A very effective Hamiltonian in nuclear physics

Atomic nuclei are complex quantum many-body systems. Effective theories have helped provide a better understanding of them. The best-known are the shell model, the (Aage) Bohr-Mottelson theory of non-spherical nuclei, and the liquid drop model. Here I introduce the Interacting Boson Model (IBM), which provides somewhat of a microscopic basis for the Bohr-Mottelson theory. Other effective theories in nuclear physics are chiral perturbation theory, Weinberg's theory for nucleon-pion interactions, and Wigner's random matrix theory.

The shell model has similarities to microscopic models in atomic physics. A major achievement is it explains the origins of magic numbers, i.e., nuclei with atomic numbers 2, 8, 20, 28, 50, 82, and 126 are particularly stable because they have closed shells. Other nuclei can then be described theoretically as an inert closed shell plus valence nucleons that interact with a mean-field potential due to the core nuclei and then with one another via effective interactions.

For medium to heavy nuclei the Bohr-Mottelson model describes collective excitations including transitions in the shape of nuclei.

An example of the trends in the low-lying excitation spectrum  to explain is shown in the figure below. The left spectrum is for nucleus with close to a magic number of nuclei and the right one for an almost half-filled shell. R_4/2 is the ratio of the energies of the J=4+ state to that of the 2+ state, relative to the ground state. B(E2) is the strength of the quadrupole transition between the 2+ state and the ground state.

The Interacting Boson Model (IBM) is surprisingly simple and successful. It illustrates the importance of quasi-particles, builds on the stability of closed shells, and neglects many degrees of freedom. It describes even-even nuclei, i.e., nuclei with an even number of protons and an even number of neutrons. The basic entities in the theory are pairs of nucleons. These are taken to be either an s-wave state or a d-wave state. There are five d-wave states (corresponding to the 2J+1 possible states of total angular momentum with J=2). Each state is represented by a boson creation operator and so the Hilbert space is six-dimensional. If the states are degenerate [which they are not] the model has U(6) symmetry.

The IBM Hamiltonian is written in terms of the most general possible combinations of the boson operators. This has a surprisingly simple form.

Note that it involves only four parameters. For a given nucleus these parameters can be fixed from experiment, and in principle calculated from the shell model. The Hamiltonian can be written in a form that gives physical insight, connects to the Bohr-Mottelson model and is amenable to a group theoretical analysis that makes calculation and understanding of the energy spectrum relatively simple.

Central to the group theoretical analysis is considering subalgebra chains as shown below

 

An example of an energy spectrum is shown below.

The fuzzy figures are taken from a helpful Physics Today article by Casten and Feng from 1984 (Aside: the article discusses an extension of the IBM involving supersymmetry, but I don't think that has been particularly fruitful).

The figure below connects the different parameter regimes of the model to the different subalgebra chains.

The nucleotide chart below has entries that have colour shading corresponding to their parameter values for the IBM model according to the symmetry triangle above.

The different vertices of the triangle correspond to different nuclear geometries and allow a connection to Aage Bohr's model for the surface excitations. 

This is discussed in a nice review article, which includes the figure above.

Quantum phase transitions in shape of nuclei

Pavel Cejnar, Jan Jolie, and Richard F. Casten

Aside: one thing that is not clear to me from the article concerns questions that arise because the nucleus has a finite number of degrees of freedom. Are the symmetries actually broken or is there tunneling between degenerate ground states?   

Colloquium on 2024 Nobel Prizes

This friday I am giving a colloquium for the UQ Physics department.

2024 Nobel Prizes in Physics and Chemistry: from biological physics to artificial intelligence and back

The 2024 Nobel Prize in Physics was awarded to John Hopfield and Geoffrey Hinton “for foundational discoveries and inventions that enable machine learning with artificial neural networks.” Half of the 2024 Chemistry prize was awarded to Dennis Hassabis and John Jumper for “protein structure prediction” using artificial intelligence. I will describe the physics background needed to appreciate the significance of the awardees work. 

Hopfield proposed a simple theoretical model for how networks of neurons in a brain can store and recall memories. Hopfield drew on his background in and ideas from condensed matter physics, including the theory of spin glasses, the subject of the 2021 Physics Nobel Prize.

Hinton, a computer scientist, generalised Hopfield’s model, using ideas from statistical physics to propose a “Boltzmann machine” that used an artificial neural network to learn to identify patterns in data, by being trained on a finite set of examples. 

For fifty years scientists have struggled with the following challenge in biochemistry: given the unique sequence of amino acids that make up a particular protein can the native structure of the protein be predicted? Hassabis, a computer scientist, and Jumper, a theoretical chemist, used AI methods to solve this problem, highlighting the power of AI in scientific research. 

I will briefly consider some issues these awards raise, including the blurring of boundaries between scientific disciplines, tensions between public and corporate interests, research driven by curiosity versus technological advance, and the limits of AI in scientific research.

Here is my current draft of the slides.

John Hopfield on what physics is

A decade ago John Hopfield reflected on his scientific life in Annual Reviews in Condensed Matter Physics, Whatever Happened to Solid State Physics?

"What is physics? To me—growing up with a father and mother who were both physicists—physics was not subject matter. The atom, the troposphere, the nucleus, a piece of glass, the washing machine, my bicycle, the phonograph, a magnet—these were all incidentally the subject matter. The central idea was that the world is understandable, that you should be able to take anything apart, understand the relationships between its constituents, do experiments, and on that basis be able to develop a quantitative understanding of its behavior. 

Physics was a point of view that the world around us is, with effort, ingenuity, and adequate resources, understandable in a predictive and reasonably quantitative fashion. Being a physicist is a dedication to the quest for this kind of understanding."

He describes how this view was worked out in his work in solid state theory and moved into biological physics and the paper for which he was awarded the Nobel Prize. 

"Eventually, my knowledge of spin-glass lore (thanks to a lifetime of interaction with P.W. Anderson), Caltech chemistry computing facilities, and a little neurobiology led to the first paper in which I used the word neuron. It was to provide an entryway to working on neuroscience for many physicists..."

After he started working on biological physics in the late 1970s he got an offer from Chemistry and Biology at Caltech and Princeton Physics suggested he take it. 

"In 1997, I returned to Princeton—in the Molecular Biology Department, which was interested in expanding into neurobiology. Although no one in that department thought of me as anything but a physicist, there was a grudging realization that biology could use an infusion of physics attitudes and viewpoints. I had by then strayed too far from conventional physics to be courted for a position in any physics department. So I was quite astonished in 2003 to be asked by the American Physical Society to be a candidate for vice president. And, I was very happy to be elected and ultimately to serve as the APS president. I had consistently felt that the research I was doing was entirely in the spirit and paradigms of physics, even when disowned by university physics departments."

2024 Nobel Prize in Physics

 I was happy to see John Hopfield was awarded the Nobel Prize in Physics for his work on neural networks. The award is based on this paper from 1982

Neural networks and physical systems with emergent collective computational abilities

One thing I find beautiful about the paper is how Hopfield drew on ideas about spin glasses (many competing interactions lead to many ground states and a complex energy landscape).

A central insight is that an efficient way to store the information describing multiple objects (different collective spin states in an Ising model) is in terms of the inter-spin interaction constants (J_ij's) in the Ising model. These are the "weights" that are trained/learned in computer neural nets.

It should be noted that Hopfield's motivation was not at all to contribute to computer science. It was to understand a problem in biological physics: what is the physical basis for associative memory? 

I have mixed feelings about Geoffrey Hinton sharing the prize.  On the one hand, in his initial work, Hinton used physics ideas (Boltzmann weights) to extend Hopfields ideas so they were useful in computer science. Basically, Hopfield considered a spin glass model at zero temperature and Hinton considered it at non-zero temperature. [Note, the temperature is not physical it is just a parameter in a Boltzmann probability distribution for different states of the neural network]. Hinton certainly deserves lots of prizes, but I am not sure a physics one is appropriate. His work on AI has certainly been helpful for physics research. But so have lots of other advances in computer software and hardware, and those pioneers did not receive a prize.

I feel a bit like I did with Jack Kilby getting a physics prize for his work on integrated circuits. I feel that sometimes the Nobel Committee just wants to remind the world how physics is so relevant to modern technology.

Ten years ago Hopfield wrote a nice scientific autobiography for Annual Reviews in Condensed Matter Physics,

Whatever Happened to Solid State Physics?

After the 2021 Physics Nobel to Parisi, I reflected on the legacy of spin glasses, including the work of Hopfield.

Aside: I once pondered whether a chemist will ever win the Physics prize, given that many condensed matter physicists have won the chemistry prize. Well now, we have had an electronic engineer and a computer scientist winning the Physics prize.

Another side: I think calling Hinton's network a Boltzmann machine is a scientific misnomer. I should add this to my list of people getting credit for things that did not do. Boltzmann never considered networks, spin glasses or computer algorithms. Boltzmann was a genius, but I don't think we should be attaching his name to everything that involves a Boltzmann distribution. To me, this is a bit like calling the Metropolis algorithm for Monte Carlo simulations the Boltzmann algorithm. 

Mental Health for Academics

Tomorrow I am giving a talk "Mental health for academics" for the ARC Centre for Engineered Quantum Systems as part of Mental Health Week.

Here is a video recording of my planned talk. As an experiment, I did a record practice versions of my talked and uploaded it on YouTube. Feedback both on content and the technology welcome.

Here are the slides.

A resource I mention at the end is the blog Voices of Academia, set up by Marissa Edwards from UQ.