Quantum geometry - some intuition

There has been a great growing interest in quantum geometry in recent years.  Last week, I heard an excellent talk by Raquel Queiroz about this that gave me a more physically intuitive interpretation  of this topic.  The more formal write-up is in this preprint from this past April, which I'd missed at the time.

Caution:  Math incoming.  I will try to give a more physical picture at the end.  I know that this won't be very readable to non-experts.    

As I've written before,  (e.g. here and a bit here), the electronic states in crystalline solids are often written as Bloch waves of the form \(u_{n\mathbf{k}}(\mathbf{r})\exp(i \mathbf{k}\cdot \mathbf{r})\), where \(u_{n\mathbf{k}}(\mathbf{r})\) is periodic in the spatial period of the crystal lattice.  For many years, the \(\mathbf{k}\) dependence of \(u_{n\mathbf{k}}(\mathbf{r})\) was comparatively neglected, but now it is broadly appreciated that this is the root of all kinds of interesting physics, including the anomalous Hall effect and its quantum version.  

We can compute how much \(u_{n\mathbf{k}}(\mathbf{r})\) changes with \(\mathbf{k}\).  The Berry connection is related to the phase angle racked up by moving around in \(\mathbf{k}\), and it's given by \( \mathbf{A}(\mathbf{k}) = i \langle u_{n\mathbf{k}}| \nabla_{\mathbf{k}}| u_{n\mathbf{k}} \rangle \).  One can define \(\mathbf{\Omega} \equiv \nabla \times \mathbf{A}(\mathbf{k})\) as the Berry curvature, and the "anomalous velocity" is given by \(-\dot{\mathbf{k}}\times \mathbf{\Omega}\).  

If we worry about possible changes in the magnitude as well, and \( |\langle u_{n\mathbf{k}}| u_{n\mathbf{k+dk}} \rangle |^{2} = 1 - g^{n}_{\mu \nu}dk_{\mu}dk_{\nu}\) plus higher order terms.  The quantity \(g^{n}_{\mu \nu}\) is the quantum metric, and it can be written in terms of dipole operators:  \(g^{n}_{\mu \nu}= \sum_{m\ne n}\langle u_{n,\mathbf{k}}|\hat{r}_{\mu}|u_{m \mathbf{k}}\rangle \langle u_{m,\mathbf{k}}|\hat{r}_{\nu}|u_{n \mathbf{k}}\rangle\).  The quantum metric quantifies the "distance between" the Bloch states as one moves around in \(\mathbf{k}\).  

That last bit is what I really learned from the talk.  Basically, if you try to consider electrons localized to a particular lattice site in real space, this can require figuring in states in multiple bands, and the matrix elements involve dipole operators.  The quantum geometric tensor \(g_{\mu \nu}\) quantifies the dipole fluctuations in the electronic density.  You can define a lengthscale \(\ell_{g}\equiv \sqrt{\mathrm{Tr} g}\), and this can tell you about the spatial scale of polarization fluctuations relative to, e.g., the lattice spacing.  Metals will have essentially divergent fluctuation lengthscales, while insulators have nicely bound charges (that give peaks in the optical conductivity at finite frequency).   The quantum geometry then influences all kinds of experimentally measurable quantities (see here).  

Neat stuff.  Someday I'd like to return to this with a nice cartoon/animation/presentation for non-experts.  The idea that there is so much richness within even relatively "boring" materials still amazes me.

Vortices everywhere

The 2026 APS Oliver E. Buckley Prize in condensed matter physics was announced this week, and it's a really interesting combination of topics that, to a lay person, may seem to be completely unrelated.  

Fig. 1 from this follow-up PRB.

On the one hand, John Reppy (at age 94!) and Dave Bishop were honored for their work examining the properties of vortices in thin films of superfluid helium-4.  Relevant papers include this one from 1977, where they used a torsion pendulum coated with the helium film to examine the transition between normal and superfluid.  When the helium becomes a superfluid, it has (at low speeds) no viscosity, so it no longer has to rotate with the torsion pendulum; this means the rotational moment of inertia goes from that of (pendulum+helium) to just (pendulum), and the period of the oscillations increases.  Really detailed measurements of the oscillations and their damping allowed Reppy and Bishop to compare with models of the superfluid transition based on work by Kosterlitz and Thouless (and Berezinskii).  See the image for a diagram of the experimental setup - very clever and intricate.  

The key idea here is the role of vortices.  Superfluidity in helium is described by an order parameter that looks like a wavefunction - it has an amplitude, \(\Psi_{0}\), and a phase \(\phi\), so that \(\Psi(\mathbf{r}) = \Psi_{0} \exp(i \phi)\).   That order parameter is supposed to be single-valued, meaning if you go around a closed loop of some kind, that phase will either remain the same or ramp by some integer multiple of \(2\pi\).  The gradient of the phase is related to the velocity of the superfluid, so if the phase winds by \(2\pi\), that implies there is a circulation of flow and orbital angular momentum that has to be an integer multiple of \(\hbar\).  In the BKT theory, the demise of the superfluid phase as the system is warmed happens through the creation and unbinding of vortex-antivortex pairs.

On the other hand, the other recipients of the Buckley Prize were Gwendal Fève and Mike Manfra for their work (experiments here and here) regarding the braiding statistics of anyons in fractional quantum Hall systems.  I'd written about anyons here.  For electrons in 2D, the wavefunctions of excitations of the fractional quantum Hall system look like vortices.  The phase of the electronic wavefunction can wind due to circulation, and because electrons are charged, the phase can also wind due to magnetic flux attached to the little whirlpool.  It's the combination of these phase effects that can lead to those excitations acting like anyons (so that when two are physically swapped or braided around one another, the wavefunction picks up a phase factor that is not just the \(+1\) of bosons or the \(-1\) of fermions).  

As my friend Dan Arovas pointed out, there was a hope back in the early 1980s that perhaps vortices in superfluid helium would also act like anyons and have fractional statistics.  However, this paper by Haldane and Wu disproved that possibility.  

Vortex shedding, from here.

Because of the relationship between quantum phase winding and actual flow of (density) currents, vortices show up in lots of places in hard condensed matter physics.  Classical vortices are also physically nontrivial objects - they're topological and often seem to have very counterintuitive properties and motions.  Heck, Lord Kelvin was so taken by this that he thought (pre-quantum) that maybe everything is really vortices of some kind.  

Perhaps it is fitting that I am posting this on the 85th anniversary of the Tacoma Narrows bridge collapse.  That classic civil engineering failure was caused by vortex shedding by the bridge coupling to its torsional resonance frequency.  Vortices can have big consequences!  

Science journalism - dark times

At this point it's old hat to decry the problems facing traditional news media.  Still, it is abundantly clear in our late stage capitalist society that there has been a collective business decision over the last 20+ years that, like local newspapers and television news, real science journalism is not a money maker.   Just a few examples:  Seventeen years ago, CNN cut its entire science, technology and environment reporting team.  In 2022, Popular Science ceased publication.  In 2023, National Geographic laid off their staff writers.  Last week, the Wall Street Journal laid off their science and health reporters.  

I have it on good authority that there is now only one science reporter left at the WSJ.  One, at a time when science and technology are more critically important to our rapidly changing society than ever, and there is enormous tumult in the US and elsewhere about how science is or is not supported and is or is not factoring into policy decisions.  All of this is happening at a time when public trust in science is falling.  (Check out this from Science Friday.)  

(updated for context) Leaving aside professional science outlets (the news sections of Science, Nature, and society publications like Physics Today, C&EN, Physics World, Chemistry World), there are some good publications out there, like Quanta and Nautilus (both founded by nonprofits). There are outstanding public writers of science, like Philip Ball, Helen Czerski, Katie Mack, Ethan Siegel, and many others (apologies for the incompleteness of this list).  There are some excellent freelance journalists.  The internet also means that there are many opportunities for great engagement.  For example, the videos from 3blue1brown are uniformly outstanding.  However, there are no filters, and the temptation to be click-baity or sensationalistic is problematic.  

I have no solutions to offer, except that I encourage you to support good science journalism and reporting when you see it.  It's important.

Interesting preprints: chirality-induced spin selectivity + quantum gravity

This continues to be a very busy time, but I wanted to point out two preprints that caught my eye this week.  Their subjects are completely disparate, but they stand out as essentially reviews written in a much more conversational tone than the usual literature.

The first is this preprint about chirality-induced spin selectivity, a subject that I've mentioned before on this blog.  There is now an extensive body of evidence (of varying quality) that there is a connection between structural chirality of molecules and their interactions with the spin angular momentum of electrons.  This includes monolayers of chiral molecules leading to net spin polarization of photoemitted electrons (here), a lot of electronic transport experiments involving chiral molecules and magnetic electrodes that seem to show spin-dependent transmission that is absent with achiral molecules, and even a chirality dependence of molecular adsorption kinetics on magnetic surfaces (here).  The preprint is a provocative discussion of the topic and possible mechanisms, and the importance of precision in the description of the various phenomena.

On a completely different topic, this preprint is a fun discussion about quantum gravity (!) and how condensed matter ideas of "the vacuum" can lead to insights about how quantum mechanics and gravity might need to play together.  One fun bit early on is a discussion of something I like to point out to my undergrad stat mech students:  A single hydrogen atom in a very very large box will apparently (if the usual stat mech formalism of partition functions is valid) be spontaneously ionized, even when the box (which presumably functions as a reservoir at temperature \(T\)) and atom are at temperatures faaaaaar below the energy scale for ionization.  This is discussed nicely in this 1966 article in the Journal of Chemical Education.  Anyway, I thought this was an interesting discussion from three condensed matter theorists.

Postdoctoral opportunity in materials

The Rice Advanced Materials Research Institute is having its 2025-2026 competition for prestigious postdoctoral fellowships - see here:  https://rami.rice.edu/rami-postdoctoral-fellowship-program  .

If you are interested and meet the criteria, I'd be happy to talk.  I have some ideas that lean into the materials for electronics direction, and other possibilities are welcome.  

2025 Physics Nobel: Macroscopic quantum tunneling

As announced this morning, the 2025 Nobel Prize in Physics has been awarded to John Clarke, Michel Devoret, and John Martinis, for a series of ground-breaking experiments in the 1980s that demonstrated macroscopic quantum tunneling. 

For non-experts: "Tunneling" was originally coined to describe the physical motion of a quantum object, which can pass through a "classically forbidden" region.  I've written about this here, and here is an evocative picture. Suppose there is a particle with a certain amount of total energy in the left region.  Classically, the particle is trapped, because going too far to the left (gray region) or too far to the right (gray region) is forbidden:  Putting the particle inside the shaded regions is "classically forbidden" by conservation of energy.  The particle bounces back and forth in the left well.  If the particle is a quantum object, though, it is described by a wave function, and that wave function has some non-zero amplitude on the far side of barrier in the middle.  The particle can "tunnel" through the barrier, with a probability that decreases exponentially with the height of the barrier and its width.

Fig. 2 from here

Clarke, Devoret, and Martinis were working not with a single particle, but with electrons in a superconductor (many many electrons in a coherent quantum state).  The particular system they chose was a Josephson junction made from an oxide-coated Nb film contacted by a PbIn electrode with a dc current flowing through it.  Instead of an x coordinate of a particle, the relevant coordinate in this system is the phase difference \(\delta\) of the superconducting wave function across the junction.  There is an effective potential energy for this system called a "washboard" potential, \(U(\delta)\), as in this figure.  At the particular DC current, which tilts \(U(\delta)\), the system can transition from one state (\(\delta\) bopping around a constant value, no voltage across the junction) to a state where \(\delta\) is continuously ramping (corresponding to a nonzero voltage across the junction).  The system can get thermally kicked from the zero voltage state to the nonzero voltage state (thermal energy doinks it over the barrier), but the really interesting thing is that the system can quantum mechanically tunnel "through" the barrier as well.

This idea, that a macroscopic (in the sense of comprising many many electrons) system could tunnel out of a metastable state like this, had been investigated by Amir Caldeira and Tony Leggett in this important paper, where they worried about the role of dissipation in the environment.  People tried hard to demonstrate this, but issues with thermal radiation and other noise in the experiments were extremely challenging.  With great care in experimental setup, the three laureates put together a remarkable series of papers (here, here, here) that showed all the hallmarks, including resonantly enhancing tunneling with tuned microwaves (designed to kick the system between the levels shown in panel (d) of the figure above).  

This was an impressive demonstration of controllable, macroscopic quantum tunneling, and it also laid the foundation for the devices now used by the whole superconducting quantum computing community.  

ACS National Nanotechnology Day webinar, Thursday Oct 9

Time for a rare bit of explicit self-promotion on this blog.  This coming Thursday, October 9, as part of the American Chemical Society's activities for National Nanotechnology Day (Why October 9?  In US convention, Oct 9 = 10/9, and 10^-9 m = 1 nm.  Look, it wasn't my idea....), I'm speaking in a free webinar titled "Illuminating the Nano Frontier", with Prof. Dongling Ma of INRS in Quebec.  The event is 11am-12:30pm EDT, and there will also be a recording for people who are unable to watch it live.  Should be a fun event. 

Update:  Here is the link to the webinar recording.  It's free and open-access.