Condensed concepts

The Times Higher Education Supplement has a nice book review  of a new text So thats how it all fits together by the celebrated mathematician John B. Conway. The book is designed for a "capstone" course at the end of a mathematics undergraduate degree. I thought the following introductory paragraph was helpful, and just as relevant to physics undergraduate education (at least in Australia):  the increasing modularisation of university curricula has meant that the connections between different areas of mathematics are not always made apparent. Indeed, for many students who attended schools in England, their pre-university experience of mathematics is that it is presented as a range of topics to be mastered for assessment, and which can then be completely forgotten as the learner moves on to the next module. Undergraduates today often express surprise when they discover that a topic covered earlier in the curriculum is required elsewhere later. This attitude is profoundly dep

The New York Times has a nice article by Thomas Friedman, The Last Person  about the Indian Institute for Technology in Rajasthan developing an extremely cheap tablet computer that is affordable [with some government subsidy] to the extremely poor.

How does someone learn to do good research? Most scientists might claim it is by osmosis and experience. Social scientists tend to make graduate students take formal courses in "research methods". In contrast, natural scientists seem quite skeptical to such approaches. Michael Marder has a new book out Research Methods for Science . It looks quite worthwhile because of the mix of background, hands on exercises, practical advice, statistics, .... It is based on a course  he has co-taught to undergraduate science majors for a number of years at UTexas. I welcome comments, particularly from people who have taught or taken such courses.

More and Different by Phil Anderson is stimulating and challenging holiday reading. Here is a paragraph from an essay, "Reflections on Twentieth Century Physics," which considers how from the 1950s to the end of the century: There was a very sharp change in the nature of a research career. The "promising" young scientists publication rate grew by factors of five to ten; the number of applications a young researcher might make for post-doctoral work or an entry-level position rose from two or three to 50. Senior scientists were overwhelmed with receiving and sending reams of letters of recommendation, which thereupon became meaningless. The numbers of meetings .. grew by factors of ten or more... Most publications became tactical in this game of jockeying one's way to the top; publications in certain prestige journals were seen as essential entry tickets or score counters rather than as serious means of communication. Great numbers of these publications were a

How does one distinguish a metal from an insulator? One signature might be the presence of a charge gap at the chemical potential. Is there a broken symmetry associated with a metal-insulator phase transition? A key concept emphasized by Anderson is that distinct phases of matter are associated with the rigidity of their order parameter. For example, for a superconductor the superfluid density is associated with the phase stiffness of the ground state wave function. It turns out that the Drude weight and the charge compressibility are both zero in a Mott insulator and non-zero in a metal. This is discussed in an interesting article by Imada.  Aside: Imada's paper assumes the metal-insulator transition is continuous (i.e. not first-order) and derives scaling relations for the transition. This assumption may not be valid. At least, in dynamical mean-field theory the Mott transition at half filling is first order. Kohn (and later Thouless) emphasized that for a metal one cou

Phil Anderson makes the following interesting comment [in the context of Cooper's treatment of the "pairing" problem] about model building in research. Actually, in almost every case where I have been really successful it has been by dint of discarding almost all of the apparently relevant features of reality in order to create a “model” which has the two almost incompatible features:   (1)  enough simplicity to be solvable , or at least understandable;   (2) enough complexity left to be interesting , in the sense that the remaining complexity actually contains some essential features which mimic the actual behavior of the real world, preferably in one of its as yet unexplained aspects.   I said dangerous, and the sense in which this is true is that one is laying a trap for the majority of one’s colleagues, who are too literal-minded to understand either the necessity or the reality of the model-building process. A really well-built model can often stand a great d

This post connects two seemingly disparate research topics of interest to me: optical properties of organic molecules and low energy excitations of strongly correlated electron materials. The connecting concept is that of an isosbestic point. The figure below shows the absorption spectrum of an organic dye  for a range of pH values. Note that there is a wavelength (around 500 nm) at which all the spectra appear to cross. This is called the isosbestic poin t. Varying the pH varies the relative concentration of two forms of the dye molecule. Each form has a characteristic absorption peak (centred at lambda_1 and lambda_2 in the figure). The isosbestic point occurs at the wavelength at which the absorption due to each of the molecular forms has the same intensity. Hence, at this wavelength varying the relative concentration does not change the absorption intensity. So what does this have to do with strongly correlated electron materials? In 1997 Dieter Vollhardt published a PRL,  Char

The Heckler is a column in the Sydney Morning Herald where "Readers are invited to send 450 words about what makes their blood boil." This week  Nick Coleman had a piece "You could do this or go fishing"  about the process for applying for research grants in Australia.

Previously I posted how one could see a signature of Berry's geometric phase in Shubnikov de Haas experiments on graphene. Similar experiments and analysis of data for topological insulators produces ambiguous results, as discussed in this recent  PRB  by Taskin and Ando. In particular, the Berry phase gamma can be extracted from a "Landau level fan diagram" where one plots the inverse of the magnetic field B vs. the Landau level index N. Taskin and Ando claim that in the topological insulator the dispersion relation E(k) is non-linear and as a result 1. the fan diagram is non-linear 2. the Berry phase gamma deviates from pi and is a function of B I am confused by 2. I would have thought that topology would require gamma to be only be pi or 0. This could be checked by looking at the eigenfunctions for the non-linear dispersion. Am I missing something? The above claims are based on eqn. 8 in the paper. It is derived from the generalised Onsager condition which is

I have just started reading Phil Anderson's new book, More and Different: notes from a thoughtful curmudgeon . It is a collection of essays on wide ranging subjects: personal reminiscences, history, philosophy, sociology, science wars, .... Some of these have been published before but many have not. The history is fascinating and the science stimulating. I highly recommend the book and will hopefully post some highlights. Last night I read the personal reminiscence "BCS and me" which gave me an even greater appreciation of just how tortuous the road to the theory was, how brilliant the solution was, and how it still left some important questions to be answered.

When and how can a large magnetic field change the ground state of a strongly correlated electron metal? An earlier post considered this question. To me, one of the most striking cases is that of heavy fermion metals in high magnetic fields. From quantum magnetic oscillations [e.g., SdH and dHvA] one can measure the effective mass of the Fermi liquid quasi-particles. The figure below shows how in CeB6 the effective mass decreases significantly with increasing magnetic field. Hence, the field destroys the heavy fermion behaviour. What is the physics behind this dramatic effect? The heavy fermion character arises from the formation of Kondo singlets between the localised spins and the conduction electrons. However, an external magnetic field breaks these singlets, reducing the heavy fermion character. The Kondo lattice temperature [coherence temperature] sets the relevant magnetic field scale  and is well described by a slave boson theory of Wasserman, Springford, and Hewson  [see eq

There is an excellent New York Times online article What is College For?  by Gary Gutting. Here are just a few quotes to stimulate you to read the whole article. the university curriculum leaves students disengaged from the material they are supposed to be learning.  They see most of their courses as intrinsically “boring,” of value only if they provide training relevant to future employment or if the teacher has a pleasing (amusing, exciting, “relevant”) way of presenting the material. As a result, students spend only as much time as they need to get what they see as acceptable grades (on average,   about 12 to 14 hour a week   for all courses combined).  Professors have ceased to expect genuine engagement from students and often give good grades (B or better) to work that is at best minimally adequate.  .... the raison d’être of a college is to nourish a world of intellectual culture ; that is, a world of ideas, dedicated to what we can know scientifically, understand humanistical

I really love the program Papers for the Mac. They just released version 2.1 and so I downloaded a trial version. However, after a week I reverted back to my old version 1. I had trouble getting the "Match" and "Query" functions to work and so gave up. I welcome alternative views and suggestions.

The renormalisation group and scaling has proven to be an extremely powerful technique in theoretical physics. It has even been succesfully  applied to turbulence. A paper by Yakhot and Orczag in the Journal of Scientific Computing (!) is widely cited.

The geometric (or Berry's) phase in quantum mechanics does have experimental manifestations in macroscopic materials. For example it can be seen in the phase of Shubnikov de Haas oscillations. The graph below is based on experimental data for graphene, taken from a 2005 Nature paper.  In this "fan diagram" the Landau level index is plotted versus 1/B [the magnetic field at which a peak in the resistance occurs]. The y-axis) intercept (shown in the upper right inset as a function of gate voltage) is related to the Berry phase, beta. For a linear Dirac spectrum, beta=1/2 and for a quadratic spectrum, beta=0. Similar experiments and analysis of data for topological insulators produces ambiguous results, which I will discuss in a forthcoming post about this recent PRB.

The Nernst effect is a thermal conduction analogue of the Hall effect for electrical conductivity, i.e., it measures the transverse electrical current induced by a longitudinal thermal current in the presence of a magnetic field perpendicular to both currents. It was considered an obscure (and very small) effect in elemental metals. However, the past 15 years it has become a powerful probe of strongly correlated metals, initially because of its sensitivity to superconducting fluctuations, as discussed by Ong. A nice helpful review is  The Nernst Effect and the Boundaries of the Fermi Liquid Picture by Kamran Behnia. He argues that the magnitude of the Nernst signal at low temperatures for a wide range of materials is proportional to the ratio of the charge carrier mobility to the Fermi energy. This is supported by the Figure below. Note the logarithmic scales. A few notes. 1. The simple Fermi liquid expression [equation (8)] gives the Nernst signal as proportional to the energ

Silver chalcogenides (e.g., Ag2Te) with slightly altered stoichiometry exhibit an unusual magnetoresistance. It is large and linear in field for magnetic fields up to about 6 tesla and temperatures between 5 and 300 K. [See this 1997 Nature paper ]. Several possible physical origins of the magnetoresistance have been proposed. 1. A PRL earlier this year proposes is the material is a topological insulator with gapless surface states described by a highly anisotropic Dirac cone. 2. In 1998 Abrikosov proposed the materials are gapless semiconductors with a linear spectrum, doped to a small carrier concentration, and that only one Landau level contributes to the conductivity. 3. Parish and Littlewood's proposal that the key physics is that of a strongly spatially inhomogeneous semiconductor which can be described by a random resistor network. 4. I also note that a band structure which produces a non-zero Berry curvature can produce a linear magnetoresistance, according to

I am enjoying re-reading the book 5 Minds for the Future by Howard Gardner. The 5 minds are Disciplined, Synthetic, Creative, Respectful, and Ethical. With regard to all of the first three he puts emphasis on the importance of considering the same topic from several angles and perspectives. In particular, creative needs to occur within a context of mastering earlier work. I wonder how might this does and might happen in condensed matter theory? Phenomenological vs. microscopic. Strong coupling vs. weak coupling treatments. Numerical vs. variational wave functions vs. field theories vs. renormalisation group. A "chemical" approach concerned with specific details vs. a "physics" approach which neglects many details. Other ideas? I actually wonder whether we actually do this more often and better than some disciplines.   But that perception may be based on ignorance and hubris!

Phil Anderson has written an interesting comment on my earlier post on this subject. His comment might be read in conjunction with two earlier posts that are revelant. Glueing together a theory  is relevant to his comment about whether the pairing interaction is instantaneous. Overdoped cuprates are an anisotropic marginal Fermi liquid  is relevant to his comment about the Anderson-Casey theory of non-Fermi liquid effects. I welcome further comments.

Sodium cobaltate (NaxCoO2) is a strongly correlated electron material which achieved a lot of attention before the mass migration to the new iron pnictide superconductors following their discovery around 2008. Some of the interest was motivated by the large thermopower, spin frustration associated with the underlying triangular lattice, and superconductivity from water! Jaime Merino, Ben Powell, and I wrote several papers on the subject. At the cake meeting today we reviewed two papers which focused on the doping x=0.5. Electronic and magnetic properties of the ionic Hubbard model on the striped triangular lattice at 3/4 filling Ionic Hubbard model on a triangular lattice for Na0.5CoO2, Rb0.5CoO2, and K0.5CoO2: Mean-field slave boson theory The latter features some really cool movies. Here are some of the outstanding questions raised by the strange ground state of the x=0.5 material. It appears to be an insulator, with a small amount of charge order, a large magnetic moment w

There is an opinion piece Up-end attendance rules for tutorial and lectures by Peter Van Onselen in the Higher Education Section of today's Australian newspaper. He is a Professor of Political Science at U. of Western Australia and a contributing editor to The Australian . He says that Arts/Humanities courses follow the "tried and true" format of two lectures and one tutorial per week. The former are optional and typically attract less than 50 per cent attendance. Tutorials are compulsory, are over-crowded, and diluted by students who have not done the assigned reading. He argues that the quality of education could be improved, without any additional costs, by making the lectures compulsory and the tutorials optional. I prefer my own proposals: Fail more students and set the pass rate at the lecture attendance rate.

Seth Olsen brought to my attention an article Examining the Relationships among Doctoral Completion Time, Gender, and Future Salary Prospects for Physical Scientists in the Journal of Chemical Education. It is based on a survey of more than 3000 Ph.D graduates of physics and chemistry in the USA. It claims there is a correlation (for men but not women) between Ph.D completion time and future salary. It also debates whether completion time is a good measure of the scientific merit of the graduate (the shorter the better) and the quality of the program (the longer the better!). I found I was rather skeptical of many of the claims, values, and assertions in the article. [I also wonder about the reliability of the statistical methodology but am not claiming I could do any better...]. Nevertheless, the article is worth reading because all of the issues it raises and the literature that it surveys. I welcome any comments on the article.

What is so unique about the optimal doping at which the superconducting transition temperature is a maximum in the cuprates? There is an interesting paper Unified electronic phase diagram for hold-doped high-Tc cuprates by Honma and Hor. It builds on their earlier work which argued the existence of a universal planar hole scale (P_pl), which can be characterised by the thermopower at T=290 K, denoted S^290. P_pl is independent of the nature of the dopant, the number of CuO2 plane layers per unit cell, the structure, and the sample quality. The figure below shows S^290 versus P_pl for a wide range of cuprates. Note that the thermopower changes sign at a doping of P_pl ~ 0.25 which is comparable to that at which  Tc is a maximum [except for Sr doped La214].  What is the significance of this sign change of the thermopower? For a simple Fermi liquid it would correspond to a change in the sign of the charge carriers, i.e., from electrons to holes. Recently  Peterson and Shastry  interpre