O God! O Montreal!

I came across a review of a couple of Penguin books on nuclear physics from 1972.  Written by Denys Wilkinson and published in the book review section of Nature, they were not especially positive.  I thought it quite telling that he disliked the comparison of the square well and oscillator potential, which is done to show how similar the level scheme is.  Not having that in nuclear structure textbooks seems like a lost battle, perhaps thanks to the ubiquity of the shell model and the nod to Meyer and Jensen's textbook on the matter, among other things.  The quote from the review is quite impassioned:

October book: Density-Matrix Renormalizatiom.

Terribly and predictably, my goal of doing something significant with one book on my shelf each month proceeds in an unsatisfactory way.  October's book is – was – Density-Matrix Renormalization ed. Peschel et al., from the Springer Lecture Notes in Physics series (#528).  

I got the book something like 20 years ago after Pittel and Dukelsky started using the DMRG method in nuclear physics.  I was a young academic looking to move beyond my PhD work, and to the extent that I felt competent about anything outside the narrow focus of my PhD, computational work was the thing, and I was intrigued to learn more about this method.  My plan was to start by applying it to the Lipkin-Meshkov-Glick (LMG) model, a simplified version of the nuclear shell model, just to get to grips with the method, and see where that led me.  In the end, I got diverted into some other methods for solving the LMG model, inspired by a collaboration with a condensed matter physics friend and we wrote up some work on that, which we ran out of energy to publish after one rejection, and now it sits as the first (but not last) of outputs I've written that have got no further than the arXiv.  

Anyway - I never did become a practitioner of DMRG, though I felt like the basic idea was a simple enough concept in terms of a numerical algorithm:  Start from a truncated space and find the eigenvalues and eigenvectors.  Keep "the most important" (suitably defined) eigenvectors and throw away the rest.  Increase the size of your problem by adding extra states that were not included in the original truncation to replace the "unimportant" states that were thrown away.  Solve for eigenstates in this new space, and keep discarding and enlarging in this way until you (hopefully) converge at a solution which is (hopefully) a good approximation to the true solution.  This keeps the size of the space you ever need to deal with limited to a manageable value.  

The "density matrix" part of the algortihm is to do with how one chooses the most important states.  In the space being dealt with, the eigenvalues of the density matrix are found and it is those with greatest values that are selected as the "important ones".  The "renormalization" is the shifting of model space through the algorithm, at least that's how I picture it, though really the words "renormalization" and "group" bind together to refer back to a method from Grassman algebras in field theory that inspired the forerunner of the DMRG method.  Anything to do with Group Theory is, at least to me, obscure by the time the DMRG has turned into a neat numerical method. 

Anyway - the book... much of what I wrote above has come to me through being reminded of the method by reading the preface article "How it all began: A Personal Account" by Stephen White.  (NB this is hidden in the "Front Matter" chapter if you try to access the book online).  As an autobiographical reminiscence, this was very readable and helped me contextualise things.  I then felt emboldened to go on to the first "real" chapter: "Wilson's Numerical Renomalization Group" by Theo Costi.  Unfortunately I found this impossible to understand, as it supposed a high level of knowledge of the kind of condensed matter systems and models that the method was originally applied to.   I gave up and tried the second chapter:  "The Density Matrix Renormalization Group" (Noack and White) which was much more text-book like and a kind of tutorial for the method, with a first example being a partice in a discretised box (which the condensed matter physicists call "a single particle on a tight-binding chain") and I was able to follow the arguments here well.  When it got to the details of exactly how the density matrix is used and the desired states projected out, I did not work through everything in detail, as I would have liked to have done in order to sketch out my own implementation on computer, but if I had given it the kind of time I hoped I would have given to each book in this project, I think I'd have got there. 

Okay.  Not long left to do November's book.  😭

... and major new discoveries in this area ceased to be made in Cambridge

In Charles Clement's biographical article on Tony Lane, the following matter of fact description of the demise of nuclear and high energy physics in Cambridge is summarized neatly as

"In the late 1950s, theorists in high-energy fundamental-particle physics in Cambridge moved out of the Cavendish Laboratory away from experiment into the Department of Applied Mathematics and Theoretical Physics (DAMTP), and major new discoveries in this area ceased to be made in Cambridge."

 - a salutory warning to us all!

 

My name will live on forever

Some time ago, I blogged about the over-use of girls' names used to name physics things.  This was prompted by a Tweet stating "we are not decoration!"

Today I learnt about a new facility in South Africa called PAUL.  This is my name, and it's a male name, so I guess it slowly helps redress the balance. 

It's an underground facility for the kind of experiment where you want a low background of cosmic radiation.  Here's a picture below of a CAD picture of part of the setup, from the website above.

Solid State Physics: Ashcroft and Mermin

 So... at the beginning of last month I posted about a plan to take one of the books I have on my physics bookshelf each month and do something worthwhile with it - learn something that I didn't know, work through some problem and see what enlightenment I get, even kick-start a research project, and then report back here with what I've learnt.

The first book, as prompted by a post on X (a website I have left in favour of BlueSky), was Ashcroft and Mermin's weighty textbook on Solid State Physics.  I should preface this whole post by saying that I'm a bit disappointed (with myself) for not carefully managing my time to get more out of the book, but, you know, life.  It's been probably a busier September than most years.  It's the one year that all 4 of my kids are in school, with the youngest starting Reception Year and the oldest in Year 13, and its freshers flu season, and all sorts of other reasons ... I got "promoted" to be the representative of nuclear physics at STFC's science board at very short notice and that knocked out a couple of days. 

Well, so much for the excuses.  What about the book?  

I picked up a copy of the book when I was a PhD student, bought from Oxford's Blackwell's bookshop.  There's still a sticker on the back of the book telling me that I paid £25.95 for it, which even in the late 90s was not a bad price for a hefty 800-page hardback advanced-level textbook.  Despite the pretty poor rate of the PhD stipend in those days, it was the first time in my life I felt rich enough to splash out £26 on textbooks.  My PhD was not in Solid State Physics but I recognised it as an interesting area that I understood to be a ripe area for a would-be theoretical physicist to do a PhD in.  Probably it was foolish of me to go for nuclear physics over solid state, but I had largely found the teaching of Solid State physics uninspiring as an undergraduate and had not been motivated to learn very much of it, and felt ill-prepared for further study.  Well, I bought the textbook to have as a reference and perhaps I thought I'd even study it and learn from it, an indication that I was not as self-aware then as I am now.  Or at least I saw a practically endless life with copious spare time stretching in front of me in a way I don't now.

Picking up the book now, I started by reading from the beginning with the Drude Theory of Metals - a picture in which mobile electrons form a gas following the laws of kinetic theory.  The theory dates back to only just after the discovery of the electron, but before the structure of atoms was understood, and before the laws of quantum mechanics, so vital for atomic and solid state structure so generally, were known.  As a model it does a reasonable job (order of magnitude or better) of giving free electron densities and resistivities of metals; it can describe the Hall effect, and thermal conductivity.  The Drude model was something I had studied once upon a time, but I would have been hard pressed to say anything about it now, before re-learning from Ashcroft and Mermin.  

I carried on skimming through the following chapters to get an idea of the broad brush of development of ideas, but then decided that I wanted to learn at least something that might be a bit more useful to me, so I jumped way ahead to near the end of the book, to the chapter on Electron Interactions and Magnetic Structure.  It contains introductions to the kinds of spin Hamiltonians familar as standard models to me as a quantum many-body physicist: the Heisenberg model and the Hubbard model.  I realise I had a very facile view of the development of these models, supposing that they started from the assumptions that you could imagine lattices in atoms with magnetic moments were fixed in place and you supposed a very simple interaction between the atoms based on the relative orientation of the neighbouring spins.  In reality, there is much more to it, and this is brought out nicely in the book.  For a start, though they are models of magnetism, the authors emphasise the electrostatic origin of the magnetic effects.  Mainly because of the required antisymmetry of the overall wave function, the spin orientation of atoms in a lattice can be determined, with the spin having to match (or "anti-match", I suppose) with the spatial part of the wave function, which itself is determined mainly by electrostatic effects.  Actual magnetic interactions between atoms are a smaller effect when it comes to how the atoms line up to give macroscopic magnetism.  Interesting!  

Of course, I would have liked to have gone further, and worked through some examples to do actual calculations, and maybe worked through a problem or two at the end of the chapters.   I am already a week late writing this up here, though, and a week late starting the October book, so alas I will leave A&M behind for now.  On the other hand, I have to submit some ideas for BSc final year projects for Physics students at Surrey, so maybe I'll set one on the Heisenberg model, and vicariously live my continuing interest in this stuff through my student. 

To my regret, this exercise resulted in a nasty splodge on the fore-edge of the book when I left it in my bag with a too-ripe banana.  Still, perhaps that's better than having the book look pristine through being barely touched since its purchase nearly 30 years ago