Condensed concepts

Can strongly correlated electrons save the planet II?

2012-01-30T17:26:00.001+10:00

Several earlier posts discussed the thermoelectric effect in strongly correlated electron materials. The Seebeck coefficient S is a quantitative measure of the effect. At low temperatures it can be orders of magnitude larger than in elemental metals.

The figure above illustrates how thermoelectric couples can be used to either perform refrigeration or generate electrical power from waste heat. It is taken from a nice Perspective in Science Smaller is Cooler by Brian Sales which reviews state of the art materials in 2002.

The thermoelectric figure of merit, ZT is a dimensionless ratio which is a good measure of how useful a material will be in thermoelectric applications.

sigma is the conductivity and kappa the thermal conductivity.

Currently used materials such as Bi2Te3 [also a topological insulator!] have values of ZT ~1. If materials can be found with ZT~4 then thermoelectric refrigerators will be competitive with traditional compressor refrigerators, which are less reliable and environmentally dirtier.

So how good are strongly correlated electron materials?

There is a nice review by Paschen on heavy fermions.

It is important to note that the thermal conductivity is the sum of electronic and phonon contributions. If one neglects the latter (for the moment) and uses the Wiedemann-Franz ratio then ZT ~ S^2 where S is in units of k_B/e. This is indeed its magnitude near the coherence temperature in strongly correlated electron materials. Hence, ZT ~ 1 (but not larger) seems possible.

BUT, this argument neglects the thermal conductivity due to phonons which is much larger that the electronic contribution in this temperature regime. So one needs to find a way to reduce this. This leads to the idea of a Phonon Glass Electron Crystal.

Candidate strongly correlated materials may be skutterudites which exhibit heavy fermion behaviour (e.g. SmPt4Ge12).

Giving up on theoretical physics?

2012-01-29T17:02:00.002+10:00

No I am not! But it seems some of our colleagues are.

Alan Lightman has a rather disappointing article The accidental universe: science's crisis of faith in the last issue of Harper's from 2011.

[It was reprinted this week as the lead article in a weekly section of the Australian Financial Review, Review: Your Guide to the world of issues, ideas, and opinion. I thank an economist friend from church for bringing it to my attention, ``Is this really what you theoretical physicists believe?"]

Here are a few choice from Lightman's article quotes I found debatable:

Dramatic developments in cosmological findings and thought have led some of the world’s premier physicists to propose that our universe is only one of an enormous number of universes with wildly varying properties, and that some of the most basic features of our particular universe are indeed mere accidents—a random throw of the cosmic dice. In which case, there is no hope of ever explaining our universe’s features in terms of fundamental causes and principles...
Theoretical physics is the deepest and purest branch of science. It is the outpost of science closest to philosophy, and religion....

Theoretical physicists are Platonists. Until the past few years, they agreed that the entire universe, the one universe, is generated from a few mathematical truths and principles of symmetry, perhaps throwing in a handful of parameters like the mass of the electron. It seemed that we were closing in on a vision of our universe in which everything could be calculated, predicted, and understood....

We are living in a universe uncalculable by science....

The most striking example of fine-tuning, and one that practically demands the multiverse to explain it, is the unexpected detection of what scientists call dark energy.

Perhaps rather than giving up on theoretical physics better options may be to

give up on Platonism,
give up on reductionism,
give up on unbridled philosophical speculation
be a little humbler and be open to the idea that there may be some new physics to be discovered in the particle physics Desert which covers 12 orders of magnitude in energy!

Low Tc but strong Cooper pair binding

2012-01-27T17:19:00.000+10:00

The plot below is an important one which I have been meaning to blog about for a while. It shows the strength of the diamagnetic signal [increasing from blue to purple to red]

from an underdoped cuprate superconductor as a function of magnetic field H and temperature T.

The figure is taken from a paper from Ong's group, discussed in an earlier post.
The superconducting transition temperature Tc is 12 K. However, a magnetic field of about 45 tesla is required to destroy the diamagnetic signal which is associated with Cooper pairing. Furthermore, this "upper critical field" H_c2 is weakly temperature dependent. This large field scale reflects the large binding energy of the Cooper pairs. This can be seen by converting H_c2 to a coherence length (~30 A) and then an energy gap (~20 meV ~ 200 K) via a Pippard type formula [see this earlier Science paper by Ong et al.]. The energy gap is comparable to the pseudogap seen in ARPES and much smaller than k_B Tc.

I cannot get excited about atomic BEC's

2012-01-25T16:28:00.001+10:00

There is an interesting article Ultracold Bose gases deviate from the textbook picture in the Search and Discovery section of the July 2011 Physics Today. [My issue just arrived by snail mail today!].
It discusses how recent experiments have quantified deviations from the non-interacting boson theory of Einstein, which is taught to undergraduates.
It seems that these deviations can be described by Hartree-Fock theory. One might argue Hartree-Fock is also rather "text book".

For all the hype, somehow I cannot get excited about atomic BECs. To me, there seems a distinct contrast to solid state systems such as strongly correlated electron materials which exhibit properties (high-Tc superconductivity, spin liquids, heavy fermions, pseudogap, non-Fermi liquid metals,...) which are such a long way from anything remotely "text book"-ish and whose explanation requires the development of new physical concepts, approximation schemes, and numerical methods.

But, perhaps I am missing something.

Probing fluctuating superconductivity

2012-01-24T11:36:00.000+10:00

An important paper for understanding the pseudogap state of the cuprates is Diamagnetism and Cooper pairing above Tc in cuprates by Lu Li, Yayu Wang, Seiki Komiya, Shimpei Ono, Yoichi Ando, G. D. Gu, and N. P. Ong.
Physics has a helpful commentary by Kivelson and Fradkin.

Diamagnetic response of the superconducting state is orders of magnitude larger than other states of matter. [Due to the Meissner effect superconductors are sometimes said to be perfect diamagnets]. A state with no long range superconducting order but large fluctuations can produce a significant diamagnetic response. The authors find that for a wide range of underdoped cuprates that there is significant diamagnetism for a wide temperature regime above Tc. Moreover, this signal co-exists with a large Nernst signal.

This is important because it tends to rule out a proposed alternative explanation for the large Nernst signal that it could be produced by quasi-particles in small hole pockets associated with a density wave state.

A key signature of superconducting fluctuations is a non-linear dependence of the magnetisation on the magnitude of the magnetic field. For small fields it must be linear in field, but there must be some field scale, of the order of the upper critical mean-field H_c2 above which there is no diamagnetic response. This means there is some field scale H_min at which the magnetisation is a minimum. The non-linearity of the field dependence is seen in the Figure below. (The different curves correspond to different temperatures).

The Nernst signal shows a similar non-linear field dependence (see this PRB). The authors argue that if it is due to quasi-particles it should be linear in field up to a much higher field scale, e.g. comparable to the band width.

Aside: Are there any measurements of non-linear diamagnetism on organic charge transfer salts?

Strongly correlated electron systems in high magnetic fields III

2012-01-23T15:57:00.000+10:00

Metamagnetism occurs when the magnetic susceptibility increases with increasing magnetic field. This generally does not occur in weakly interacting systems. For example, if the susceptibility is enhanced by magnetic fluctuations, these are generally decreased by a magnetic field. However, DMFT calculations show this can occur for intermediate coupling. This is discussed in detail in the following paper:

Quasiparticle properties of strongly correlated electron systems with itinerant metamagnetic behavior by J. Bauer

In a Fermi liquid picture the susceptibility can either increase due to an increase in the effective mass or increase due to the quasi-particle interaction F0a (the Landau Fermi liquid parameter which determines the Sommerfeld-Wilson ratio).

Possibly the most promising candidate material for some of this physics is CeRu2Si2. The susceptibility increasses by a factor of more than 8, whereas the specific heat coefficient gamma only increases by a factor of 1.6.

Other candidate heavy fermion materials include YbT₂Zn₂₀ (T : Co, Rh, Ir) which is the same family described in an earlier post because it has particularly interesting thermopower.

Do conical intersections really matter?

2012-01-20T09:52:00.002+10:00

Conical intersections [where the potential energy surfaces of two electronic states touch] are ubiquitous in photochemistry. Their most important role is that they can explain why some photochemical reactions proceed so fast (i.e, on the scale of 10's of femtoseconds). However, an important outstanding question is:

Does the Berry's phase [geometric phase] associated with the conical intersection [CI] have important observable consequences?

There is a nice 2005 Perspective in Science by David Clary which discusses this for the specific case of the simplest possible chemical reaction H2 + H to H + H2. It seems that [contrary to what was claimed in the 90's] molecular scattering experiments associated with the ground state surface below are not sensitive to the geometric phase due to cancellations of terms associated with different angular momentum channels. However, showing this cancellation is rather subtle!

Aside 1 : the CI here arises due to the degeneracy of the E irreducible representation associated with the C_3 symmetry of the transition state which occurs when the three H atoms form an equilateral triangle.

Aside 2: For more background see the nice Physics Today article by Berry, Anticipations of the Geometric Phase.

I thank Seth Olsen and Ben Powell for discussions stimulating this post.

Learn from others mistakes

2012-01-19T16:38:00.000+10:00

The latest issue of APS News has some useful career advice in an article Ten Mistakes for Physicists to Avoid by James D. Patterson.

Top 10 popular mathematics books

2012-01-19T16:20:00.000+10:00

In the Guardian, Ian Stewart (one of the most successful authors of maths books for the general public) lists his top 10 choices. It is interesting that number 10 is Newton's Principia!

Violating a text book rule

2012-01-18T21:52:00.000+10:00

Solid state physics text books tell us that Matthiessen's rule is obeyed by simple metals: the temperature dependent resistivity is the sum of a temperature independent term due to elastic scattering off impurities and an impurity independent term which is temperature dependent due to inelastic scattering. I teach this to undergrads, but have struggled to actually find experimental data to show them.

Yesterday I came across a 1944 paper by Fairbank which contained the figure below for copper with tin impurities:

It looks pretty convincing. However, the text points out that the temperature dependence actually varies significantly with the impurity concentration, in violation of Matthiesens rule!

Does anyone know of better data?

Any simple explanations for these deviations?

The writing grind

2012-01-18T21:24:00.000+10:00

I am currently writing a grant application. This is hard going even for old hands. The most irritating bit is all the messing around with formats and fonts. As to text I find the best strategy is like for most writing: just get something (almost anything!) down on paper and then rewrite and polish. This first step is the hardest.

The insatiable greed of commercial journals

2012-01-16T18:12:00.002+10:00

Last week there was an excellent Op-Ed piece in the New York Times, Research Bought, Then Paid For, by Michael Eisen. It is critical of a Bill before the U.S Congress which would stop the current policy of research results from National Institute for Health funded research being available free to the general public. Apparently publisher commercial journals have lobbied for the bill as they see it a way to increase their revenues. The following paragraph is particularly poignant:

the journals receive billions of dollars in subscription payments derived largely from public funds. The value they say they add lies primarily in peer review, the process through which works are assessed for validity and significance before publication. But while the journals manage that process, it is carried out almost entirely by researchers who volunteer their time. Scientists are expected to participate in peer review as part of their employment, and thus the publicly funded salaries most of them draw through universities or research organizations are yet another way in which taxpayers already subsidize the publishing process.

Eisen makes the radical proposal:

Researchers should cut off commercial journals’ supply of papers by publishing exclusively in one of the many “open-access” journals that are perfectly capable of managing peer review (like those published by the Public Library of Science, which I co-founded). Libraries should cut off their supply of money by canceling subscriptions.

Mundane thermal equilibrium not quantum exotica

2012-01-14T13:38:00.000+10:00

Advocates of the highly speculative notion of "quantum biology" like to invoke the case of superconductivity as a "proof of principle" that macroscopic quantum effects can play a role in biology.

In 1994, Phil Anderson wrote a devastating critique of Roger Penrose's book Shadows of the Mind: A search for the missing science of consciousness. Anderson's review was entitled Shadows of Doubt, and contains the following relevant paragraph:

The review is also reprinted in More and Different. The preamble states the Penrose Fallacy: "all problems too difficult to be solved by the great brain of the author must be identical." (p. 186)

BTW: a review of the book by David Mermin just appeared in Physics Today.

Deconstructing the metal-insulator transition in 2DEGs

2012-01-13T21:28:00.000+10:00

There is an interesting preprint, Wigner-Mott scaling of transport near the two-dimensional metal-insulator transition by M. M. Radonjic, D. Tanaskovic, V. Dobrosavljevic, K. Haule, and G. Kotliar.

They argue that the density dependent metal-insulator transition seen in Silicon MOSFETs and other two dimensional electron gases (2DEGs) in semiconductor heterostructures is not driven by disorder (which has been claimed for many years) but rather by electronic correlations. Furthermore, the relevant experimental data can be described by a Dynamical Mean-Field Theory (DMFT) treatment of the Wigner-Mott transition in an extended Hubbard model on a lattice.

This means that the non-monotonic temperature dependence of the resistivity is associated with the crossover from a Fermi liquid at low temperatures to a bad metal at higher temperatures. I think thermopower measurements may be the most effective way to test this claim (see the previous post).

Thermopower reveals destruction of quasi-particles

2012-01-12T20:03:00.001+10:00

There is a nice preprint, Thermoelectric Power of the YbT2Zn20 (T = Fe, Ru, Os, Ir, Rh, and Co) Heavy Fermions by E. D. Mun, S. Jia, S. L. Bud’ko, and P. C. Canfield

The figure below shows the temperature dependence of the thermoelectric power for the title compounds. Note that it non-monotonic, being linear at low temperatures, reaching a maximum magnitude of order k_B/e ~ 80 microV/K at a temperature T_min.

The next figure shows that T_min (left scale) is correlated with the single ion Kondo temperature (horizontal scale) and the temperature at which the resistivity is a maximum (right scale).

The magnitude of the linear temperature dependence at low temperatures is simply related to that for the specific heat (see also this earlier post) as shown in the Figure below.

Aside: the inset on the lower right considers the Kadowaki-Woods ratio but does not make use of recent work concerning its universality.

All of the above features seem to be characteristic of broad classes of strongly correlated electron metals, as emphasized by Jaime Merino and I, in a PRB published in 2000. The temperature scales in the middle figure are characteristic of that at which there is a crossover from a Fermi liquid at low temperature to a bad metal at higher temperatures.

A challenging New Year's resolution

2012-01-11T19:31:00.001+10:00

I never reported back on how I fared on my New Year's resolution for 2010 (n.b. not 2011):

Spend the first half hour of each day thinking and writing in a notebook about the important science questions I am interested in and want to try and answer. And, specifically coming up with multiple alternative hypotheses and devising ways to distinguish them.

I tried hard but found this very difficult, more than I thought. The tyranny of the urgent often takes over and makes even carving out half an hour at the beginning of each day difficult. Maybe I have done this on 2-3 days per week (on average) over the past two years. However, the multiple alternative hypothesis bit is extremely hard, much harder than just carving out the time. I am hard pressed to think of a single example where I have successfully done this, to the level of actually being able to rule out one alternative hypothesis. On the other hand, I do feel the struggling process is sharpening my scientific thinking.

I welcome others to take up this challenge and let me know how they do! After all, surely this is really what good science is about.

Flawed genius

2012-01-07T08:29:00.002+10:00

In 2006 Times Higher Education published a nice review by Phil Anderson of Broken Genius: The Rise and Fall of William Shockley, Creator of the Electronic Age. Here are a few extracts:

Shockley called his method of thinking "try simplest cases". But that is really what many good theoretical physicists do when confronted with a complex problem; they try to find a simple model. Only Shockley elevated it to a mantra. He could see his way through the first few stages of any problem very quickly, but he rarely employed his talents to look further below the surface or to check his models against reality. Those who had this ability - for whom Shockley undoubtedly had a good eye - would almost inevitably go off on their own because Shockley could never allow them to follow their own instincts. He saw them as competition and a threat to his authority. I know, because it happened to me in a small way in 1950, and, like others, I survived and may have been the better for it.

.... What comes out of the book well is Shockley's importance. He arguably saved Britain from the U-boats during the battle of the Atlantic. He is certainly the true father of the age of silicon. He is even the inventor of the graphite-moderated nuclear reactor. Yet the public will remember his name as that of the nutty Nobellist who donated his sperm to a genius bank. Who would not want to hear his whole story?

This review is reprinted, along with several other fascinating ones, in More and Different.

Taming the bulge

2012-01-06T10:53:00.000+10:00

I am told one of the most common (and least kept) New Year's resolutions is to "tame the bulge", i.e. diet and exercise. That is not my subject. Rather, it is taming the academic bulge of every expanding papers, reports, filing cabinets, books, and book cases. Of course, those who do everything electronic do not have to worry about this. It is all on their laptop or iPad. I am not there yet. How do I cope? Not well, but there is a simple rule I have been following at both home and work the past few years that seems to be helping.

Do not buy any new filing cabinets or book cases.

Hence, whenever one buys a new book (or two or three) one has to throw out or give away one (or two or three).

Similarily, when one wants to force new files into a filing cabinet one is forced to throw away some old ones. It is amazing actually how easy this is.

Some cautions about mathematics in theoretical physics

2012-01-03T15:57:00.001+10:00

In a short essay, [reprinted in More and Different] Phil Anderson cautions about overplaying the role of mathematics in physical sciences. He states three cautions:

I. In my experience, interesting and relevant mathematics is more often stimulated by interesting experimental results or questions than vice versa.

II. Interesting mathematical ideas can misdirect you into scientific dead ends - they can become answers in search of a problem.

III. Complicated or lengthy mathematical procedures are very easy to use as a cover-up for shoddy or dishonest science.

He gives the following examples:

I. random matrices, general relativity, quantum Hall effect, localisation, Kondo effect

II. "particle democracy" = "self-consistent dispersion theory"
quantum critical points!

III. Attempts of an unnamed "honored professor" [presumably W. Goddard III's] to calculate superconducting high-Tc using computational quantum chemistry.

I agree with Anderson's concerns, but offer one minor disagreement about the title and context of his essay. It was meant to be a response to Eugene Wigner's famous essay "The unreasonable effectiveness of mathematics in the natural sciences". However, I think Wigner's paper was wrestling with profound philosophical questions not singing the praises on a highly mathematical approach to theoretical physics.

On the other hand, Phil said he embarked on the exercise "with a very negative attitude" and with his "usual determination to put the cat among the pigeons" due to his difficult relationship with his Princeton colleague Wigner. (This is described at several points in the book.)

Capping it all off

2011-12-31T10:02:00.000+10:00

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 depressing: do students really think that we are encouraging them to put enormous effort into learning things that are going to be of no future value?

The review also recommends several other texts for Capstone courses. I would be interesting in any recommendations for physics courses.

iPoverty

2011-12-29T18:42:00.000+10:00

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.

Teaching scientific research methods

2011-12-29T13:44:00.000+10:00

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.

Careers should be driven by scientific reality not metrics

2011-12-28T13:38:00.001+10:00

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 about simulations of dubious realism or relevance. Essentially, in the early part of the post-war period the career was science-driven, motivated mostly by absorption with the great enterprise of discovery, and by genuine curiosity as to how nature operates. But the last decade of the century far too many, especially of the young people, were seeing science as a competitive interpersonal game, in which the winner was not the one who was objectively right as the the nature of scientific reality but the one who was successful at getting grants, publishing in PRL, and being noticed in the news pages of Nature, Science, or Physics Today.

More and Different, page 100.

I consider this is a painfully accurate description of the current reality. It is amazing to see how few papers [one or two per year] the leaders published back in the 50s and 60s.

Is there anyway out of this undesirable situation? We can't turn back the clock. However, we can all [both junior and senior scientists] exercise some critical judgement and self control and not be completely conformed to the system and our colleagues.
Keep the science at the forefront of our minds and pre-occupations, don't jump on the latest band-wagon or become obsessed with the latest "metric" of productivity.

Photo is Anderson and Richards looking at apparatus for experiments on superfluid 4He at Bell labs (in the 1960s) taken from the AIP archives.

What is a metal?

2011-12-23T14:08:00.000+10:00

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 could calculate the Drude weight from the size and twist dependence of the ground state energy in the presence of twisted boundary conditions.

Imada considered a system at finite temperature and considered twisted boundary conditions in the imaginary time direction [anti-periodicity for fermions is required by the Kubo-Martin-Schwinger condition] showed that the charge compressibility determines the twist dependence.

I would be interested to see how the Drude weight and charge compressibility vary as one increases the temperature through the Fermi liquid - bad metal crossover.

Dialectic model building

2011-12-22T14:57:00.001+10:00

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 deal of weight if used judiciously, but it can never hold up against being taken completely literally.

P.W. Anderson, "BCS" and Me, More and Different, page 38.

I would argue that this approach has largely been abandoned in theoretical chemistry due to the dominance of computational chemistry.

A signature of a "two fluid" picture for a strongly correlated electron system

2011-12-21T15:44:00.000+10:00

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 point. 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, Characteristic Crossing Points in Specific Heat Curves of Correlated Systems which pointed out that in liquid 3He, some heavy fermion materials, and the Hubbard model (solved by dynamical mean-field theory (DMFT)) there was a temperature at which the specific heat curves for different pressures (or Hubbard U or magnetic field) all crossed. Below is shown the data for liquid 3He. The crossing temperature is approximately that at which Fermi liquid theory breaks down.

Below are the curves for the Hubbard model.

This is not an artefact of DMFT since later Chandra, Kollar, and Vollhardt showed it is also present for the exact solution of the one-dimensional model.

I find this quite amazing!

Vollhardt made no mention of isosbestic points which I think would have been well known to many chemists. But, in 2007 there is Isosbestic Points in the Spectral Function of Correlated Electrons by Martin Eckstein, Marcus Kollar and Dieter Vollhardt.
It contains a nice discussion of the essential physics, including the connection to the spectra of dye molecules.
The paper contains the figure below of the spectral density for a Hubbard model (calculated with DMFT). The horizontal scale is frequency (omega) and the dashed curve is the non-interacting density of states.

So what does this all mean?

The key idea is one of "two fluids" i.e., that as one varies the relevant parameter (pressure, magnetic field or Hubbard U) all one is doing is varying the relative concentration of the two fluids. A sum rule requires one fluid must be converted into another. In the case of the Hubbard model the total number of electrons (area under A(omega)) is conserved. As one increases U one decreases the spectral weight in the Fermi liquid component (centred at omega=0) and increases the weight of the Hubbard bands (centred at omega=+-U/2).

One view from the trenches

2011-12-20T15:22:00.001+10:00

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.

Observing the geometric phase in magnetoresistance II

2011-12-20T08:56:00.000+10:00

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 a semi-classical condition relating the area of the quantised cyclotron orbit to Landau level index N.

I welcome clarifications.

More and Different

2011-12-17T14:47:00.001+10:00

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.

Strongly correlated electron systems in high magnetic fields II

2011-12-16T16:03:00.000+10:00

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 eqn. 17 for m*(H)]. It also contains the above figure.

A more sophisticated treatment [connecting to recent NRG (Numerical Renormalisation Group) treatments of the single impurity Anderson model] of the case of YbRh2Si2 has recently been discussed by Zwicknagl. I am not clear on why she (and Hewson!) do not reference this earlier slave boson theory in their latest work.

The ultimate purpose of your teaching

2011-12-16T09:03:00.001+10:00

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 humanistically, or express artistically.
Teachers need to see themselves as, first of all, intellectuals, Students, in turn, need to recognize that their college education is above all a matter of opening themselves up to new dimensions of knowledge and understanding.
It is more a matter of students moving beyond their interests than of teachers fitting their subjects to interests that students already have. Good teaching does not make a course’s subject more interesting; it gives the students more interests — and so makes them more interesting.

I thank my lovely wife for bringing the article to my attention.

Is Papers 2.1 an improvement?

2011-12-15T16:15:00.000+10:00

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.

RG theory of turbulence

2011-12-15T16:08:00.000+10:00

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.

Observing the geometric phase in magnetoresistance I

2011-12-14T11:09:00.000+10:00

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 in strongly correlated electron materials

2011-12-13T13:18:00.000+10:00

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 energy derivative of the scattering time. For a Fermi liquid form of the scattering rate, the energy dependence is quadratic, and the signal will vanish. Essentially Behnia's replacement of the the energy derivative by the ratio of the scattering time and the Fermi energy means he is assuming that the scattering has a marginal Fermi liquid form. This is worth considering in more detail.

2. As the temperature increases there should be a crossover from a Fermi liquid with coherent quasi-particles with well-defined wavevectors to a "bad metal" with incoherent excitations. How this is manifested in the Nernst signal is an outstanding question.

3. A PRB by Kontani has given a general expression for the Nernst coefficient in a Fermi liquid including vertex corrections. A detailed analysis (within the framework of FLEX) claims that vertex corrections are important and due to antiferromagnetic fluctuations a large Nernst signal is possible in the pseudogap phase. [See Section 5.2 of this review].

What is the origin of magnetoresistance in silver chalcogenides?

2011-12-12T16:44:00.001+10:00

Silver chalcogenides (e.g., Ag2Te) with slightly altered stoichiometry exhibit an unusual magnetoresistance. It is large and linear in field for fields up to about 6 tesla and temperatures between 5 and 300 K. [See this 1997 Nature paper].
There are several possible physical origins of the magnetoresistance that 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 inhomogeneous semiconductor which can be described by a random resistor network.

4. I also note that a band structure which produces a constant Berry's phase curvature can produce a linear magnetoresistance, according to p. 1984 of this Rev. Mod. Phys.

The challenge is to come up with experimental signatures which can distinguish between the four different theoretical proposals.

From every angle

2011-12-10T18:18:00.000+10:00

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!

Seeking a unified theory for unconventional superconductors II

2011-12-10T16:01:00.000+10:00

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.

Deconstructing sodium cobaltate

2011-12-08T17:40:00.000+10:00

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 which antiferromagnetically orders, and very small Fermi surface which produces quantum oscillations.
All these properties cannot be described by the strong coupling ground state [an antiferromagnetic insulator with charge order] shown below.

What is the ground state of the ionic Hubbard model on the triangular lattice at 3/4 filling for small Delta/t where Delta=measure of ionicity between the two sublattices?
Is it a covalent insulator? Does such a state have experimental signatures which are distinct from a charge ordered insulator?

How can an "insulating" state co-exist with a very small Fermi surface?

A simple concrete proposal to improve the quality of Australian undergraduate education

2011-12-07T23:27:00.001+10:00

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.

Ph.D completion time as a statistical variable

2011-12-05T10:22:00.000+10:00

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.

Optimal doping corresponds to maximum entropy

2011-12-01T13:17:00.001+10:00

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 interpreted this sign change in terms of the Kelvin formula for the thermopower, which gives the thermopower as -1/e times the derivative of the entropy with respect to the particle number. This can be related to the temperature dependence of the chemical potential via the Maxwell relation,

Here s=specific entropy, c_h=hole density, mu_h = chemical potential.

It is rather surprising that a transport property can be expressed in terms of a thermodynamic property.

Thus the change in sign of the thermopower means that the entropy is a maximum as a function of hole doping.

Indeed, a maximum in the entropy near optimal doping is was found for the t-J model via

Finite temperature Lanczos calculations on small lattices of up to 20 sites and summarised in a 2000 review by Jaklic and Prelovsek.

The paper also considers a Kelvin type relation for the thermopower, but does not mention Kelvin, and shows how the maximum in the entropy vs. doping is associated with a change in sign of the thermopower.

[Peterson and Shastry do not mention this earlier work.]

A recent preprint by Garg, Shastry, Dave, and Philips, argue that the sign change reflects an underlying quantum critical point at optimal doping. However, I wonder about the extent that one can see a quantum critical effect on lattices as small as 20 sites.

This is related to a 2009 PRB by Mark Jarrell and collaborators who calculated the temperature and doping dependence of the entropy using the cluster dynamical approximation. They found entropy was a maximum at optimal doping [~0.15-0.2]. They also don't mention the earlier work by Jaklic and Prelovsek.

A very strange metal

2011-11-30T14:40:00.000+10:00

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 also reports a rather strange angular dependent magnetoresistance.

How much money does a "World class" university cost/need/want?

2011-11-28T17:05:00.000+10:00

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.

How does the Mott insulating phase differ from the metallic phase near it?

2011-11-25T16:54:00.003+10:00

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 metal-insulator transition is relatively small, being about a 15 per cent change for T=0.1t, and less at higher temperatures. To me this suggests that in some sense the character of the metallic and insulating phases near the transition are not that different, just like a liquid and gas are hard to distinguish near the critical point.

3. These results are in contrast to Brinkmann-Rice theory [which ignores J] which gives D=0 in the Mott phase.

Covalent character of hydrogen bonds III

2011-11-24T17:11:00.001+10:00

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 ~ 300 Hz, which is comparable to the actual value. Basically, when one flips one proton spin the A flips the electron spin, converting the ground state spin singlet into the excited triplet state.

The figure is taken by a nice webpage by Hans Reich

Deconstructing vertex corrections

2011-11-22T16:44:00.000+10:00

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 interaction.

Neglecting vertex corrections is equivalent to making the relaxation time approximation (RTA) when solving the Boltzmann equation. Then the quasi-particle lifetime equals the transport lifetime because one ignores dependence of the scattering rate on momentum transfer. Below is some helpful text from a review by Kontani:

....we have to take the Current Vertex Correction [CVC] into account correctly, which is totally dropped in the Relaxation Time Approximation [RTA]. In interacting electron systems, an excited electron induces other particle– hole excitations by collisions. The CVC represents the induced current due to these particle–hole excitations. The CVC is closely related to the momentum conservation law, which is mathematically described using the Ward identity [28–31]. In fact, Landau proved the existence of the CVC, which is called backflow in the phenomenological Fermi liquid theory, as a natural consequence of the conservation law [28]. The CVC can be significant in strongly correlated Fermi liquids owing to strong electron–electron scattering.

For specific types of interactions Ward identities allow one to relate the vertex function to derivatives of the self energy. Mahan's book (Section 8.1.3) discusses this in detail.

In the limit of infinite dimensions [in which Dynamical Mean-Field Theory (DMFT)] becomes exact, vertex corrections can be neglected.

In a recent PRB, Bergeron, Hankevych, Kyung, and Tremblay calculated the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. They found that at "high" temperatures (T > 0.2t) vertex corrections did not matter much, but were significant at lower temperatures near a quantum critical point.

Its all in the title?

2011-11-21T14:38:00.001+10:00

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.

Should you follow a textbook?

2011-11-19T09:16:00.000+10:00

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.

Postdoc in theoretical chemical physics available at UQ

2011-11-18T17:21:00.000+10:00

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.

Competing phases are generic to strongly correlated electron systems

2011-11-17T16:31:00.000+10:00

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 "dimerisation".

The authors argue these phases will also be stable for the actual isotropic square lattice.

It is fascinating to me that one can produce such rich and diverse broken symmetry states starting from such a relatively simple model which has no competing interactions in the Hamiltonian.

An earlier PRB by Yao, Tsai, and Kivelson, Myriad Phases of the checkerboard Hubbard model suggests competing phases are a generic feature of strongly interacting systems.

Writing your first grant proposal

2011-11-15T19:48:00.001+10:00

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?

Should cats be herded?

2011-11-14T17:15:00.002+10:00

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 examples in science that have been fruitful or that I personally find inspiring. Maybe I am jaded but ones such as "nuclear fusion in our lifetime", "build a quantum computer", "discover a room temperature superconductor", "cheap organic solar cells to save the planet", or "lets make our university number one" just don't seem that achievable via highly managed research teams.

Furthermore, it seems that most Nobel Prize discoveries did not result from such programs, but rather from curiousity driven research by "independent" research groups. One obvious exception are Nobel Prizes for discoveries in elementary particle physics.

What do you think?

Two persistent misconceptions about hydrogen bonding?

2011-11-11T13:50:00.001+10:00

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 mechanics. It is essentially based on a static view derived from symmetrical one-dimensional double-well potentials V(δ), such as the one shown on the left in Figure 3 [see below], thus completely omitting the possibility of a barrierless scenario, as sketched in the central panel of this figure.

For a discussion of the origin [in terms of covalent interactions!] of these three different potentials, see my preprint.

Deconstructing the chemical potential of the cuprate superconductors

2011-11-09T13:50:00.002+10:00

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 [approximately] to the thermopower via a relationship [equation 8.6], which is essentially a restatement of the Kelvin formula [discussed by Peterson and Shastry].
The latter means the thermopower should change sign around optimal doping, as is indeed observed [more on that later].
The large entropy near optimal doping emerges from the interplay of the localised spins [from the remnants of the Mott insulator] and frustration of the antiferromagnetic spin interactions via doping.

I would be interested to see a similar calculation for the Hubbard model on the anisotropic triangular lattice at half filling to see how the chemical potential varies as a function of U/t as the Mott insulator is approached from within the metallic phase.

When the data is "too good to be true"

2011-11-09T09:53:00.000+10:00

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..... Stapel and a colleague or student came up with a hypothesis, and then designed an experiment to test it. Stapel took responsibility for collecting data through what he said was a network of contacts at other institutions, and several weeks later produced a fictitious data file for his colleague to write up into a paper. On other occasions, Stapel received co-authorship after producing data he claimed to have collected previously that exactly matched the needs of a colleague working on a particular study.....
The data were also suspicious, the report says: effects were large; missing data and outliers were rare; and hypotheses were rarely refuted. Journals publishing Stapel's papers did not question the omission of details about where the data came from.

This is part of a Nature News piece which has the misleading title "Report finds massive fraud at Dutch universities". A more responsible and accurate title would be "Report finds massive fraud by one Dutch professor of social psychology". In the comments section several Dutch researchers rightly object to the title.

Chemistry driven by conical intersections

2011-11-08T16:18:00.000+10:00

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.

Should university be fun, fun, fun!

2011-11-07T15:14:00.001+10:00

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 thank my wife for bringing the article to my attention.

A sign of something important

2011-11-04T07:30:00.013+10:00

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 reference, optimal doping is around x ~ 0.2, and for x larger than 0.3 there is no superconductivity. Note the sign change with increasing x.

The authors emphasize how this is a tricky measurement because one has to be careful that the current paths that are measured [to get both sigma_xx and sigma_xy needed for the Hall coefficient] really do lie in the plane of the layers and do not contain spurious contributions (see this earlier post about the challenge of electronic transport measurements in highly anisotropic materials).

The sign change may be an important signature of strong electronic correlations. I find it interesting (and surprising) that the observed sign change at x=0.3 is obtained in a high temperature series expansion of the high frequency Hall coefficient for the t-J model [in this 1994 PRL by Shastry, Shraiman, and Singh (SSS!)]. [An earlier post discusses Shastry's approach]. [Note: this calculation does not have a t' hopping term, which may be relevant. For example, it has a significant effect on the shape and curvature of the Fermi surface and the proximity to van-Hove singularities. See below].

An alternative explanation of the sign change in terms of Mott physics was given by Stanescu and Phillips.

There may be a more mundane explanation in terms of changes in the Fermi surface associated with the proximity of the van Hove singularity in LSCO. Indeed ARPES experiments do find an electron-like Fermi surface for x~0.3. Furthermore, experiments on Tl2201 [which does not have a close van Hove singularity] do not see any hint of a decreasing Hall coefficient [or sign change] as one increases the doping on the overdoped side towards samples with Tc=0. [Higher dopings seem problematic for Tl2201].
Furthermore, one can quantitatively describe the temperature dependence of data for x=0.3 [including the sign change with temperature] if one uses a realistic Fermi surface and assumes that the impurity scattering rate is anisotropic over the Fermi surface. See this PRB; I thank Nigel Hussey for bringing it to my attention.

I thank Jure Kokalj for some helpful discussions.

Converging to the "right" answer

2011-11-03T07:47:00.000+10:00

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 years the value of the speed of light and the charge on the electron have drifted, but at any particular time peoples values have always been within a standard deviation of the latest measurements.

Just remember Feynman's warning: "The easiest person to fool is yourself."

The challenge of a simple measurement

2011-11-01T10:49:00.000+10:00

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 sometimes referred to as quasi-two-dimensional metals.
Accurately, measuring the resistivity (and its temperature dependence) in the three different directions though is a highly non-trivial exercise. Basically, this is because you have to be sure that the current is going through the sample in the direction you think it is.

This is highlighted in a recent Nature Communications article from Nigel Hussey's group. They state:

In a quasi-1D conductor, it is especially problematic to measure the smallest of the resistivity tensor components, because even a small admixture of either of the two larger orthogonal components can give rise to erroneous values and distort the intrinsic temperature dependence of the in-chain resistivity. In Li_0.9Mo₆O₁₇, reported room-temperature values for the in-chain (b axis) resistivity range from 400 μΩ cm²³ to more than 10 mΩ cm^34, 35.

Reported values for the ratio of the a to b axis resistivity vary from about 2 to 100!
This is a very large discrepancy!

I wrote this post because I thought I had come up with a fancy theoretical explanation of why in one paper the resistivity anisotropy ratio was only ~4, whereas band structure predicts a much larger value. However, when I surveyed the literature I discovered the result I was so proud of explaining is probably an artefact!

From quantum chemistry to many-body theory of frustrated antiferromagnets

2011-10-31T11:47:00.001+10:00

There are a wide range of organic charge transfer salts that can be described by a Hubbard model at half filling on the anisotropic triangular lattice. [See this review for a full discussion]. This model can also be viewed as a square lattice with hopping t' along one diagonal.
An important parameter is the ratio t'/t. The values t'=0, t'=t, and t=0, correspond to the square lattice, isotropic triangular lattice, and decoupled chains, respectively.

The actual value of t'/t is critical because the ground state of the Heisenberg model for the Mott insulating state [Neel, spiral order, valence bond crystal, spin liquid] is quite sensitive to the value of J'/J=(t'/t)^2. For example, as J'/J increases from 0.6 to 0.9 the order can change from Neel to valence bond crystal or spin liquid to spiral order.

So, what value do specific materials have? Do they vary with the counter anion?

Previously the parameters t and t' have been estimated from Huckel theory. An earlier post discusses recent progress at calculating these parameters for BEDT-TTF materials from computational methods based on Density Functional Theory (DFT).

My UQ colleagues Edan Scriven and Ben Powell recently reported new results for dmit materials in this preprint. J'/J varies from 0.4 to 1.4 with the counterion. Furthermore, if the calculated values are combined with the results of many-body theories of the corresponding Heisenberg model one obtains a consistent picture between theory and experiment.

The results should be compared with the figure below (taken from a recent review article by Kanoda and Kato) attempts to present a unified picture of the relationship between the ground state and the value of t'/t for a range of materials. However, the values used are based on Huckel calculations and the DFT calculations give significantly different values.

Outstanding challenges include

finding a simple physical explanation for the origin of the parameter variation with counterion.
calculating the pressure dependence of t'/t
calculating Hubbard U values consistent with experiment [it seems screening is important]
the previous two need to be combined to describe the pressure-temperature phase diagram, including the transition from a Mott insulator to a superconducting state

Universal magnetic excitations in the cuprates

2011-10-27T16:54:00.001+10:00

I heard a nice talk by Bernhard Keimer which included a discussion of recent RIXS [Resonant Inelastic X-ray Scattering] results on the cuprates. The relevant Nature Physics article states

As a major result, we demonstrate the existence of spin-wave-like dispersive magnetic excitations (paramagnons) deep inside the electron–hole spin-flip continuum (up to ~300 meV), for all the investigated doping levels, with spectral weights comparable to those of magnons in the undoped parent compounds.

The work is nicely put in context in a News and Views article by Matthias Vojta.

Extracting Fermi surface information from magnetoresistance

2011-10-26T07:05:00.000+10:00

Today I am giving a talk Interlayer magnetoresistance as a probe of Fermi surface anisotropies at the GG conference on Strong Correlations and Spins.

Interlayer charge transport in the pseudogap state

2011-10-25T21:21:00.000+10:00

Previously I have posted about the unusual anisotropies present in the temperature dependence of the resistivity in the pseudogap phase of the cuprates. In particular, the intralayer resistivity exhibits a metallic temperature dependence whereas the the interlayer resistivity exhibits a semi-conductor like temperature dependence. Much had been made of this previously with exotic explanations in terms of spin-charge separation. However, I posted about recent work showing how the data has a natural explanation in terms of the presence of a pseudogap with nodes in the same direction as that at which the interlayer hopping vanishes.
[This interlayer hopping anisotropy is quite important in understanding Angle-Dependent-MagnetoResistance (ADMR)].

I recently became aware of earlier phenomenological work by Tao Xiang and collaborators which also gives this explanation. In particular, this PRB gives a nice "universal" phenomenological form for the temperature dependence of the interlayer conductivity. This form is compared to data from a wide range of cuprates.

Seeking a unified theory for unconventional superconductors

2011-10-25T07:03:00.001+10:00

I am currently in Sydney at the Gordon Godfrey workshop on Spins and Strong Correlations.
Yesterday Rajiv Singh began his talk on orbital effects in the iron pnictide superconductors with some general remarks about the relationship between magnetism and superconductivity. The two were once considered inimical (one of Bernd Matthias' rules). But we now see classes of superconductors (heavy fermions, organics, cuprates, and pnictides) where they appear to be intimately connected.

Rajiv then took a philosophical position: there should be a common physics for all of these non-electron-phonon superconductors.

This certainly reflects a physicists desire for universality and simplicity. This is in distinct contrast to a chemists focus on particularity.

I am not convinced that it is or should be the case that there is some common underlying physics. Here are a few disordered thoughts.

For pnictides it seems that orbital (multiple band) effects matter. In contrast, in cuprates and organic it seems a single band is sufficient.
If there is a unified theory I think the strongest candidate is a weak-coupling spin fluctuation RPA picture (with renormalised interactions and Fermi liquid quasi-particles). To hold to this one will have to show that "exotica" (e.g., non-Fermi liquid effects) are just some higher order perturbative effects.
Similar sentiments of a unified picture of cuprates and pnictides is presented by Basov and Chubukov.
To me the two biggest problems for a "common physics" scenario are the pseudogap state in the cuprates and the spin liquid states in organics. They represent a "discontinuity" from the other materials and from any weak-coupling picture.

I welcome comments. Is there a common physics for non-electron-phonon coupled superconductors?

Should you use Turnitin?

2011-10-22T15:54:00.000+10:00

Turnitin is commercial software that detects student plagiarism by comparing submitted assignments to everything on the web and a vast database of other assignments.

A few years ago I would have thought this was might be relevant to people teaching large courses of undergraduates in the humanities. However, I was wrong. Unfortunately, experience has shown that plagiarism does occur, even in physics courses, and at the graduate level. There are cases of students submitting Ph.D proposals and literature reviews that involve cutting and pasting text from papers and the internet. Although certainly not confined to them this can be more of a problem with students from non-Western countries. There do seem to be some "cultural" differences as to what is acceptable practice and what is not. This does not excuse it, but does mean that sometimes first-time offenders need to be gently cautioned and educated.

A range of offences can occur, ranging from sloppy referencing to blatantly copying large swathes of text and presenting them as ones own.

So if you don't use Turnitin (or something equivalent) try it. You won't know if there really is a problem until you check.

If you do detect plagiarism make sure you report it to the relevant academic authority. Do not just give the student a private warning. It is important that someone is keeping track. Otherwise repeat offenders may not really understand the severity of their offence and get appropriately disciplined.

Another issue, which is harder to detect, is that of ghost writing.

Superconductivity video goes viral

2011-10-21T21:22:00.001+10:00

Today Ben Powell gave a great colloquium at UQ on 100 years of superconductivity. During it he showed this great video. A shorter version went viral on the internet receiving more than 3 million views within less than a week of being posted!
It is great to see superconductivity is so popular.

An energy gap is not necessary for superconductivity

2011-10-20T16:49:00.001+10:00

A common misconception about superconductivity is that the presence of an energy gap at the Fermi energy is fundamental to the phenomenon. This is not correct. The key property is long range phase coherence.
I give several counter-examples to the necessity of an energy gap.

If in BCS theory you take an s-wave superconductor and add magnetic impurities there is a critical range of impurity concentration for which there is no energy gap (i.e. the density of states is non-zero at the Fermi energy) but superconductivity (i.e. a non-zero Cooper pairing amplitude and superfluid stiffness exists).

Superfluid 3He (p-wave triplet pairing) and high-Tc cuprates (d-wave singlet pairing) have nodes in the energy gap on the Fermi surface.

A gap is also not sufficient. A charge density wave state can have a gap at Fermi energy but is not be a superconductor.

A victory for curve fitting

2011-10-19T20:11:00.001+10:00

I have previously written many posts (rants?) about the dangers of curve fitting. In particular in one of the first posts The naked truth versus self deception I stated

I believe that any significant physical effect/discovery should be able to be seen by the naked eye in the experimental (or computational) data and should not require curve fitting.

However, the graphs below represent a significant counter example to my view.

The data is considered to be consistent with the upper solid curve.
This is the data which ultimately led to the 2011 Nobel Prize in Physics.
When I see such graphs I am immediately skeptical. But the authors and the broader community were not and subsequent measurements supported the original claims and parameters deduced from this data. In particular the graph below [taken from a Physics Today article Supernovae, Dark Energy, and the Accelarating Universe by Saul Perlmutter] shows how several independent methods constrain the parameters.

I am curious to hear what others think.

Deconstructing Transport properties of strongly correlated electron metals

2011-10-18T19:22:00.002+10:00

I have been working through a really nice article Electrothermal transport coefficients at finite frequencies by Sriram Shastry. The key idea is that there are certain transport coefficients [the Hall coefficient (Hall resistivity), Lorenz ratio, and Thermopower] for which have a weak frequency dependence and so one can obtain a reliable estimate of the dc value from the high frequency value. This greatly simplifies the computation because the latter is determined by the expectation value of a specific operator in the ground state (or thermal ensemble). Unlike the dc transport coefficient this expectation value is not particularly sensitive to finite size effects and so can be evaluated from Lanczos (exact diagonalization) on a small lattice. Alternatively it can be evaluated from a high temperature series expansion.

Is this high frequency approximation justified? It can motivated in a heuristic manner from the fact that in the Drude model the relevant transport coefficients [the Hall coefficient, Lorenz ratio, and Thermopower] are all independent of the relaxation time.
For the t-J model on the triangular lattice Shastry also compares explicit evaluations of the Hall coefficient at zero, non-zero, and infinite frequency and finds there is little variation between them.

Here are a few highlights of the paper.

1. Strong correlations cause qualitative differences. Consider the Hubbard model as a function of doping. There are three changes in the sign of the Hall and Seebeck coefficients, in contrast to the one change in sign (at half filling) that occurs for the uncorrelated (U=0) band. In particular one can have a "hole-like" band structure and Fermi surface but an "electron-like" Hall coefficient.

[I think the solid line in the above graph is the Heikes formula which holds in the infinite temperature limit and is related to the entropy of the charge carriers in the Hubbard model in the atomic limit U >> |t|].

2. On the triangular lattice changing the sign of the hopping t can lead to significant changes in the magnitude and temperature dependence of the thermopower. [Although I wonder if some of this difference is related to the relative proximity to van Hove singularities and the associated differences in the non-interacting density of states near the Fermi energy as discussed here].

3. On the triangular lattice at high temperatures there are contributions to the thermopower and Hall resistance which are first order in t/T. In contrast on the square lattice the leading terms are of order (t/T)^2. This arises because on the triangular lattice one can perform closed loop hops involving only 3 lattice sites.

4. A connection is made [with some interesting history] to the expression of Thomson [Lord Kelvin] for the thermopower in terms of entropy. This is relevant to this post.

I thank Subroto Mukerjee for helping me gain a better understanding of Shastry's work.

A "big picture" book worth re-reading

2011-10-17T11:30:00.001+10:00

Almost 2 years ago I posted about how much I liked the book Five Minds for the Future by Howard Gardner. That post enunciated how his "5 Minds" relate to development of a research career. I also later mentioned how the book is helpful for thinking through what the goal of a physics undergraduate education might be. I am now reading the book with my son to help him think through how do get the most of his undergraduate education which he starts next year. It is one of those books that I find that each reading is like the first one because each reading produces new insights and understanding.

If when teaching you struggle with filling out a "Course profile" which requests information about "Learning goals" and "Graduate attributes" you may also find the book helpful. These can be an exercise in "mumbo jumbo". I used to think they were a complete waste of time. But, if done thoughtfully and in a concrete manner I believe that they can be helpful to both students and teacher. The book helped me see the value of these exercises and how to do them in a more effective manner.

Essential ingredients in the high-Tc puzzle

2011-10-14T15:25:00.001+10:00

On monday I had a nice meeting in Bangalore with T.V. Ramakrishnan who told me about recent work on a Phenomenological Ginzburg-Landau-like theory for superconductivity in the cuprates done together with Sumilan Banerjee and Chandan Dasgupta.

They assume that there is singlet pairing along each bond on the square lattice at some temperature which linearly decreases with increasing doping x. A key ingredient is the coupling between neighbouring horizontal and vertical bonds. This coupling C is assumed to have a negative sign to produce d-wave pairing. Furthermore, to produce a superfluid density which for small x scales x it is assumed that C is also proportional to x. [They also give a simple argument suggesting that this C scales with t' the diagonal hopping in an underlying band structure].

From this simple model they can extract some of the key phenomenology of the cuprates, including the doping dependence of the transition temperature. Coupling the order parameter fluctuations to electrons produces a self energy and spectral density consistent with ARPES experiments, including the existence of Fermi arcs.

A few things I found interesting

Quantum fluctuations are argued to be important in the underdoped andoverdoped regime. Indeed Tc goes to zero for small non-zero dopings.
Past the optimum Tc the superfluid density decreases with decreasing doping.
I would be interested to see these relatively simple ideas extended to the half-filled case corresponding to organic charge transfer salts. Then, x will have to replaced with something like U-Uc where Uc is the critical value of the Hubbard U for the Mott transition.

A concrete test of the Morse potential in a complex molecule

2011-10-13T19:27:00.000+10:00

Just how accurate is the Morse potential? A key feature is the equidistance of adjacent energy levels.

This graph below illustrates the high quality of the Morse potential for describing a C=O bond within a protein. The data (taken from this PNAS paper) is via a technique which induces transitions from the v'th vibrational energy level to the (v+1) level.

Thus we see the Morse potential describes the true anharmonic potential at least up to v=7, which corresponds to energies of about 1.5 eV above that of the potential minimum.

This success also underscores the local character of these bond stretching vibrations.

Different routes to the Mott insulator

2011-10-10T22:31:00.000+10:00

Today I am giving a seminar Destruction of quasi-particles near the Mott transition (slides) at the Physics Department of the Indian Institute of Science in Bangalore.
The talk is largely based on a PRL which shows how Dynamical Mean-Field Theory (DMFT) can give a quantitative description of the optical conductivity of a family of organic charge transfer salts close to the Mott transition.
For a much broader context see this review. The talk briefly stresses that the nature of the quasi-particles in the metallic phase near the Mott transition for the band-width controlled and the filling-controlled Mott transitions are distinctly different.

Remembering Coulson

2011-10-08T19:26:00.000+10:00

Charles Coulson is definitely one of my heroes and I found this nice memorial talk by Roy McWeeny on a history of quantum chemistry site. It emphasizes how Coulson made the transition from applied mathematics to theoretical chemistry:

The broadening of his philosophy is summed up in a sentence .... in a lecture on mathematical models, .... "It is likely that one cannot be a good applied mathematician unless one has the ability to simplify to the point of absurdity!" — recalling at the same time a remark by Eddington that "If the model is right, the rest is easy".

Two trivia about Coulson I also learned:
He was chairman of Oxfam from 1965-1971.
He was Ph.D supervisor of Peter Higgs (of Higgs boson fame).

Seminar on hydrogen bonding

2011-10-07T14:04:00.000+10:00

Today in Bangalore I am giving the theory seminar at the Jawaharlal Nehru Centre for Advanced Scientific Research. I will be talking about my recent preprint on hydrogen bonding. Here are the slides.

2011 Nobel Prize in Chemistry

2011-10-06T22:46:00.003+10:00

I was pleased to see that the Chemistry prize was awarded to Dan Shechtman for the discovery of quasi-periodic materials that led to a new definition of crystal (quasi-crystals). Back in 2003 APS News published a nice discussion of the background to the paper which appeared in PRL in 1984 and is the 8th most cited PRL of all time.

An earlier post discusses why I teach undergraduates about quasi-crystals. It illustrates the fundamental logical principle that A implies B does not mean that B implies A. Specifically, just because a periodic arrangement of atoms is a sufficient condition for a discrete X-ray pattern does not mean that it is a necessary condition. It also shows students that what they read in textbooks may be wrong.

Shechtman is not the first condensed matter physicist to be awarded the Nobel Prize in chemistry. Other recent examples include Walter Kohn and Alan Heeger.

Directionality is a key property of hydrogen bonds

2011-10-05T22:53:00.000+10:00

Today I re-read a nice paper The hydrogen bond: a molecular beam microwave spectroscopist’s view with a universal appeal, by Mausumi Goswami and E. Arunan

I found the following sentences very helpful

one aspect about hydrogen bonding that is widely accepted is the directionality, i.e. X–HY is found to be linear in most cases. Although secondary interactions in a system could force X–HY away from linearity, it is the directionality in hydrogen bonding resulting in an anisotropic intermolecular potential that separates it from the more general ‘van der Waals forces’, which are expected to be isotropic.

The main point of the paper is

For a ‘hydrogen bonded complex’, the zero point energy along any large amplitude vibrational coordinate that destroys the orientational preference for the hydrogen bond should be significantly below the barrier along that coordinate so that there is at least one bound level. These are vibrational modes that do not lead to the breakdown of the complex as a whole. If the zero point level is higher than the barrier, the ‘hydrogen bond’ would not be able to stabilize the orientation which favors it and it is no longer sensible to characterize a complex as hydrogen bonded.

Next week I will be visiting the Indian Institute for Science in Bangalore and look forward to meeting the authors then.

This directionality of is incorporated in my effective Hamiltonian for hydrogen bonding via the directional dependence of the matrix element which couples the two diabatic states. It is also responsible for the hardening of vibrational modes associated with rotation of the D-H unit relative to the acceptor atom A (discussed in my last H-bond post).

2011 Nobel Prize predictions

2011-10-04T09:50:00.002+10:00

It is that season again. In 2009 I posted about this but forgot last year. There is an article in Science (which was wrong about the Medicine prize)
Here are a few sites with predictions.

David Pendlebury at Thomson Reuters
Although I disagree with the methodology, based on citations, I think their choice of Aspect-Clauser-Zeilinger for quantum entanglement is an likely and good one.

Paul Braker at Chembark lists odds for different people/topics for the chemistry prize. For physicists this is worth reading because it is a brief education in significant discoveries in chemistry.

Everyday scientist predicts Moerner for single molecule spectroscopy, whose name comes up for chemistry in some other lists.

The curious wave function has predictions for all fields.

Some of these lists suggest a prize for biomolecular dynamics simulations. But, what is the really significant new discovery that goes with this technique?

Aharonov and Berry's name comes up for quantum phase effects. This could be combined naturally with Duncan Haldane or David Thouless for the role of topological phases in many-body systems.

I welcome your suggestions and comments.

More hydrogen bond correlations

2011-10-03T23:19:00.000+10:00

Previous posts have considered correlations between different physical properties of hydrogen bonded complexes (D-H...A). A key correlation is that as the donor-acceptor (D-A) separation R decreases the D-H stretch vibrational frequency (corresponding to varying r in the figure below) decreases.

There is a second vibrational mode associated with angle phi in the figure above. This can also be viewed as a torsional or bending mode. Hydrogen bonding leads to a stiffening of this mode.

The figure below shows how the frequency of this mode (horizontal axis) is correlated with the D-H stretch frequency (vertical axis).

The figure is taken from a classic 1974 review by Novak.

Low-tech solutions change the world

2011-10-03T01:21:00.000+10:00

The New York Times has an excellent section Small Fixes: Low cost innovations that can save thousands of lives. One example are Liquid Lens, self-adjustable eyeglasses, developed by Joshua Silver, a physicist at Oxford.

Quantifying breakdown of Born-Oppenheimer

2011-10-01T14:19:00.001+10:00

Yesterday I read a nice paper An unusually large nonadiabatic error in the BNB molecule by John Stanton.
Here are a few choice quotes:

A simple analysis shows that significant nonadiabatic corrections to energy levels should occur only when the affected vibrational frequency is large enough to be of comparable magnitude to the energy gap involved in the coupling.

The results provide evidence that nonadiabatic corrections should be given as much weight as issues such as high-level electron correlation, relativistic corrections, etc., in quantum chemical calculations of energy levels for radicals with close-lying and strongly coupled electronic states even in cases where conical intersections are not obviously involved.

One would be tempted to prepare a manuscript based on this result, but such an action would be premature.

Well. [Yes. This is a sentence in the paper!]

One thing of particular interest to me is that the adiabatic potential energy curves of both the ground and lowest excited state can be described very accurately by the eigenvalues of a two-state effective Hamiltonian [ascribed here to Koppel, Domcke, and Cederbaum]. This is identical [modulo a 45 degree rotation] to the Hamiltonian considered in a recent paper by Laura McKemmish, Jeff Reimers, Noel Hush, and myself.

The paper makes no mention of the "twin state" concept or how the vibrational frequency is exalted in the excited state relative to the ground state.

This article was one of the 20 most downloaded articles from Journal of Chemical Physics this month. I found it encouraging that people aren't just reading papers about the latest DFT functional recipe book!

Why you might worry about classical models of water

2011-09-30T23:06:00.001+10:00

A nagging question for the whole field of (classical) molecular dynamics simulations of biomolecules is whether they can have an adequate description of water. This is particularly important because almost all biomolecular processes involve subtle interactions of the biomolecule of interest with its aqueous environment.

I learnt a lot from reading the article On the origin of the redshift of the OH stretch in Ice Ih: evidence from the momentum distribution of the protons and the infrared spectral density, by C. J. Burnham, G. F. Reiter, J. Mayers, T. Abdul-Redah, H. Reichert and H. Dosch.

They highlight several problems and state:

Clearly there is something missing from water models. All of the above difficulties have been consistently tackled by experimentalists, but they remain either unrecognized or unacknowledged by much of the simulation community.

Here are the 4 main difficulties they list concerning the differences between the properties of a water monomer and the water molecule in ice Ih [most common phase of ice]:

1. the magnitude of the measured anharmonicity parameter of the OH stretch X_OH (=difference between 1/2 of the second overtone frequency and the fundamental) of the OH stretch in the condensed phase is increased from 87 cm-1 in the gas-phase to 134 cm-1 in ice Ih. This behavior cannot be reproduced by a simple anharmonic oscillator (such as a Morse oscillator), for which elongation of the OH stretch by the ice H-bond results in a decrease in |XOH|.

2. the enormous observed increase (25 fold) of the integrated IR intensity in the OH stretch mode in ice compared to that of the gas-phase monomer. Most water models (even including polarizable ones) predict almost no increase at all.

3. The gas-phase molecular dipole moment derivative with respect to the OH stretch is observed to be in a direction some 25 degrees outside of the OH stretch vector. In contrast, it is observed that this derivative becomes nearly parallel to the OH vector in ice.

4. The HOH angle increases from the gas-phase value of 104.5 to 107 degrees in ice. This is in contrast to virtually all empirical water models, which predict a lowering of the HOH angle from the gas-phase value.

The authors propose their own (classical) solution to these problems.
I am not in a position to judge the validity or reasonableness of the solution.
However, I have a prejudice that ultimately the origins of these problems is that the H-bond has a significant covalent and quantum character.

Scepticism and caution is usually the best response

2011-09-28T19:32:00.000+10:00

All the media attention to recent anomalies in the speeds of neutrinos raises some interesting scientific and sociological questions. Should you go to the media before you have results published in a peer reviewed journal?

A few things to bear in mind.

These deviations from the speed of light are one part in 100 thousand. They require measuring the distance travelled to the same precision, i.e. an accuracy of 20 cm!

Relative measurements rather than absolute measurements are usually more reliable. As the Nature News (n.b. not paper) article said

Most troubling for OPERA is a separate analysis of a pulse of neutrinos from a nearby supernova known as 1987a. If the speeds seen by OPERA were achievable by all neutrinos, then the pulse from the supernova would have shown up years earlier than the exploding star's flash of light; instead, they arrived within hours of each other.

An earlier anomaly in neutrino physics was of the 17 keV neutrino that was "discovered" in 1985. It is worth reading the historical reviews in Reviews of Modern Physics and Nature which discuss what went wrong. These articles should sober theorists who jump on a bandwagon once some apparent experimental anomaly is observed.

On the other hand, the solar neutrino problem is a case where experimental anomalies did lead to interesting new physics. But note, there the anomalies were by a factor of three in the observed neutrino flux.

A quantum chemist tries to solve cuprate superconductivity

2011-09-27T08:16:00.000+10:00

Journal of Physical Chemistry Letters has a paper Origin of the Pseudogap in High-Temperature Cuprate Superconductors by Jamil Tahir-Kheli and William A. Goddard, III

I can't say I really follow the details here. The doping dependence of the pseudogap they calculate seems to be a percolation effect. A key issue is whether they can produce the anisotropy in the pseudogap in momentum space. It is not clear they do since everything seems to be done in real space.

There is some interesting history here, going back to the early days of high-Tc.
Sparks fly over conflicting theories from Chemical and Engineering News in 1988, discusses conflict between Phil Anderson and Goddard.
A clash erupts between scientific subculture is a New York Times article from 1989 which quotes a Dr. Anderson who is presumably Phil Anderson.

I thank Seth Olsen for bringing some of these articles to my attention.

Quantum computational matter

2011-09-23T19:37:00.000+10:00

Stephen Bartlett gave a colloquium on this topic today.
The ground states of quantum antiferromagnets contain substantial amounts of entanglement (although how much is hard to quantify). Hence, one might hope they are a resource that might be used in quantum computation. Stephen described some recent work [see this PRL and this PRL ] which considers how this might be done. The focus seems to be on gapped systems which have hidden symmetries and can be described as tensor network states. The prime example is the Haldane spin-1 antiferromagnetic Heisenberg chain which is adiabatically connected to the AKLT model.

I was reminded of some work I was involved in a few years ago (described in this PRL) which showed how one could take any spin singlet state [not just the ground state!] from a spin-1/2 chain and perform projective Bell measurements (on pairs of spins) along the chain and teleport quantum states with perfect fidelity along the chain. I was wondering what the relationship (similarities and differences) of our work was with this more recent work.

Where is theoretical chemistry going?

2011-09-23T09:39:00.000+10:00

There is an Editorial Theoretical Chemistry - Quo Vadis? by Walter Thiel in Angewandte Chemie International Edition.
["Quo Vadis" is latin for "where are you going?"]

It is worth reading. It is particularly good that he offers some concrete precautions for experimentalists running computational codes.
I thought the following characterisation of theoretical chemistry was disappointingly narrow:

Theoreticians are primarily interested in testing the performance and limitations of newly developed computational methods, for example by systematic validation on established benchmarks or through proof-of-principle calculations.

Overall I prefer the articles I highlighted in 5 papers every computational chemistry student should read, together with Hoffmann's 1974 article on theory in chemistry and Zewail's article on the future of chemical physics.
Those articles place a much greater emphasis on theory providing unifying concepts.

I thank Seth Olsen for bringing the article to my attention.

Writing effective personal statements

2011-09-22T09:26:00.000+10:00

Most applications for scholarships or admission to graduate school require a personal statement. I found a very useful site at Penn State, Writing Personal Statements online.

I read the two sample statements from applicants for a Marshall Scholarship. Both were very impressive. They illustrate several key ingredients

Distinctly personal. It is about you. Only you could write this. Avoid generics "I am really interested in subject X and want to study at University Y because it is a world class university."
Personally engaging. A natural outcome is that the reader should want to meet you.
Specific connection between applicant and the target institution/program. Mentioning specific courses, faculty, and research projects is key.
Polished and well written. This means writing and rewriting and getting feedback on drafts.

For those of us reading and reviewing applications I see two important consequences of this material being freely available.

First, the bar is higher. Any student with a little "get up and go" can Google "personal statements scholarship applications" and find material such as this. They should then aim to produce something of comparable quality.

Second, we need to be wary of plagiarism and so running applications through Turnitin or Googling suspect sentences may be a necessary precaution.

As an aside, I mention the movie Spanglish has an amusing opening scene [which I could not find on YouTube] where an admissions officer at Princeton is reading personal statements from undergraduate applicants.

The case for effective Hamiltonians

2011-09-21T17:30:00.001+10:00

When trying to understand complex molecular materials, the dominant approach in chemistry to is to do DFT-based calculations for the system of interest. However, a case needs to be made for an alternative "physics" approach. Recent Anthony Jacko, Ben Powell and I wrote an article Models of organometallic complexes which makes the case below for effective Hamiltonians.

Another approach to modeling the optoelectronic properties of organometallic complexes is to construct a model with fewer states but an accurate treatment of the electronic correlations. This contrasts with first principles calculations, which include several basis states for each atom but neglect some electronic correlations. The small number of degrees of freedom in such semi-empirical models allows one to make fewer approximations on the interactions and correlations in the model. It also allows one to identify key trends that describe broad classes of materials. This approach has proven itself incredibly powerful in wide areas of materials science. For example, the Anderson single impurity model can describe a wide range of systems including magnetic impurities in metals, quantum dots in semiconductor heterostructures, carbon nanotubes , and single molecule transistors.47,48

In principal an effective model Hamiltonian is found by starting with the exact Hamiltonian and ‘integrating out’ high energy states.49 This procedure is computationally expensive,50 so often one simply chooses a reduced basis set, motivated by the physical processes one wishes to capture.49 DFT can be used to estimate the values (or trends in values) of some of the parameters of these effective models. The model Hamiltonian can then be solved, retaining correlations that the approximate DFT functional does not include.51–55

Identifying the frontier orbitals which dominate the photophysics is one of the most significant steps of the effective model approach. In this reduced basis set one can define an effective Hamiltonian with just a few parameters. Conjugated polyenes have been investigated in this way via the Hückel, Hubbard, Heisenberg and Pariser-Parr-Pople models.49 This approach has been applied to organometallic complexes, for example mixed valence binuclear systems including magnetic atoms in proteins (Hubbard and double exchange models; Ref. 56), molecular magnets (Ref. 57), Anderson impurity models for cobalt based valence tautomers (Ref. 58), and a series of metal-coredbipyridine complexes (Ref. 22). It has also been shown recently that this approach naturally explains the sensitivity of the photophysical properties of organometallic complexes to small chemical changes.18

To correctly describe the character of the excited states the model must capture the key interactions. There are many important features of the system that might be included in such a model, for example electronic ‘hopping’ terms between the frontier molecular orbitals, direct Coloumb interactions between electrons in those orbitals, spin interactions, and relativistic effects such as spin–orbit coupling. The relative energy scales of these various interactions will define the composition of the excited states and therefore their properties.

I welcome comments and suggestions on any review articles that persuasively make the case for effective Hamiltonians as an important tool in materials modelling.

Seeking a new phase of matter in organic charge transfer salts

2011-09-19T18:45:00.001+10:00

An important finding of recent cluster DMFT studies concerns the nature of the metallic state in the Hubbard model at half filling. Near the Mott transition it is found that the scattering rate is anisotropic over the Fermi surface (a Fermi liquid with momentum space differentiation) just as it is in the doped Hubbard model.
Emanuel Gull showed a general phase diagram (U/t vs. doping) in his talk at the Ringberg meeting.

(I could not find it in any of his papers, such as this PRB, but he kindly provided a copy). This "momentum space differentiated" phase is intermediate between the pseudogap state and the isotropic Fermi liquid, both as a function of doping and U/t.

This (temperature dependent) anisotropy should be observable in angle dependent magnetoresistance (ADMR) measurements on organic charge transfer salts in the kappa-(BEDT-TTF)2X family. These materials are all at half filling. A candidate material is X= Cu[N(CN)2]Br which lies close to the Mott transition. [By deuteration it can be tuned into the Mott phase]. Previous papers [e.g. see Section 3.4 in a recent review article I wrote with Ben Powell] have considered evidence for a pseudogap state in the organics. However, I am unaware of any significant attention being paid to this distinct idea of an anisotropic scattering rate in the organics.

In a 2007 PRL Singleton et al. found that that experimental data for X=Cu(SCN)2 can be adequately modelled by an isotropic scattering rate with a Fermi liquid temperature

dependence. Is the co-efficient for the quadratic temperature dependence consistent with what Dressel finds for the quadratic frequency dependence of the scattering rate from

the optical conductivity? [This earlier post discusses the general issue of the relation between the temperature and frequency dependence of the scattering rate in Fermi liquid theory].

Thus, in order to see the variation in the scattering rate one may have to go even closer to the Mott insulating phase by considering X= Cu[N(CN)2]Br, (as in this Nature paper on the Nernst effect).

Topological insulators in prime time TV

2011-09-19T15:46:00.001+10:00

The popular sit-com The Big Bang Theory continues to break new ground by introducing prime time TV viewers to the latest developments in condensed matter physics. This clip from an episode The Thespian Catalyst begins with Sheldon attempting to teach a class of Caltech graduate students about topological insulators. Note that the equations on the board are authentic, as usual.

Sheldon finds, as many of us do, there is a weak correlation between our perceptions of our teaching performance and our students perceptions of the quality of our performance. The show also discusses important issues about what initiatives we should take to address this issue and to improve the quality of our teaching.

Effective Hamiltonians for fluorescent proteins

2011-09-16T17:39:00.001+10:00

This morning I am giving a seminar at the Laboratoire de Chimie Quantique in Strasbourg. Here are the slides.

What is the "mechanism" of superconductivity in the cuprates?

2011-09-15T20:02:00.001+10:00

Thomas Maier gave a nice talk at the conference. He considered this question [or at least what is the mechanism of superconductivity in the Hubbard model?] via a Bethe-Salpeter equation (some of the results are described in this PRB and this PRB).

where Gamma^pp is the irreducible particle-particle vertex and chi_0 is

Their calculations are for a cluster DMFT treatment of the doped Hubbard model. They find that the irreducible particle-particle vertex peaks at a wavevector of (pi,pi) as does the spin susceptibility chi(K-K'). Indeed they find that this vertex is to a good approximation related to the spin susceptibility via an RPA type relation.

where U(T) is a temperature dependent renormalised Hubbard U.

But why does the superconductivity go away as one approaches the Mott insulator? After all, the spin fluctuations are increasing! This is because chi_0 ~ GG [Thomas had a name for this that I did not catch] is decreasing because of the suppresion of quasi-particle weight as the Mott insulator is approached.

I would be interested to see this approach applied to an extended Hubbard model on the square lattice at one quarter filling, near the charge ordering transition [see this PRL and PRB for the context]. Two questions the approach could answer are

Is there superconductivity? If so, is it d_xy symmetry?
Is it mediated by spin or charge fluctuations?

The latter question is relevant because there is spin ordering associated with the charge ordering and when the charge order melts so will the spin order, potentially leading to significant spin fluctuations.

The long road from the Hubbard model to experiment

2011-09-14T16:48:00.001+10:00

An outstanding theoretical challenge remains (reliably) calculating thermodynamic and transport properties of the one band Hubbard model. This is necessary to answer questions such as: is it the relevant effective Hamiltonian for cuprate superconductors?

Emanuel Gull gave a nice talk at the conference about work on cluster DMFT treatment of the doped Hubbard model. This PRB paper contains a nice comparison of how the results vary with cluster size. They consider cluster geometries up to 16 sites. At least 8 sites are necessary to cover the important region around wavevector (pi/2,pi/2) where one hopes to find nodes in a pseudogap, superconducting gap, and cold spots on the Fermi surface.
The calculations produce four distinct phases:

Mott insulator
An isotropic Fermi liquid (FL)
A FL with momentum space differentiation
A momentum selective Mott insulator (a pseudogap state)

Of particular interest to me is the existence of the third state which may correspond to the anisotropic Marginal Fermi liquid state seen in overdoped cuprates. It is desirable to compare the doping, temperature, and frequency dependences of the self energies they calculate [Figures 8-11] to the model form that Jure Kokalj and I have considered.

Mapping out Fermi surface anisotropies in overdoped cuprates

2011-09-13T21:45:00.000+10:00

Here are the slides for the talk, Angle-dependent magnetoresistance as a probe of Fermi surface anisotropies I am giving this afternoon at the conference.
Much of the talk is based on this preprint.

Key elements of Iron pnictide superconductors

2011-09-13T04:05:00.000+10:00

David Singh gave a nice talk today about the iron pnictide superconductors. Here are just a few of the points that he emphasized.

This is a big family of materials - high Tc is a very robust phenonema. Common features

Fe atoms are in a square lattice
near divalent Fe
tetrahedral coordination (2 Fe per unit cell)

The connection of the pnictides to the cuprates is weak, contrary to what some people asserted in the early days [because of the proximity of superconductivity and antiferromagnetism]
-doping not essential to superconductivity
-Mott physics is not relevant (no Mott insulating state)
-magnetic order & superconductivity do co-exist?
-multiple orbitals are present [they may be important for avoiding Mott insulator]
-many materials are electronically "three-dimensional" rather than two-dimensional

[ARPES and dHvA see significant corrugation of the Fermi surface].

Proximity of superconductivity to a magnetically ordered phase should not be viewed as necessarily implying a magnetic mechanism for Cooper pairing. He gave a summary [in the form of a phase diagram] of the 1966 paper by Berk and Schrieffer who argued that Pd was actually not superconducting due to proximity to a ferromagnetic state.

An important case is fullerene superconductors. They are due to electron-phonon pairing, but can be close to an antiferromagnetic Mott insulating phase.

How long will it take us to solve high-Tc superconductivity?

2011-09-13T02:40:00.000+10:00

How long after the experimental discovery of a quantum many-body phenomena does it take to develop a theory that will be eventually accepted as the correct one?

Superconductivity was first observed in 1911 and BCS theory arrived in 1957.

At the conference in the castle today, Sasha Lichtenstein pointed out that the Kondo effect was first observed in 1933, Kondo described some of the relevant physics in 1964, but it wasnt until the 1980s that the problem was really solved. Thats 50 years. He pointed out that it is 25 years since the discovery of high-Tc superconductivity in the cuprates and suggested maybe we have 25 more years to get the theory right!

On the other hand, Laughlin found the correct theory of the fractional quantum Hall effect within a year of its experimental discovery!

How bad can a metal get?

2011-09-12T02:15:00.002+10:00

A striking feature of strongly correlated electron materials is the existence of bad metallic behaviour. The resistivity can monotonically increase with temperature to values which are much larger than the Mott-Ioffe-Regel limit, a value corresponding (in a simple Drude model) to a value where the mean-free path is smaller than a lattice constant.
Key questions are

Does the resistivity ever "saturate" at high temperatures?
If so, to what value is the maximum possible resistivity?
Could this be related to the large spectral weight spread out over a broad spectral range in the optical conductivity? (see the next figure below).

There is a nice summary of the issues in section VIIH of this RMP by Basov and Timusk.

Gunnarsson and collaborators have come up with a simple argument to address these questions [see this nice RMP colloquium].

At high enough temperatures there is no Drude peak in the optical conductivity. However, the latter must still satisfy the f-sum rule, which relates the total "low energy" spectral weight to the average kinetic energy, E_K. The spectral weight may be spread over the non-interacting band width W.
These observations can be summarised in the figure and equation below which provides an estimate for the dc conductivity sigma(0) on the scale e^2/(h d) where d is the interlayer spacing.

A recent PRB by Bergeron, Hankevych, Kyung, and Tremblay calculates the optical conductivity for the Hubbard model at the level of a two-particle self-consistent approach, including the constraint of the f-sum rule. The curves below is for a doping p=0.2 , close to optimal, and a temperature of T=0.2t.

Qualitatively, it seems consistent with the predictions of Gunnarsson et al.

A unified description of hydrogen bonding

2011-09-10T06:41:00.002+10:00

I have just finished a paper Unified description of hydrogen bonding by a two-state effective Hamiltonian. I welcome feedback. Many of the ideas in the paper have been discussed on this blog before over the past year. The paper sits right at the boundary of physics and chemistry and exemplifies the associated tensions. Some physicists may like it because of the simplicity and transparency. Some chemists may consider it glosses over and misses too many details.

Practically all the ideas in the paper have been discussed in some form before. I see the key novelty as synthesis: providing a conceptual and semi-quantitative framework to understand a wide range of experimental and theoretical results. Consequently, I consider the paper may be the most significant one I have ever written...

I welcome feedback.

I will submit the paper to Physical Review Letters in about a week, after I have received more feedback.