Sunday, May 15, 2022

Flat bands: Why you might care, and one way to get them

When physicists talk about the electronic properties of solids, we often talk about "band theory".  I've written a bit about this before here.  In classical mechanics, a free particle of mass \(m\) and momentum \(\mathbf{p}\) has a kinetic energy given by \(p^2/2m\).  In a crystalline solid, we can define a parameter, the crystal momentum, \(\hbar \mathbf{k}\), that acts a lot like momentum (accounting for the ability to transfer momentum to and from the whole lattice).  The energy near the top or bottom of a band is often described by an effective mass \(m_{*}\), so that \(E(\mathbf{k}) = E_{0} + (\hbar^2 k^2/2m_{*})\).  The whole energy band spans some range of energies called the bandwidth, \(\Delta\). If a band is "flat", that means that its energy is independent of \(\mathbf{k}\) and \(\Delta = 0\).  In the language above, that would imply an infinite effective mass; in a semiclassical picture, that implies zero velocity - the electrons are "localized", stuck around particular spatial locations.  

Why is this an interesting situation?  Well, the typical band picture basically ignores electron-electron interactions - the assumption is that the interaction energy scale is small compared to \(\Delta\).  If there is a flat band, then interactions can become the dominant physics, leading potentially to all kinds of interesting physics, like magnetism, superconductivity, etc.  There has been enormous excitement in the last few years about this because twisting adjacent layers of atomically thin materials like graphene by the right amount can lead to flat bands and does go along with a ton of cool phenomena.  

How else can you get a flat band?  Quantum interference is one way.  When worrying about quantum interference in electron motion, you have to add the complex amplitudes for different electronic trajectories.  This is what gives you the interference pattern in the two-slit experiment.   When trajectories to a certain position interfere destructively, the electron can't end up there.  

It turns out that destructive interference can come about from lattice symmetry. Shown in the figure is a panel adapted from this paper, a snapshot of part of a 2D kagome lattice.  For the labeled hexagon of atoms there, you can think of that rather like the carbon atoms in benzene, and it turns out that there are states such that the electrons tend to be localized to that hexagon.  Within a Wannier framework, the amplitudes for an electron to hop from the + and - labeled sites to the nearest (red) site are equal in magnitude but opposite in sign.  So, hopping out of the hexagon does not happen, due to destructive interference of the two trajectories (one from the + site, and one from the - site).  

Of course, if the flat band is empty, or if the flat band is buried deep down among the completely occupied electronic states, that's not likely to have readily observable consequences.  The situation is much more interesting if the flat band is near the Fermi level, the border between filled and empty electronic states.  Happily, this does seem to happen - one example is Ni3In, as discussed here showing "strange metal" response; another example is the (semiconducting?) system Nb3Cl8, described here.  These flat bands are one reason why there is a lot of interest these days in "kagome metals".

Saturday, May 14, 2022

Grad students mentoring grad students - best practices?

I'm working on a physics post about flat bands, but in the meantime I thought I would appeal to the greater community.  Our physics and astronomy graduate student association is spinning up a mentoring program, wherein senior grad students will mentor beginning grad students.  It would be interesting to get a sense of best practices in this.  Do any readers have recommendations for resources about this kind of mentoring, or examples of departments that do this particularly well?  I'm aware of the program at UCI and the one at WUSTL, for example.

Sunday, May 01, 2022

The multiverse, everywhere, all at once

The multiverse (in a cartoonish version of the many-words interpretation of quantum mechanics sense - see here for a more in-depth writeup) is having a really good year.  There's all the Marvel properties (Spider-Man: No Way Home; Loki, with its Time Variance Authority; and this week's debut of Doctor Strange in the Multiverse of Madness), and the absolutely wonderful film Everything, Everywhere, All at Once, which I wholeheartedly recommend.  

While it's fun to imagine alternate timelines, the actual many-worlds interpretation of quantum mechanics (MWI) is considerably more complicated than that, as outlined in the wiki link above.  The basic idea is that the apparent "collapse of the wavefunction" upon a measurement is a misleading way to think about quantum mechanics.  Prepare an electron so that its spin is aligned along the \(+x\) direction, and then measure \(s_{z}\).  The Copenhagen interpretation of quantum would say that prior to the measurement, the spin is in a superposition of \(s_{z} = +1/2\) and \(s_{z}=-1/2\), with equal amplitudes.  Once the measurement is completed, the system (discontinuously) ends up in a definite state of \(s_{z}\), either up or down.  If you started with an ensemble of identically prepared systems, you'd find up or down with 50/50 probability once you looked at the measurement results.    

The MWI assumes that all time evolution of quantum systems is (in the non-relativistic limit) governed by the Schrödinger equation, period.  There is no sudden discontinuity in the time evolution of a quantum system due to measurement.  Rather, at times after the measurement, the spin up and spin down results both occur, and there are observers who (measured spin up, and \(s_{z}\) is now +1/2) and observers who (measured spin down, and \(s_{z}\) is now -1/2).  Voila, we no longer have to think about any discontinuous time evolution of a quantum state; of course, we have the small issues that (1) the universe becomes truly enormously huge, since it would have to encompass this idea that all these different branches/terms in the universal superposition "exist", and (2) there is apparently no way to tell experimentally whether that is actually the case, or whether it is just a way to think about things that makes some people feel more comfortable.  (Note, too, that exactly how the Born rule for probabilities arises and what it means in the MWI is not simple.) 

I'm not overly fond of the cartoony version of MWI.  As mentioned in point (2), there doesn't seem to be an experimental way to distinguish MWI from many other interpretations anyway, so maybe I shouldn't care.  I like Zurek's ideas quite a bit, but I freely admit that I have not had time to sit down and think deeply about this (I'm not alone in that.).  That being said, lately I've been idly wondering if the objection of the "truly enormously huge" MWI multiverse is well-founded beyond an emotional level.  I mean, as a modern physicist, I already have come to accept (because of observational evidence) that the universe is huge, possibly infinite in spatial extent, appears to have erupted into an inflationary phase 13.6 billion years ago from an incredibly dense starting point, and contains incredibly rich structure that only represents 5% of the total mass of everything, etc.  I've also come to accept that quantum mechanics makes decidedly unintuitive predictions about reality that are borne out by experiment.  Maybe I should get over being squeamish about the MWI need for a zillion-dimensional hilbert space multiverse.  As xkcd once said, the Drake Equation should include a factor for "amount of bullshit you're willing to buy from Frank Drake".  Why should MWI's overhead be a bridge too far?  

It's certainly fun to speculate idly about roads not taken.  I recommend this thought-provoking short story by Larry Niven about this, which struck my physics imagination back when I was in high school.  Perhaps there's a branch of the multiverse where my readership is vast :-)

Monday, April 25, 2022

Science Communications Symposium

 I will be posting more about science very soon, but today I'm participating in a science communications symposium here in the Wiess School of Natural Sciences at Rice.  It's a lot of fun and it's great to hear from some amazing colleagues who do impressive work.   For example, Lesa Tran Lu and her work on the chemistry of cooking, Julian West and his compelling scientific story-telling, Scott Solomon and his writing about evolution, and Kirsten Siebach and her work on Mars rovers and geology.

(On a side note, I've now been blogging for almost 17 years - that makes me almost 119 blog-years old.)

Friday, April 08, 2022

Brief items

It's been a while since the APS meeting, with many things going on that have made catching up here a challenge.  Here are some recent items that I wanted to point out:

  • Igor Mazin had a very pointed letter to the editor in Nature last week, which is rather ironic since much of what he was excoriating is the scientific publishing culture promulgated by Nature.  His main point is that reaching for often-unjustified exotic explanations is rewarded by glossy journals - a kind of inverse Occam's Razor.   He also points out correctly that it's almost impossible for experimentalists to get a result published in a fancy journal without claiming some theoretical explanation.
  • We had a great physics colloquium here this week by Vincenzo Vitelli of the University of Chicago.  He spoke about a number of things, including "odd elasticity".  See, when relating stresses \(\sigma_{ij}\) to strains \(u_{kl}\), in ordinary elasticity there is a tensor that connects these things: \(\sigma_{ij} = K_{ijkl} u_{kl}\), and that tensor is symmetric:  \(K_{ijkl} = K_{klij}\).  Vitelli and collaborators consider what happens when there is are antisymmetric contributions to that tensor.  This means that a cycle of stress/strain ending back at the original material configuration could add or remove energy from the system, depending on the direction of the cycle.  (Clearly this only makes sense in active matter, like driven or living systems.)  The results are pretty wild - see the videos about halfway down this page.
  • Here's something I didn't expect to see:  a new result out of the Tevatron at FermiLab, which is interesting since the Tevatron hasn't run since 2011.  Quanta has a nice write-up.  Basically a new combined analysis of FermiLab data has a new estimate out for the mass of the W boson along with a claimed improved understanding of systematic errors and backgrounds.  The result is a statement that the W boson is heavier than expectations from the Standard Model by an amount that is estimated to be 7 standard deviations.  The exotic explanation (perhaps favored by the inverse Occam's Razor above) is that the Standard Model calculation is off because it's missing some added contributions from so-far-undiscovered particles.  The less exotic explanation is that the new analysis and small error estimates have some undiscovered flaw.  Time will tell - I gather that the LHC collaborations are working on their own measurements. 
  • This result is very impressive.  Princeton investigators have made qubits using spins of single electrons trapped in Si quantum dots, and they have achieved fidelity in 2-qubit operations greater than 99%.  If this is possible in (excellent) university-level fabrication, it does make you wonder whether great things may be possible in a scalable way with industrial-level process control.
  • This is a great interview with John Preskill.  In general the AIP oral history project is outstanding.
  • Well, this is certainly suggestive evidence that the universe really is a simulation.

Friday, March 18, 2022

APS March Meeting 2022, Day 4 and wrap-up

I gave my contributed talk this (Fri) morning, and I will head to the airport shortly, so this is the end of my March Meeting blogging.  A few highlights from yesterday:

  • Konrad Lehnert gave a very nice, pedagogical talk about the possibility of detecting axionic dark matter using quantum sensing.  The super short version:  it is thought that axions if they exist can, in the presence of a large magnetic field, convert at some rate into photons with energy \(\hbar \omega = m_{\mathrm{a}}c^2\).  In a microwave cavity, it is possible to detect such excess photons, and by doing clever things with "squeezing", it is possible to beat the standard quantum limit and to examine parameter space more rapidly than otherwise.  There is still a lot of room for improvement if one wants to be able to look across the whole range of potential axion masses and not have it take years and cost a gazillion dollars.  One approach using entanglement can eliminate a number of confounding factors.
  • I saw two very clear talks, one by Kevin Nuckolls and one by Stevan Nadj-Perge about using STM and tunneling (and point contact) spectroscopy to examine superconductivity in magic-angle twisted bilayer and trilayer graphene, respectively.  In the former, one challenge is to decide how much of the observed gap features in tunneling are due to superconductivity, and then using the functional form of that superconducting part to consider pairing mechanisms.  It is also possible to see how band flattening increases the density of states even at angles away from the magic angle.
  • In a different session, Inti Sodemann spoke about whether and how it is possible to get current rectification in semiconductors when they are illuminated by light with energy below the band gap, so that there is no absorption.  There are thermodynamic restrictions that come in - you can't get energy from nowhere, and you can't break the second law.  Thanks to Berry curvature effects, it is actually possible to have this kind of rectification under some circumstances.
  • There was another extremely clear talk by N. Peter Armitage about Co-containing compounds as Kitaev spin liquid candidates.  There was some really great THz absorption data as a fn of temperature and magnetic field for CoNb2O6 that had amazing agreement with theory, and newer results looking at a more 2D system, BaCo2(AsO4)2.
  • Unfortunately I was unable to attend the Kavli Symposium.  I hope to be able to watch the talks later, as these are typically of very high quality and general interest.
Closing thoughts:
  • It was nice and kind of weird to finally see a good number of people in person.  Really great to catch up with old friends, though I think my conference stamina has waned since the 2019 meeting.
  • When the participants skew younger, as seemed to be the case this year, the crowd definitely looks more diverse.  It would be interesting to know the demographics of the attendees.
  • I don't think pre-recorded short talks work well.  The inability to ask/answer questions is a problem.  
  • I wonder if we will have hybrid meetings in general from now on.  There are definitely environmental impact reasons to go that way, and it would help solve the APS's problem that prior to covid the meeting had grown so large that it was difficult to plan or host.

Wednesday, March 16, 2022

APS March Meeting 2022, Day 3

Highlights are brief today, because I spent more of my time seeing talks from my group and chatting with people:

  • Started the day with the Keithley Prize session, and Dan Rugar talking about the history of magnetic resonance force microscopy.   Very interesting and educational.  It is inspiring to see the evolution of a technique, from the genesis of the idea (an early paper here) to initial testing to advanced developments.
  • Later I saw Marcel Franz give a very clear talk about how to try to build a topological superconductor (fully gapped with topologically protected chiral edge modes) by stacking individual cuprate layers rotated by 45 degrees with respect to each other.  
  • There was a neat talk by Naomi Ginsberg on her group's pump-probe interferometric technique ("stroboSCAT") that allows them to visualize and separate the diffusion of heat and the diffusion of charge in various materials.  For a review, see here.
  • Later in the day I bopped back and forth a bit between the Buckley/Isakson/Onsager Prize session and a session about the BCS/BEC crossover in condensed matter systems.  It was pretty neat hearing Emmanuel Rashba speak.  
Now to figure out what to see tomorrow....