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

The linear chain compound Li0.9Mo6O17 exhibits a subtle competition between superconductivity, a "bad" metal, and a strange "insulating" phase. Recently large deviations from the Weidemann-Franz law were reported by Nigel Hussey's group. The graph below shows the temperature dependence of the electrical resistance for current parallel to the chain direction. It has a "metallic" temperature dependence above about 30 K, and an "insulating" temperature dependence between the superconducting transition temperature around 1 K and 30 K. This is rather unusual and puzzling since one normally sees a direct transition from a metallic phase to a superconducting phase. Although there are other cases such as reported in this PRB  [see Fig. 2 inset] for an organic charge transfer salt where a superconducting state occurs close to a charge ordered insulator [see also the Table in this PRL ]. The data is taken from a Europhys. Lett. by Chen et al. which

A lot! This is a fact that I feel politicians who fund public universities just do not appreciate. Here is a statistic that I find mind boggling. Princeton University now has an endowment of $17.1 billion dollars! Aside: the graph above shows how the endowment is now above pre-GFC levels. What does that mean? Well, the university only has 5,000 undergraduates and 2,500 grad students. That means the average endowment per student is more than $2 million! [This is the highest per student endowment in the world]. The university aims to spend the endowment at a rate of 4-5.75% on the annual operating budget. That means about $100 K per year is being contributed (indirectly) towards each students education. For reference annual tuition is about $36 K. Room and board are a further $12K per year.

For organic superconductors there is a first-order phase transition from a Mott insulator to  a superconductor with increasing pressure. This post concerns the relevant Hubbard model, that on an anisotropic triangular lattice at half filling, as discussed in this review. With increasing U/t there is a first-order transition from a metal to an insulator. This leads to a discontinuity in the double occupancy at the transition, illustrated in the sketch above. The double occupancy D is shown below versus U/t for t'=0.8t. For reference D=0.25 for a half-filled system at U=0. The figure is taken from a 2008 PRL by Ohashi et al. D is calculated with Cluster Dynamical Mean-Field Theory (DMFT) 1.  A rough estimate of the  magnitude of D can be found from the Hellman-Feynman theorem D= dE_0/dU where E_0 is the ground state energy. In the Mott phase this is dominated by the antiferromagnetic Heisenberg exchange J ~ 4t^2/U. Hence, D ~ (t/U)^2 2. The discontinuity in D at the meta

Following up on an earlier post about how indirect spin couplings in NMR (Nuclear Magnetic Resonance) may be a signature of the covalent character of hydrogen bonds I have been reading a range of papers on the subject. The Figure below shows how the calculated O-O nuclear coupling J correlates with the donor-acceptor distance [another example of an empirical correlation I have been highlighting]. The figure is taken from a 2000 JACS by Del Bene, Perera, and Bartlett. One thing that is frustrating about reading most of these chemistry NMR papers is that they never explain the basic physics involved. The Oxford Chemistry primer on NMR by Peter Hore has a useful section on Indirect coupling. He gives a nice simple argument explaining how [from 2nd order perturbation theory] the H-H coupling in the hydrogen molecule is roughly J ~ A^2/E  where A is the proton hyperfine interaction and E is the energy gap between the ground state and the lowest triplet state. This estimate gives J ~ 30

Ultimately much of quantum many-body theory concerns calculating correlation functions which are measurable. For example, the conductivity can written as a current-current correlation function [Kubo formula]. The simplest approximation neglects vertex corrections and just calculates the "bubble" diagram consisting of the product of Green's functions. What are vertex corrections? When do they matter? What sort of robust or general results are available about them? Many people, including myself, often just ignore them. I fear this is partly motivated by difficulty rather good scientific criteria. Below are a few things I am slowly learning, re-learning, and digesting. Migdal showed that for the electron-phonon interaction the vertex corrections are small due to the smallness of the ratio of the electronic mass to the nuclear mass [alternatively the ratio of the speed of sound to the Fermi velocity]. But, Migdal's argument breaks down for an electron-magnon inte

An earlier post, Entitled to a reading , pointed out the value of carefully choosing engaging titles for your papers. Contrast the titles of the following two papers. The subject and conclusions of the papers are similar. Benzene forms hydrogen bonds with water  published in Science. Low-J rotational spectra, internal rotation, and structures of several benzene-water dimers  published in Journal of Chemical Physics. Arunan brought this contrast to my attention.

Yes. Closely. This is the conclusion I have slowly come to over the years. Furthermore, the more junior the class the more closely you should follow a text. Often I have struggled to find a text I thought suitable or have drawn on material from several books. This has meant giving out lecture notes. It seems closely following a book is most effective if you can actually get students to read it ! This appears to be a major goal of people who use methods such as Peer Instruction. Having said all that you can expect student complaints. "You are just telling me what it is in the book". "There is too much reading". "Why are we paying you?" "Don't you have any ideas of your own!" I welcome your thoughts. I would be curious to learn of systematic studies which showed whether student learning (rather than satisfaction and comfort) was actually enhanced by closely following a text.

Seth Olsen and I have just advertised  for a postdoc to work with us at UQ. The flavour of our interests and approach can be seen in posts on this blog under labels such as  organic photonics , quantum chemistry , conical intersections, and Born-Oppenheimer approximation.

One might think that the Hubbard model on the square lattice with infinite U would be relatively boring. For example, a "simple" theory like Brinkman-Rice [or equivalently slave bosons] would predict that the ground state is a metal, except at half filling where it is a Mott insulator. There is an interesting preprint Phases of the infinite U Hubbard model by Liu, Yao, Berg, and Kivelson. Here are a just a couple of the results concerning the ground state on a ladder, that I found interesting. For 3/8 filling [3 electrons per 4 sites] they find the ground state is an insulator with a charge gap (0.24t) and plaquette bond order. The spin degrees of freedom are equivalent to those of a spin-3/2 antiferromagnetic Heisenberg model. I think this can be "understood" this by starting from the limit of weakly coupled placquettes with 3 electrons per plaquette. For 1/4 filling the ground state is an insulator with a charge gap (~0.1t) and a small spin gap and "dimeris

For most people this is one of the hardest and most tedious things to do. Some helpful thoughts are  Writing the first proposal  by John Wilkins [who mentored me through my first proposal!]. There is also a useful chapter in A Ph.D is not Enough , by Peter Feibelman. Three basic questions you need to make sure your clearly and convincingly answer: Is the proposed project important? Is it feasible? Are you the best person to do the work?

Everyone agrees that you need to herd cattle and sheep. But what about cats? Cats are best enjoyed and fulfil their purpose if they are left alone and allowed to be what they are. What is the relevance of this? It is sometimes claimed that academic researchers are like cats. They are fiercely independent and groups of them are very difficult to manage. Hence, books such as Herding Cats: Being advice to aspiring academic and research leaders   by Geoff Garrett and Graeme Davies. I became aware of the existence of the book because Geoff Garrett, who is now Queensland's Chief Scientist, gave the UQ Physics Colloquium on friday. I have not read the book. Afterwards a colleague expressed reservations about the ideas presented, saying, "this is relevant to engineers, not physicists!" One idea that was presented was the importance of having a " Big Hairy Audacious Goal " which creates team spirit. Although laudable on some level, I am hard pressed to think of exa

Misconception 1?          "H-bonding interactions are dominated by electrostatics " This is stated in  Assessment of the Performance of DFT and DFT-D Methods for Describing Distance Dependence of Hydrogen-Bonded Interactions ,  one of the most downloaded recent papers in Journal of Chemical Theory and Computation. Earlier posts  discuss the importance of covalent interactions, particularly in shorter (stronger) Hydrogen bonds. Misconception 2? “One of the most fundamental, yet enigmatic, of all chemical processes is the transfer of protons in liquid water, which occurs via ultrafast quantum tunneling in the hydrogen bonded network”. This is stated in a 2003 Science article  and  challenged by Dominic Marx in  Proton Transfer 200 Years after von Grotthuss: Insights from Ab Initio Simulations proton transfer has often readily been explained by taking recourse to the appealing concept of “proton tunneling”, 27 – 29  which was begotten shortly after the birth of quantum m

I have been reading through the nice review Finite temperature properties of doped antiferromagnets by Jaklic and Prelovsek from 2000. They summarise their studies of the t-J model by the Finite temperature Lanczos method. At first sight the graph below of the temperature and doping dependence of the chemical potential does not look particularly interesting [at least to me]. However, they highlight its significance. Here are a few points. In a simple Fermi liquid the chemical potential has a positive, quadratic and weak temperature dependence. This is only seen for doping c_h=x=0.3 For a wide doping range [0.05 < c_h < 0.3] the temperature dependence is approximately linear. The slope changes sign for approximately optimal doping (c_h ~ 0.15). The weak temperature dependence for c_h ~ 0.15 means that optimal doping corresponds to maximum entropy!  [This can be deduced via the Maxwell relation below. Don't you love thermodynamics!] This relation is also related [approxima

Remember Hendrik Schon ! A decade ago he published a string of very impressive Nature and Science papers that eventually turned out to be "too good to be true". It seems a similar thing has been happening in the field of social psychology. The AP reports   three graduate students grew suspicious of the data Stapel had supplied them without allowing them to participate in the actual research. When they ran statistical tests on it themselves they found it too perfect to be true and went to the university's dean with their suspicions. In the future, the university plans to require raw data from studies to be preserved and made available to other researchers on request - a practice already common in most disciplines. Nature News reports The commission found that co-authors of Stapel's papers seem to have been unaware of the fraud, naively trusting in Stapel's reputation and fooled by elaborate preparations for tests that were never actually carried out .....  S

Nonadiabatic Quantum Chemistry is a nice Chemical Reviews article by David Yarkony. It is quite succinct but covers a significant number of specific chemical systems where non-adiabatic effects [including conical intersections] are important and have been treated theoretically. Here I just mention one example for which theory has failed so far, the vibrationally mediated photodissociation of NH3 (ammonia) to NH2 + H. Experiments find that if the excited state contains a symmetric (asymmetric) N-H  stretch the dominant decay channel is to the NH2 ground state (excited state). Yarkony says that calculations [e.g. this one from Truhlar's group, which contains the figure below] have not yet captured this vibrational selectivity. I thank Seth Olsen for bringing the article to my attention.

There is an interesting (and somewhat depressing) article in the New York Times  Why Science Majors Change Their Minds (It's just so darn hard).  It discusses how in the US there is a big push to have more STEM (Science, Technology, Engineering, and Mathematics) graduates but even if many start these degrees they do not finish. One contributing factor is that these courses are graded harder than humanities courses. The article also discusses initiatives, particularly in engineering courses, to make the courses more "fun" and "relevant", especially via projects. I think this is all commendable and valuable. However, I have a sneaking discomfort that people [students, faculty, and administrators] just don't want to face the painful reality that engineering and science education does involve a certain amount of tedious hard work and that ultimately a lot of jobs (in any field) just aren't that exciting or satisfying. Or am I just a grumpy old man? I

The Hall coefficient is a fundamental property of metals. In simple Fermi liquid metals it is temperature independent and inverse proportional to the charge carrier density. It has the same sign as the charge carriers (electrons or holes). A major triumph of the Bloch model of metals is that it could explain the sign of the Hall coefficient for simple metals in terms of their Fermi surface. In contrast, the Hall coefficient of cuprate superconductors has a complex temperature and doping dependence which defies a simple description. Basic questions about the Hall coefficient are: What determines its sign? What is the origin of its temperature dependence? What is the relationship between it and the structure (or absence) of the Fermi surface?   A 2006 PRB by Tsukada and Ono  describes measurements of the Hall coefficient in the cuprate LSCO. The graph below shows the temperature dependence of the Hall coefficient for a range of dopings x of La2-xSrxCuO4 in the overdoped region. For r

This week I had an interesting experience. I was doing a calculation and comparing my result to experiment. The comparison was poor, with a discrepancy of a factor of about two. This was disappointing, but then I decided that the theory was just too simple and one should not experiment anything better than qualitative agreement... I just had to accept this. But then I found a mistake in my Mathematica code. I realised I had to check everything more carefully. .. One of my variables I had defined incorrectly... I redid the plot. The agreement of theory and experiment was excellent. But, now there is a real danger. I could stop checking for errors. Afterall, given I already found a couple there may be another one which will lead to new discrepancies. I will let you know if I find any. But, I have to confess the motivation to find errors is less than it was.. I wonder how often this happens in science.  I think I recall that there are some famous historical examples, e.g. that over

Just because an experimentalist claims to have measured a specific physical quantity does not mean they actually have measured the desired quantity. Theorists need to be particularly wary at uncritically accepting data. To most people, especially theorists, measuring the electrical resistivity of a metal sounds like an almost trivial measurement! Surely, you just stick a sample of the metal between the leads of an ohm-meter and read off the resistance! The temperature dependence of the resistance can provide significant information about scattering of quasi-particles in the metal and any decent theory should be able to describe it. A famous case it the "linear in T" resistivity of optimally doped cuprate superconductors, a signature of non- Fermi liquid behaviour. Most of the interesting strongly correlated metals (cuprates, organic charge transfer salts, iron pnictides, ....) have layered crystal structures leading to anisotropic electronic properties. These are sometim