoms to look at systems rather analogous to the ones Prof. Bird had mentioned.  They can create blobs of fermionic cold atom fluids of unequal populations, and link them by a carefully controlled constriction, and then they can image transport.  If they squeeze the contact to be effectively one dimensional, they can see quantized conductance of atoms (just as solid state folks can do with charge in a quantum point contact).  They can use atomic physics methods to dial around the interactions between the particles, and can then look at how this affects dissipation in the out of equilibrium situation.  Gorgeous stuff.
• Takashi Oka of the Max Planck Institutes in Dresden talked about Floquet theory and using lasers to control the topology of the band structure of materials.  There was a lot to this talk, and it's not easy to summarize.  In Floquet theory, you apply a periodic driving potential to a quantum system.  Just like a spatially periodic potential energy picks out certain spatial periodicities and gives you a compact way of looking at band structure, temporal periodicity creates what you could call replicas of the band structure but shifted in energy by multiples of \( \hbar \omega\), where \(\omega\) is the driving frequency.  If you do this right, the driven system can have topological edge states.  You can also use periodic driving to reorient the magnetization of materials as if you had a whopping huge effective magnetic field.
• Andrew Millis of Columbia University has worked on many relevant topics, and in this case chose to speak about theory he and collaborators have done regarding a recent experiment looking at vanadium dioxide.  That material has a structural phase transition at 65 C that separates a low temperature, monoclinic, insulating state from a high temperature, tetragonal, metallic state.  In the experiment, optical excitation puts the material into a metallic state without actually leaving the monoclinic crystal structure.  The theory suggests that this is a correlation effect - scoop electrons out of the lower Hubbard band and drop them into the upper band, and interorbital interaction effects can stabilize a new, metastable electronic structure that's a metal.
• Alessandra Lanzara of Berkeley gave a really nice talk about her group's work on time-resolved angle-resolved photoemission.  You hit a material of interest with an ultrafast, time-resolved pump pulse of near-infrared light (1.5 eV photons), and then at some known delay you smack the system with a 6 eV probe pulse at a particular polarization and orientation, and measure the energy and momentum distribution of the electrons that get kicked out.  This lets you measure the transient electronic structure.  They've been able to use this approach to study the dynamics of quasiparticles in cuprate superconductors, how Cooper pairs respond to such pumping, etc.
• N. Peter Armitage at Johns Hopkins articulated nicely three reasons to "go nonequilibrium":  to learn about elementary excitations of an equilibrium phase; to access "phases" not possible in equilibrium material configurations; and to look for new "phases" that have no equilibrium analog.  He then gave a fun talk about using optical spectroscopy techniques to look at many-body relaxations (older paper here) in the Coulomb glass phase of lightly doped semiconductors - when there are strongly interacting, localized electrons in a disordered configuration so that screening is poor.  Interestingly, these systems relax more slowly when the carrier densities get higher, in physics related to the orthogonality catastrophe. 
• My faculty colleague Jun Kono from Rice spoke about so-called Dicke phenomena (such as superradiance, superfluorescence) in semiconductors.  These effects are great examples of nonequilibrium physics, when a driven system (say a semiconductor in a magnetic field illuminated by THz radiation that spans the energy scale of the cyclotron resonance, \(\omega_{\mathrm{c}} = e B/m^{*}\)) spontaneously develops coherence among the many electron-hole excitations in the system.  You can put such a system in a clever kind of 1d optical cavity, and approach the "strong coupling" regime so that the energetic coupling between the charge carriers and the photons in the cavity is comparable to the cyclotron energy.
• Christof Weitenberg from Hamburg then spoke about exciting results in simulating condensed matter systems using cold atoms in optical lattices.  One piece of physics that's very in vogue right now because of the rise of topology and various 2d materials is Berry curvature.  It's hard to explain this in brief - if you look at how the energy bands of a material as a function of crystal momentum \(E(\mathbf{k})\) are curved, the wavefunction of a particle traversing some closed trajectory in \(\mathbf{k}\)-space can pick up a phase factor related to that curvature.  In Weitenberg's experiments, cleverly arranged laser beams can create designer lattices.  Shaking the lasers periodically as a function of time can lead to the same Floquet physics discussed above, changing the effective band structure for atoms confined in those lattices, and through cool imaging techniques the experimentalists can reconstruct the Berry curvature that they have designed into that effective band structure.
• Another colleague Kaden Hazzard from Rice gave a nice theoretical talk about different nonequilibrium collective phenomena in ultracold atomic matter.  One aspect involved dilute molecules with electric dipoles (KBr) trapped in an optical lattice.  Because of their dipole moments, the molecules interact with each other over long ranges (dipole-dipole interactions scale like \(1/r^{3}\)), and their relaxation after getting dinged is governed by many-body interaction effects.  Another system is trapped Rydberg atoms, where dipolar interactions scale like the principal quantum number to the eleventh power (!).  
• Andrea Cavalleri from the Max Planck in Hamburg (and also spending time at Oxford) spoke about his group's very high profile work that I've already described here.  The central question here is really can driving a quantum material stabilize collective states like superconductivity that have coherence, correlations, and remarkable physical properties that would be absent without the drive.  Both Cavalleri and Oka made reference to this video, which shows how driving a classical pendulum can render the inverted position of the pendulum stable.  The experiments themselves are truly remarkable.
• In the last talk of Day 1, Sarang Gopalakrishnan of Cal Tech gave a theory talk again examining the response of driven many-body quantum systems, focusing particularly on the issue of many-body localization.  That is, when do the quantum dynamics of a many-body system lead to a real breakdown of quantum ergodicity, so that the degrees of freedom get "stuck", having large variability of local observables (instead of things being smoothed out and looking thermally smeared) and comparatively weak entanglement (which grows more slowly with system size than in the effectively thermal case).  He pointed out experimental challenges, that experiments probe dynamics rather than quantum eigenstates and that everything really is coupled (however weakly) to some thermal "bath", but argued that these issues aren't fatal to the interesting physics.

Updates coming - Interacting Quantum Systems Driven Out of Equilibrium

As I'd advertised, the Rice Center for Quantum Materials is hosting a two-day workshop on interacting quantum systems driven out of equilibrium.  This event brings together people from roughly three different perspectives:  people who worry about (solid state) systems driven out of equilibrium by electrical bias; people who worry about quantum systems driven out of equilibrium by light (often ultrafast and/or very intense); and people who leverage the amazing cleanliness and tunability of cold atom systems to examine driven quantum many-body systems.   I've been taking notes, and after the workshop wraps up today I'll post some highlights.

Technical help question: Quantum Design magnet power supplies

I'd like to ask my readers that own Quantum Design PPMS or MPMS instruments for help regarding a technical glitch.  My aging PPMS superconducting magnet power supply (the kind QD calls the H-plate version) has developed a problem.  For high fields (say above 7 T) the power supply fails to properly put the magnet in persistent mode and throws up an error in the control software.  After talking with QD, it seems like options are limited.  They no longer service this model of power supply, and therefore one option would be to buy a new one.  However, I have a sense that other people have dealt with this issue before, and I would feel dumb buying a new supply if the answer was that this is a known issue involving a $ 0.30 diode or something.  Without a schematic it's difficult to do diagnostics ourselves.  Has anyone out there seen this issue and knows how to correct it?

Oxide interfaces for fun and profit

The so-called III-V semiconductors, compounds that combine a group III element (Al, Ga, In) and a group V element (N, As, P, Sb), are mainstays of (opto)electronic devices and condensed matter physics.  They have never taken over for Si in logic and memory like some thought they might, for a number of materials science and economic reasons.  (To paraphrase an old line, "GaAs is the material of the future [for logic] and always will be.")  However, they are tremendously useful, in part because they are (now) fortuitously easy to grow - many of the compounds prefer the diamond-like "zinc blende" structure, and it is possible to prepare atomically sharp, flat, abrupt interfaces between materials with quite different semiconducting properties (very different band gaps and energetic alignments relative to each other).  Fundamentally, though, the palette is limited - these materials are very conventional semiconductors, without exhibiting other potentially exciting properties or competing phases like ferroelectricity, magnetism, superconductivity, etc.

Enter oxides.  Various complex oxides can exhibit all of these properties, and that has led to a concerted effort to develop materials growth techniques to create high quality oxide thin films, with an eye toward creating the same kind of atomically sharp heterointerfaces as in III-Vs.  A foundational paper is this one by Ohtomo and Hwang, where they used pulsed laser deposition to produce a heterojunction between LaAlO₃, an insulating transparent oxide, and SrTiO₃, another insulating transparent oxide (though one known to be almost a ferroelectric).  Despite the fact that both of those parent constituents are band insulators, the interface between the two was found to play host to a two-dimensional gas of electrons with remarkable properties.  The wikipedia article linked above is pretty good, so you should read it if you're interested.   

When you think about it, this is really remarkable.  You take an insulator, and another insulator, and yet the interface between them acts like a metal.  Where did the charge carriers come from?  (It's complicated - charge transfer from LAO to STO, but the free surface of the LAO and its chemical termination is hugely important.)  What is happening right at that interface?  (It's complicated.  There can be some lattice distortion from the growth process. There can be oxygen vacancies and other kinds of defects.  Below about 105 K the STO substrate distorts "ferroelastically", further complicating matters.)   Do the charge carriers live more on one side of the interface than the other, as in III-V interfaces, where the (conduction) band offset between the two layers can act like a potential barrier, and the same charge transfer that spills electrons onto one side leads to a self-consistent electrostatic potential that holds the charge layer right against that interface?  (Yes.)

Even just looking at the LAO/STO system, there is a ton of exciting work being performed.  Directly relevant to the meeting I just attended, Jeremy Levy's group at Pitt has been at the forefront of creating nanoscale electronic structures at the LAO/STO interface and examining their properties.  It turns out (one of these fortunate things!) that you can use a conductive atomic force microscope tip to do (reversible) electrochemistry at the free LAO surface, and basically draw conductive structures with nm resolution at the buried LAO/STO interface right below.   This is a very powerful technique, and it's enabled the study of the basic science of electronic transport at this interface at the nanoscale.

Beyond LAO/STO, over the same period there has been great progress in complex oxide materials growth by groups at a number of universities and at national labs.  I will refrain from trying to list them since I don't know them all and don't want to offend with the sin of inadvertent omission.  It is now possible to prepare a dizzying array of material types (ferromagnetic insulators like GdTiO₃; antiferromagnetic insulators like SmTiO₃; Mott insulators like LaTiO₃; nickelates; superconducting cuprates; etc.) and complicated multilayers and superlattices of these systems.   It's far too early to say where this is all going, but historically the ability to grow new material systems of high quality with excellent precision tends to pay big dividends in the long term, even if they're not the benefits originally envisioned.

The Pittsburgh Quantum Institute: PQI2016 - Quantum Challenges

For the last 2.5 days I've been at the PQI2016:  Quantum Challenges symposium.  It's been a very fun meeting, bringing together talks spanning physical chemistry, 2d materials, semiconductor and oxide structures, magnetic systems, plasmonics, cold atoms, and quantum information.  Since the talks are all going to end up streamable online from the PQI website, I'll highlight just a couple of things that I learned rather than trying to summarize everything.

• If you can make a material such that the dielectric permittivity \( \epsilon \equiv \kappa \epsilon_{0} \) is zero over some frequency range, you end up with a very odd situation.   The phase velocity of EM waves at that frequency would go to infinity, and the in-medium wavelength at that frequency would therefore become infinite.  Everything in that medium (at that frequency) would be in the near-field of everything else.  See here for a paper about what this means for transmission of EM waves through such a region, and here for a review.  
• Screening of charge and therefore carrier-carrier electrostatic interactions in 2d materials like transition metal dichalcogenides varies in a complicated way with distance.  At short range,  screening is pretty effective (logarithmic with distance, basically the result you'd get if you worried about the interaction potential from an infinitely long charged rod), and at longer distances the field lines leak out into empty space, so the potential falls like \(1/\epsilon_{0}r\).  This has a big effect on the binding of electrons and holes into excitons in these materials.
• There are a bunch of people working on unconventional transistor designs, including devices based on band-to-band tunneling between band-offset 2d materials.
• In a discussion about growth and shapes of magnetic domains in a particular system, I learned about the Wulff construction, and this great paper by Conyers Herring on why crystal take the shapes that they do.  
• After a public talk by Michel Devoret, I think I finally have some sense of the fundamental differences between the Yale group's approach to quantum computing and the John Martinis/Google group's approach.  This deserves a longer post later.  
• Oxide interfaces continue to show interesting and surprising properties - again, I hope to say more later.
• On a more science-outreach note, I learned about an app called Periscope (basically part of twitter) that allows people to do video broadcasting from their phones.  Hat tip to Julia Majors (aka Feynwoman) who pointed this out to me and that it's becoming a platform for a lot of science education work.

I'll update this post later with links to the talks when those become available.
Update:  Here is the link to all the talk videos, which have been uploaded to youtube.

Sci-fi time, part 2: Really big lasers

I had a whole post written about laser weapons, and then the announcement came out about trying to build laser-launched interstellar probes, so I figured I should revise and talk about that as well.

Now that the future is here, and space-faring rockets can land upright on autonomous ships, it's clearly time to look at other formerly science fiction technologies.  Last August I wrote a post looking at whether laser pistols really make practical physics sense as weapons.  The short answer:  Not really, at least not with present power densities.

What about laser cannons?  The US military has been looking at bigger, high power lasers for things like anti-aircraft and ship defense applications.  Given that Navy ships would not have to worry so much about portability and size, and that in principle nuclear-powered ships should have plenty of electrical generating capacity, do big lasers make more sense here?  It's not entirely clear.  Supposedly the operating costs of the laser systems are less than $1/shot, though that's also not a transparent analysis.

Let's look first at the competition.  The US Navy has been using the Phalanx gun system for ship defense, a high speed 20mm cannon that can spew out 75 rounds per second, each about 100 g and traveling at around 1100 m/s.   That's an effective output power, in kinetic energy alone, of 4.5 MW (!).  Even ignoring explosive munitions, each projectile carries 60 kJ of kinetic energy.  The laser weapons being tested are typically 150 kW.  To transfer the same amount of energy to the target as a single kinetic slug from the Phalanx would require keeping the beam focused on the target (assuming complete absorption) for about 0.4 sec, which is a pretty long time if the target is an inbound antiship missile traveling at supersonic speeds.   Clearly, as with hand-held weapons, kinetic projectiles are pretty serious in terms of power and delivered energy on target, and beating that with lasers is not simple.

The other big news story recently about big lasers was the announcement by Yuri Milner and Stephen Hawking of the Starshot project, an attempt to launch many extremely small and light probes toward Alpha Centauri using ground-based lasers for propulsion.  One striking feature of the plan is the idea of using a ground-based optical phased array laser system with about 100 GW of power (!) to boost the probes up to about 0.2 c in a few minutes.  As far as I can tell, the reason for the very high power and quick boost is to avoid problems with pointing the lasers for long periods of time as the earth rotates and the probes become increasingly distant.  Needless to say, pulling this off is an enormous technical challenge.  That power would be about equivalent to 50 large city-serving powerplants.   I really wonder if it would be easier to drop the power by a factor of 1000, increase the boost time by a factor of 1000, and use a 100 MW nuclear reactor in solar orbit (i.e. at the earth-sun L1 or L2 point) to avoid the earth rotation or earth orbital velocity constraint.  That level of reactor power is comparable to what is used in naval ships, and I have a feeling like the pain of working out in space may be easier to overcome than the challenge of building a 100 GW laser array.  Still, exciting times that anyone is even entertaining the idea of trying this.

"Joulies": the coffee equivalent of whiskey stones, done right

Once upon a time I wrote a post about whiskey stones, rocks that you cool down and then place into your drink to chill your Scotch without dilution, and why they are rather lousy at controlling your drink's temperature.  The short version:  Ice is so effective, per mass, at cooling your drink because its melting is a phase transition.  Add heat to a mixture of ice and water, and the mixture sits there at zero degrees Celsius, sucking up energy (the "latent heat") as the solid ice is converted into liquid water.  Conversely, a rock just gets warmer.

Now look at Joulies, designed to keep your hot beverage of choice at about 60 degrees Celsius.  Note:  I've never used these, so I don't know how well-made they are, but the science behind them is right.  They're stainless steel and contain a material that happens to have a melting phase transition right at 60 C and a pretty large latent heat - more on that below.  If you put them into coffee that's hotter than this, the coffee will transfer heat to the Joulies until their interior warms up to the transition, and then the temperature of the coffee+Joulies will sit fixed at 60 C as the filling partially melts.   Then, if you leave the coffee sitting there and it loses heat to the environment through evaporation, conduction, convection, and radiation, the Joulies will transfer heat back to the coffee as their interior solidifies, again doing their level best to keep the (Joulies+coffee) at 60 C as long as there is a liquid/solid mixture within the Joulies.  This is how you regulate the temperature of your beverage.  (Note that we can estimate the total latent heat of the filling of the Joulies - you'd want it to be enough that cooling 375 ml of coffee from 100 C to 60 C would not completely melt the filling.  At 4.18 J/g for the specific heat of water (close enough), the total latent heat of the Joulies filling should be more than 375 g  \( \times \) 40 degrees C  \( \times \) 4.18 J/g = 62700 J. )

Unsurprisingly, the same company offers a version filled with a different material, one that melts a bit below 0 C, for cooling your cold beverages.  Basically they function like an ice cube, but with the melting liquid contained within a thin stainless steel shell so that it doesn't dilute your drink.

Random undergrad anecdote:  As a senior in college I was part of an undergrad senior design team in a class where the theme was satellites and spacecraft.  We designed a probe to land on Venus, and a big part of our design was a temperature-regulating reservoir of a material with a big latent heat of melting and a melting point at something like 100 C, to keep the interior of the probe comparatively cool for as long as possible.  Clearly we should've been early investors in Joulies.

Interacting Quantum Systems Out of Equilibrium - Workshop at Rice

The Rice Center for Quantum Materials will be hosting a workshop, "Interacting Quantum Systems Driven Out of Equilibrium", at Rice University in Houston on May 5-6, 2016.

  A central challenge of condensed matter and atomic physics today is understanding interacting, quantum many-body systems driven out of thermal equilibrium.  Thanks to recent advances in both experimental and theoretical techniques, this is an exciting, active area that is seeing new emergent results.  The Rice Center for Quantum Materials is hosting a workshop that will bring together the diverse community of researchers examining the various facets of the nonequilibrium quantum many-body problem.  Experimental systems include:  quantum materials driven by electronic bias beyond the linear regime; optical pump/probe methods to examine dynamic and steady-state nonequilibrium response; ultracold atoms in response to quench conditions and probed with far-from-equilibrium spectroscopy.  Theoretical issues include: coherent many-body dynamics; many-body localization; Floquet states and dynamics in driving potentials; and thermalization/dissipation with driven quantum dynamics.

  For more details, including a speaker list and draft program, please see our website.  Attendance by students/postdocs from traditionally underrepresented groups is encouraged.

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.