What are heavy fermions?

I'm surprised that I haven't written about heavy fermions as a separate post before, so here we go. (It's a break from thinking about science and politics, anyway.)

I've written before about "effective mass" for electronic excitations in solids (wiki page here).  From classical physics, we are used to the idea that inertial mass \(m\) is the ratio between an external force \(\mathbf{F}\) and the acceleration \(\mathbf{a}\) of some object, \(\mathbf{F} = m\mathbf{a}\), which is also the rate of change of momentum, \(d\mathbf{p}/dt\).  Kinetic energy (for a nonrelativistic particle) is \(p^{2}/2m\).  Electrons in crystalline solids "feel" the lattice, so in general their kinetic energy \(\epsilon\) can be a more complicated function of their (crystal) momentum, and we can try do define an effective mass as \(1/m* \equiv d^{2}\epsilon/dp^{2}\).  So, if the kinetic energy is very weakly dependent on \(p\), this corresponds to having a very large effective mass.  TL/DR:  the periodic lattice can strongly alter how an electronic excitation accelerates in the presence of a force from, e.g., an electric field, compared to a free particle.  This isn't too surprising.  

Interestingly, in most semiconductors and metals, \(m*\) for electrons in the conduction band (or holes in the valence band) is not thaaaaat different than the free electron mass \(m_{0}\).  The lightest effective mass I know (leaving aside graphene and other Dirac systems when \(\epsilon\) is approximately linear in \(p\)) is electrons in InSb, about \(0.014 m_{0}\).  Holes tend to be a bit heavier.  Also, \(m*\) in molecular organic semiconductors like pentacene tends to be a bit larger, since hopping from molecule to molecule is comparatively weak.  There are ways to measure effective mass, including cyclotron resonance, electronic transport including Shubnikov-de Haas oscillations, magnetic susceptibility and de Haas/van Alphen oscillations, and specific heat measurements.  The electronic specific heat contribution for a metal is linear in the temperature at low \(T\), and the constant of proportionality includes the density of electronic states at the Fermi energy, which can be written in terms of \(m*\).  I've left out a lot of the complications of real anisotropic materials with complicated band structures, but generally the different measurements give consistent results. 

Therefore, it was a big surprise in 1975 when investigators found a material, CeAl3, in which the heat capacity implied an effective mass tens to hundreds of times larger than \(m_{0}\).  They knew right away that this had something to do with the very localized \(4f\) electrons of the Ce atoms.  Because those electrons are very localized, their energy is almost independent of \(p\), implying a large effective mass.  (Some heavy fermion materials also superconduct at temperatures surprisingly high given their effective masses.)

Heavy fermions, adapted from here.  (a) At high temperatures, the 

conduction  electrons are not well coupled to the unpaired local 4f 

moments.  (b) At low enough temperatures, Kondo scattering

hybridizes the f electrons with the conduction  electrons, boosting 

the carrier density.  (c) The hybridized energy-momentum relation 

is much flatter near the Fermi energy leading to a large effective mass.  

So what's the physics?  I wrote about the Kondo effect here, where "ordinary" conduction electrons scatter in a nontrivial way from local magnetic moments (such as partially filled \(4f\) states), and well below a characteristic temperature \(T_{\mathrm{K}}\), the conduction electrons hybridize with the impurities, screening out the unpaired spin.  In the heavy fermion compounds, instead of impurities, there is a whole crystal lattice of local magnetic moments. At sufficiently low temperatures, thanks to that Kondo scattering process, those otherwise localized electrons hybridize with the conduction electrons, boosting the effective density of charge carriers (see figure) and greatly increasing the effective mass.  See this figure, adapted from excellent lecture notes by Piers Coleman.  

So, two key ingredients for heavy fermions are itinerant conduction electrons and a periodic array of comparatively localized, unpaired electrons that have magnetic moments. It turns out that this combination can also be achieved in moiré lattice materials.  There are no \(f\) electrons here, but the moiré lattice can localize spins.  Apologies for not linking to all the relevant papers, but a couple of key theory results are here, here, and here, and a key experimental result is here.  The tunability of the 2D material-based systems is an excellent feature for digging down into the detailed physics.

Update:  Now some added insight from Prof. Andrew Millis:

Hi Doug:

An addendum to your very nice post on heavy fermions, to draw attention to what I think were important experimental results: Frank Steglich’s 1979  Phys. Rev. Lett. 43, 1892–1896 reporting superconductivity in CeCu2Si2 and Louis Taillefer and Gil Lonzarich’s 1988 determination of the quasiparticle mass and fermi surface in UPt3.

Prior to Steglich’s paper we knew that some rare earth/actinide intermetallics (e.g. CeAl3) had a very enhanced specific heat coefficient at low temperatures and that the entropy implied by  this specific heat was  derived from the magnetic moments of the rare earth ions. But  while it was plausible, there was no direct evidence that this enhanced specific heat was associated with heavy-mass fermions, so the physical relevance of the Kondo lattice concept remained uncertain.

Steglich observed that in CeCu2Si2 the specific heat jump at the superconducting transition (which in BCS theory is basically the same size as the electronic specific heat at Tc) was about as big as the normal state specific heat coefficient, thus showing that the spin entropy had been transmuted into something that could go superconducting. Then (I think in subsequent experiments) Steglich observed that the rest of the superconducting thermodynamics in Cecu2Si2 was also consistent with pairing of heavy mass entities. This, I believe, is what convinced everyone that the spin entropy from the rare earth moments had been converted into heavy mass electrons—in other words, that the lattice Kondo effect was real. 

A few years after this, Louis Taillefer and Gil Lonzarich’s quantum oscillation study of UPt3 (Phys. Rev. Lett. 60, 1570 ) showed indeed that the U-f electrons (which appear as local moments at higher temperatures) were included in the Fermi surface at low T and had heavy masses, providing direct experimental confirmation of the Kondo lattice concept.

Cheers

Andy Millis

NSF, National Science Board, and the politics of staying quiet

As I mentioned previously, the National Science Board was summarily fired on April 25.  The NSB nominally advises the National Science Foundation.  There have been a number of pieces written about this:

• Going back in time to 2022, this essay is interesting to read, about the history of the NSF and the NSB, and the compromises put in place with the administrative structure.  Short version: Initially there was a real tension between the Director (reporting to the President) and the NSB.  Over time, the NSB was made subordinate to the director (1968).  Senatorial confirmation of board members was waived by the Senate in 2011.  
• Many professional organizations issued statements expressing grave concern about this wholesale dismissal of the board.  This AIP news article has a summary.  The CEO of the APS wrote this, the ACS leadership wrote this, the AAS wrote this, etc.
• The presidents of the National Academy of Sciences, National Academy of Engineering, and National Academy of Medicine issued this joint statement.  That has to set some kind of record for blandness, as it somehow does not even mention that the NSB was fired.  I fully understand that the Academies have a number of federal contracts, as one of their key responsibilities is leveraging their membership to do authoritative studies, with federal agencies usually being the customers.  I have no inside knowledge, but it sure looks like they are trying to walk a line of not raising the administration's ire.  (Surely this raises the question:  If it's never acceptable to say anything that might upset the administration, then how can the objectivity of their reports relating to policy ever be trusted?)
• In contrast to the leadership, a lot of Academy membership has signed an open letter to Congress demanding the reinstatement of the board.
• Scientific American has very good reporting on this, including a no-holding-back statement by my colleague Neal Lane.
• Update:  Here is Dan Garisto's reporting in Science about letters sent by House Democrats and by Senate Democrats demanding action on this.  That article includes a statement by the fired head of the NSB, basically saying they were dismissed for defending the NSF budget from OMB.  I'm glad these letters were sent, but without the R majority signing on, I'm not holding my breath.

Meanwhile, the pace of NSF awards continues to be glacial, even compared to last year.  See this plot from Grant Witness. 

We are 7 months into the fiscal year, and obligated dollars are less than half at this time last year, and more like 27% of those at this time in "normal" year.  It's hard to look at that and not wonder whether someone is aiming for a pocket rescission, regardless of what Congress appropriated.  NSF looks like an outlier here, by the way.  As badly hit as NIH has been, their award curves look much closer to last year.

Other related things worth reading:  

• This is a sobering and interesting article about what the author describes as the anti-science movement.
• When is staying quiet effectively giving tacit support to administration policies?  There is an article in The Nation (not readable without a free signup) talking about what they term "Vichy Science".
• Update:  Essay by Holden Thorp, EIC of Science, including a link to a half-hour interview with Tim Snyder, author of "On Tyranny", about how resistance can work.  

Back to science in my next post.

Energy storage in the internal states of molecules - old and new

A science story first, then a US research ecosystem story later.

When we think about using molecules to store energy, it's usually in the context of food or fuel, so that chemical reactions take place - bonds are broken and remade, and in an exothermic reaction, the products end up with more kinetic energy (center of mass motion, molecular vibrations and rotations) than the initial reactants.  However, there are other ways that molecules can store energy.  I read about a cool example of this last week, but first I want to give tell you an old and very quantum mechanical story that I learned about in grad school when I did very low temperature physics.

Diatomic hydrogen, H2, is the simplest molecule there is, just two electrons and two protons.  Roughly speaking, the \(1s\) orbitals of the H atoms hybridize to form \(\sigma\) bonding and \(\sigma*\) antibonding molecular orbitals.  The lowest electronic state is the two electrons in a spin singlet, \((1/\sqrt{2})(|\uparrow \downarrow\rangle - |\downarrow \uparrow\rangle)\) in the \(\sigma\) molecular orbital.  Remember, the electrons are fermions, so the electronic wavefunction has to be antisymmetric (pick up a minus sign) under exchange of the electrons. The spin singlet is antisymmetric under exchange, the \(\sigma\) orbital is spatially symmetric under exchange, so the full electronic wavefunction (product of the spin and spatial components) is appropriately antisymmetric.  

That's not all there is to it, though, as explained thoroughly here.  The protons (while being made up of quarks and gluons, etc.) are (composite) fermions, so we have to think about the quantum wavefunction that describes them, too.  There are two possibilities.  In the "para" configuration, the proton spins are in a singlet (antisymmetric), meaning that the spatial wavefunction of the protons must be symmetric under exchange.  The spatial state of the bound protons can have some orbital angular momentum \(\mathbf{L}\), and the simplest, lowest energy situation is with quantum numbers \(\ell =0\) and therefore \(m_{\ell} = 0\).  In contrast, in the "ortho" configuration, the proton spins form a triplet state (symmetric under exchange), meaning that the spatial wavefunction must be antisymmetric, \(\ell = 1\).  Approximating the H2 molecule as a rigid barbell-like rotor with some moment of inertia \(I\), then ortho molecule has a rotational energy \(\hbar^2/2I\) larger than the para case.  That works out to about 15 meV of energy per molecule.  So, para-hydrogen is the true ground state.  It turns out that the ortho/para spin isomer energy difference makes liquefying hydrogen a challenge, since the latent heat of vaporization for H2 is only 9.4 meV.  That is, every time an ortho-hydrogen molecule converts to para-hydrogen through some collisional process, it releases enough energy to kick a hydrogen molecule out of the liquid.  I learned about this in my thesis work playing around at ~ 1 mK temperatures - any H2 adsorbed or otherwise stuck in the apparatus could result in detectable long-term heating effect as it slowly converted from ortho to para.  Bottom line:  Energy can be stored in the internal states of molecules.

From Fig. 1 of this paper.

This seems very esoteric, but the idea of storing energy in some internal state of a molecule for later release shows up elsewhere.  Last week, in this article in Science, for example, the authors report a molecule inspired by aspects of DNA that can be put via UV exposure into a distorted form ("Dewar isomer") where it is metastable at room temperature (half-life of 481 days).  It can be induced to pop back into the undistorted isomer by heat, acid exposure, or via a catalyst, and when it does, each molecule releases the stored energy (2.36 eV per molecule!) into vibrations and rotations that heat its surroundings.  The stored energy density in this stuff is about 4% of the releasable energy density of gasoline, which is not too shabby.  The authors propose a system where exposure to sunlight can store energy in the molecules, and this can later be released on demand via catalyst.  They demonstrated that the heat release from enough dissolved molecules can readily boil water.  Very neat stuff.

News items and essays to read

First, some inside-baseball US funding discussion.  Apologies to my international readers, who likely don't care much about this except in the abstract.

• Breaking news:  According to journalist Dan Garisto, as of April 25, 2026, the president has fired the entire National Science Board.  The NSB helps oversee the National Science Foundation.   From the outside, it had sure looked to me like the NSB had tried verrrrrrry hard to stay on the administration's good side.  That was fraying recently, as this February article in Science included comments implying that not everyone was thrilled with the executive branch strongly shifting NSF priorities.  It sure looks like their reward for not speaking out strongly about the importance of continued support for research was apparently to be terminated with prejudice.  If any readers have first-hand knowledge of what happened, please post in the comments.
• I had already been planning to point out this article in Nature, which says that NSF funding may finally be about to flow, after a long, murky back-and-forth between the agency, Congress, and OMB.  It's worth noting that Congress had stated in their guidance that no directorate within NSF should be cut by more than 5%, while OMB has mandated (apparently) different spending levels which, among other things, would cut Mathematics and Physical Sciences by 15% and Engineering by 18%.  It sure doesn't look obvious to me, with everything else going on right now, that Congress is willing to truly push back on this.  Surrendering microscopic spending authority to OMB seems like a complete abdication of congressional authority, but what do I know.

One bit of science:

• I participated in a virtual workshop this week, Targeted Questions: Strange Metals.  I have written about strange metals before (see here and here).  The workshop was streamed, including the discussion sessions, and here is the youtube link if you are interested.

Some essays this week:

• There was a report by a Yale University committee about the erosion of trust in higher education in the US earlier this month.  It certainly spurred a lot of conversation, and it raises many important issues.  That said, it does seem to downplay the fact that there has been a decades-long campaign by some with the goal of eroding trust in higher education, as pointed out in this essay.
• A friend pointed me to this essay by Santiago Schnell about higher education in the era of AI - I found it very thoughtful, even though I don't agree with every aspect.
• Speaking of AI, this essay ("The future of everything is lies, I guess: Bullshit about bullshit machines") by Kyle Kingsbury was provocative and worth reading, again even if I don't agree with every aspect.
• Speaking of thoughtful commentary, here is an essay by a billionaire that is worth reading.

Floating magnets to sense magnetic fields

We've all seen a traditional compass.  A ferromagnetic, magnetized needle is mounted on a rotating bearing (or floated on the surface of a liquid) so that it can rotate in the \(x-y\) plane.  If there is an in-plane magnetic field \(\mathbf{B}\), the needle will rotate to align with that component of the field.  (It stops in the aligned state because of friction; otherwise it would "librate", oscillating back and forth about the field direction.)  In first-year undergrad physics, we learn a simple model of why this happens.  The magnetized needle can be modeled as a magnetic dipole \(\mathbf{m}\).  We learn that a magnetic dipole in a uniform magnetic field generates a torque \(\boldsymbol{\tau} = \mathbf{m}\times \mathbf{B}\).  If both \(\mathbf{m}\) and \(\mathbf{B}\) are in the \(x-y\) plane, any torque must be directed along \(z\), and the torque goes to zero when \(\mathbf{m} || \mathbf{B}\).  The simplest result of \(\boldsymbol{\tau} || z\) is an angular acceleration that would cause an otherwise at-rest compass needle to rotate in the plane counterclockwise about the \(z\) axis. 

Left: A compass needle out of alignment with an in-plane 
magnetic field produces a torque along \(z\), and angularly
accelerates to reorient toward the field . Right: Under the 
right circumstances, the spin angular momentum of the 
needle could also precess around the field.

So, how sensitive of a magnetic field detector could you make using this basic approach?  This week I came upon this paper from a few years ago, which looks at this problem from the theoretical modeling side, and then this paper from last year that does the experiment.  The magnet in question is a little (21 \(\mu\)m diameter) sphere of Nd2Fe14B, a rare-earth magnet.  The authors put that inside a lead chamber with a rounded bottom, and they cool the lead down to 4.2 K, well below its superconducting transition temperature.  As a result, the sphere is magnetically levitated inside thanks to the Meissner effect, with its magnetization lying in the \(x-y\) plane.  There is some residual magnetic flux trapped in the setup that does lead to a preferred field direction.  The authors can use cleverly wound pickup coils inside the chamber to detect the orientation of the sphere, as well as apply AC magnetic fields.  The authors are primarily concerned in thinking about energy resolution of detection, because they are thinking about detecting unusual particles (e.g. dark matter, axions), but they point out that it should be possible to achieve tens of atto-Tesla per Hz\(^{1/2}\) field sensitivity per unit bandwidth - pretty wild.

But wait, there's more!  The magnetic moment of the magnetized needle originates from the spins of electrons in there.  This is gyromagnetism, so \(\mathbf{m} \propto \mathbf{S}\), the total spin angular momentum of the electrons in the magnet.  This means that in the presence of \(\boldsymbol{\tau} || z\), if mechanically possible the needle could start swinging up out of the \(x-y\) plane to project a component of \(\mathbf{S}\) along \(z\).  This is gyroscopic precession.  For macroscopic magnets, it's hard to be in the regime where this is the dominant effect, because that would require the precessional angular momentum to be small compared to \(\mathbf{S}\), and that's tough to achieve.  Maxwell (!) tried to do it in 1861 (!!), with no success.

In a very recent paper, this precessional response was finally observed, again in Nd2Fe14B microspheres.  (For a uniformly magnetized sphere of radius \(R\), the moment of inertia \(I \propto R^{5}\), and \(|\mathbf{S}| \propto R^{3}\), so it's easier to get into the precessional regime with smaller \(R\).)  This precession approach is a pathway to even higher sensitivity measurements of magnetic fields.

I think this is very cool, and it is a strong reminder that spin angular momentum is just as real as any "mechanical rotation of solids" angular momentum.  

Disorder and illumination

No, this is not another grim post about the chaotic US research funding environment.  Instead I wanted to write a bit about a good example of empiricism in experimental condensed matter physics, the use of illumination to (somewhat but not entirely mysteriously) improve electronic transport in 2D electronic systems.  

This story goes back decades, and it's all about the role of "disorder" and its effects on electronic conduction.  It's been appreciated since the 1930s that, at low temperatures so that lattice vibrations are frozen out, conduction in ordinary crystalline metals and semiconductors is limited by the charge carriers (let's work with electrons rather than holes to make discussion simpler) scattering from disorder, deviations from an infinite periodic crystal lattice.  Grain boundaries, vacancies, impurities - these all can scatter electrons that would otherwise propagate ballistically through the material, and this is often modeled as a "disorder potential", a spatially varying potential energy \(V(\mathbf{r})\).  If you want the best transport properties (longest elastic mean free path), you want \(V(\mathbf{r})\) to be small in magnitude and as smooth as possible.  This is even more important if you want to study some delicate many-body state that is expected to arise at very low temperatures - you need the disorder potential to be small compared to the energy scale of that state to avoid messing it up.

In semiconductors, where the carrier density is low and screening is therefore not as good, charged defects are particularly effective at scattering.  In modulation doping, the dopants that are the source of the charge carriers in some nearby semiconductor 2D interface or quantum well are spatially distant from where the current is going to be flowing, to minimize the scattering from those ionized donors.  

For decades it has been known that, to get the best transport properties in GaAs-based (and other) semiconductor structures, it can be good to illuminate the devices at cryogenic temperatures with a red LED.  See, for example, this paper trying to explore the upper limits of charge mobility in GaAs 2D electron gas (2DEG), where the authors say "For measurement, our samples are loaded into a 3He cryostat, where a red light-emitting diode (LED) is used to illuminate the samples for 5 min at \(T \sim \) 10 K. Following illumination, we wait for 30 min at \(T  \sim \) 10 K after the LED has been turned off before resuming the cool down to base temperature."   The qualitative explanation for this is that the photons provide enough energy to excite charge carriers, and those mobile carriers can occupy trap states, rearrange themselves, and generally set up a better screened disorder potential.  In GaAs 2DEG, the result is higher mobilities (as inferred from conductivity + Hall effect) and much cleaner fractional quantum Hall effect data, showing that the post-illumination disorder is now sufficiently weak that more delicate states can form - see Fig. 1 of this paper (arXiv version) for the before/after.  As far as I know, there is not a deep, rigorous theory of how this works, but it is known empirically.  

Fig. 2 from this paper, showing electronic magnetotransport

before/after illumination by a UV LED at low temperatures.

This preprint on the arXiv this week shows that a similar improvement in transport properties can be found in structures where graphene is encapsulated by hexagonal boron nitride (hBN).  Sandwiching graphene and other 2D materials in hBN has been known since 2010 as a way of drastically improving the charge disorder situation compared to just putting 2D materials on top of SiO2.  (That paper has 8800+ citations on google scholar btw.)  Now, it is shown that if you shine 5 eV photons (deep UV = 248 nm wavelength) on such a sandwiched structure, the already-good charge environment can become even better.  Even though that energy scale is below the band gap of hBN, the light is able to kick enough charges around to smooth out some residual disorder.  Very cool.  

FY27 Presidential budget request

To the surprise of no one at all, the 2027 presidential budget request is extremely bad for science.  Remember, this is largely a political document, and Congress does not have to follow this.  In the past year, Congress largely ignored the recommendations and appropriated a much flatter budget (though agency priorities are still set by the PBR for executive agencies).  This new request shows that Vought et al. still would prefer to kill much public funding for science.

• NSF: request to cut from $8.8B (FY26 enacted) to $4B, a 56% cut that would eviscerate the agency.
• DOE: request to cut $1.1B from $8.4B (FY26 enacted) Office of Science budget.
• NASA: request to cut $5.6B from $24.4B (FY26 enacted), including $3.7B from science programs and $1.1B from the ISS.
• Commerce: This one shocks me. Request to cut $993M from NIST's $1.184B (FY26 enacted) budget. That would be an 84% cut (!!), seemingly destroying NIST. This needs to get headlines.  Either the people making this recommendation have no idea what NIST does (seems plausible), or someone has a personal grudge against the standard kilogram. Update:  Dan Garisto on bluesky points out that the enormous cut topline number is not consistent with the budget appendix, which implies a much smaller cut.  Unclear what the answer is here - it'd be quite a goof to have a topline number that far off.
• NIH: Proposed $5.5B cut from $47.2B (FY26 enacted).
• DOD: It's very hard to tell, especially since they're proposing hundreds of billions of dollars in additional spending including for missile defense. The proposed DOD increases vastly outweigh the cuts described above. 

These cuts are proposed despite constant fretting that China is surpassing the US scientifically. This past year it took aggressive lobbying to ensure that Congress pushed back against these kinds of cuts. For those who favor continued public investment in science and engineering research in the US, the task of arguing against these kinds of cuts begins again now.  As I've written before, this is a marathon not a sprint, and this will be an annual exercise under this administration.