Condensed concepts

This week in the Quantum Many-Body Theory reading group we look at Chapter 6 of Advanced Solid State Physics by Philip Phillips, concerning local magnetic moments in metals and the Anderson model. Here are my random notes. The essential physics is captured by the Hartree-Fock solution. The Coulomb repulsion (Hubbard U) leads to qualitatively different behaviour: the formation of local moments. This is embodied in the zero-temperature phase diagram. I suspect that when Anderson published this diagram in 1961 this was a new way of looking at a quantum many-body problem. It is all about competition between the three energy scales: U, the hybridisation energy, and the energy separation of the Fermi energy and the localised energy level. Anderson shared the Nobel Prize in 1977 with Mott and Van Vleck. Anderson was cited for this work on local moments and localisation due to disorder. The model and the associated physics has a relevance that goes far beyond the question of local moments i

Glenn T. Seaborg was the father of heavy element chemistry. It is interesting how we was so driven by a passion for science and for educating the next generation. Even though he had a Nobel Prize (1951) and had been Chair of the Atomic Energy Commission (equivalent to Secretary of the Department of Energy today) in 1971 he returned to Berkeley to teach and to lead a research group. In the National Academy Biographical Memoir by Darleane C. Hoffmann, who was his successor at Berkeley, says: I learned so many things from him just by observing how he ran the weekly brown bag lunches with his graduate students and later mine--listening with great interest as they described their research progress. He asked insightful and penetrating questions, but not in a threatening manner, made suggestions, and frequently went to visit the labs late in the day to see what was going on. He also hosted many undergraduate research students. He was devoted to education and student training and would prep

Next month I am going to India to speak at a School and Conference on “Emergent Properties and Novel Behavior at the Nanoscale” organised by I2CAM and the Jawaharlal Nehru Centre for Advanced Scientific Research ( JNCASR ) in Bangalore. In the school I will give a one hour long lecture. Here is the abstract I have submitted. Any feedback welcome. Some related material is discussed in this talk I gave last year in the Black Forest. Quantum design principles for functional electronic materials In a complex material how does one optimise the quantum efficiency of the transition between two different quantum states when there are many alternative transitions available to a system? Regardless of whether or not it is explicitly stated this is the question which is at the heart of a wide range of research. Prominent examples include understanding biomolecular function, designing organic photovoltaic cells, and catalysis. I will discuss how this optimisation problem involves a subtle inter

Today I was at the Department of Nuclear Physics at ANU with collaborators discussing quantum decoherence in heavy ion collisions. Kouichi Hagino was visiting from Japan and gave us a nice overview of the coupled channels approach to nuclear collisions. He made use of some really nice lecture notes he gave at a summer school in 2006.

Seth Olsen has a really nice paper that just appeared ASAP online in the Journal of Chemical Theory and Computation. A real challenge of high-level quantum chemistry is to connect it to chemical concepts, chemical trends and semi-empirical theories. This seems to rarely happen. There is a tendency to just throw ones arms up in the air and say, "It is too hard! One just has to calculate the properties of every molecule from scratch." However, Seth has shown how correlated quantum chemical calculations for an important class of methine dyes can illuminate, clarify, and provide justification for resonance theories of dye colour that go back to the forties, including from John R. Platt. The figure below compares excitation energies calculated ab initio with those from a parametrisation of Platt's model.

I just read this advice: 1. Be an outsider . Don't be unduly awed by authority,... 2. Be fact driven. An obsession with facts can help you avoid common pitfalls and the bending of facts to fit your theories.... 3. Respect the unknowable . Work on the assumption that too many people are too certain about the inherently uncertain things. 4. Always question your own convictions . Most of us look for verification of our own beliefs. Instead you should be open to having your ideas criticised. Only through this kind of stress testing can the true validity of your beliefs be determined. 5. Employ meta-cognition. Consciously practise all the steps above to avoid falling into overconfidence when success is experienced. Actually, this advice is not about doing science. It is from Kathryn Schutz's reflections Five lessons for investors, from Michael Lewis's The Big Short . I read the above summary in The Week. But, it seems to me it is good advice also for doing better science!

Previously, I asked is organic semiconductors a misnomer ? One needs to be very careful about trying to apply the same concepts from inorganic semiconductors to organic materials used to make devices such as photovoltaic cells. An important example is excitons. In inorganic semiconductors these arise from the Coulomb interaction between an electron and a hole and that have binding energies of much less than an electron Volt and they have a spatial extent of many lattice constants . They are sometimes called Wannier-Mott excitons. Because they are so large variation of the dielectric constant of the material has a significant effect on the binding energy and spatial extent. As a first approximation one solves the hydrogen atom problem with the band effective mass and material dielectric constant. However, the excitons (i.e., low lying singlet excited states are produced by photon absorption and can decay radiatively) in organic materials are VERY different. They are usually spatially l

Today is Ada Lovelace day [I thank Ben Powell for bringing this to my attention] and so I asked myself, "Which woman has made the greatest contribution to quantum many-body theory?" Maria Goeppert-Mayer is best known for receiving the Nobel Prize in Physics for her role in developing the nuclear shell model. However, she did more. In 1938, together with Sklar, she did one of the first non -empirical quantum chemistry calculations: the excited states of benzene. This is in a widely cited Journal of Chemical Physics paper.

This weeks reading from Advanced Solid State Physics by Philip Phillips is Chapter 5, Interacting Electron Gas. Some key ideas: The Hartee-Fock equations for the Jellium model [equations 5.4 to 5.6] have plane wave solutions. These correspond to quasi-particles and have the energy dispersion relation 5.1. The exchange hole refers to the fact that electrons of like spin avoid one another due to the Pauli exclusion principle. This means the energy is less than the Coulomb repulsion energy of an electron moving in a uniform charge density. The total energy of all the occupied Hartree-Fock states gives an "exchange energy" which only depends on the electron density (5.14). [This expression is sometimes used in the X_alpha method of Slater and the Local Density Approximation (LDA) of Density Functional Theory (DFT)]. The HF excitation spectrum is qualitatively incorrect. It predicts that the low-temperature specific heat of metals should go like T/ln(T) whereas it is always observ

In 1987 Anderson made the radical proposal concerning doping Mott insulators: "The preexisting magnetic singlet pairs of the insulating state become charged superconducting pairs when the insulator is doped sufficiently strongly." The material SrCu2(BO3)2 has been argued to be a Mott insulator on the Shastry-Sutherland lattice (see Figure below). In the corresponding Heisenberg model there is an exchange interaction J along all vertical and horizontal bonds and a diagonal interaction J' along every second plaquette. It can be shown that for J'/J > 1.44 +/- 0.02 that the exact ground state is a product of singlets along the diagonals. Liu et al. studied the corresponding t-J model (including three site hopping terms) away from half-filling using a projected BCS wave function. They considered the model with t'=+/- 1.25 t and J=0.3t. They found the following (summarised in the Figure below). (i) There is significant particle-hole asymmetry. This is because one s

When we teach undergraduates laboratory subjects we are always on the case of the students about using the appropriate number of significant figures and always quoting error bars. However, it seems the scientific community does not apply the same criteria to themselves when quoting and comparing quantities such: average scores on student evaluations of teaching impact factors of journals citation rates per paper For example consider the graph below of impact factors of physical chemistry journals from 2001 This site reports "large changes" in impact factors of Chemistry Journals from 2007 to 2008. It reports that of ChemPhysChem changed from 3.502 to 3.636. J. Phys. Chem. B changed from 4.086 to 4.189. This suggests to me most of the comparisons are in the noise. I would suggest that what we would tell our first year undergrads to do with data like this is something like: "take the data from 5 years, average it and find the standard deviation. use the latter as your err

Just as one can discuss geometrical frustration of spins in antiferromagnets one can also discuss geometrical frustration of the kinetic energy of electrons on similar lattices. But it should also be noted that this is a strictly quantum mechanical effect arising from quantum interference. This is in distinct contrast to geometrical frustration in antiferromagnets which can occur for purely classical spins. A few years ago, Jaime Merino, Ben Powell, and I discussed trying to quantify this kinetic energy frustration. (See Section IIC of this long PRB paper ). In a non-interacting electron model the only proposal we were aware of for a quantitative measure of this frustration of the kinetic energy was due to Barford and Kim : I am wondering if the width of the lowest lying band of triplet excitations may be a good measure of frustrations in a quantum antiferromagnet. More on that later...

When you give a talk, listen carefully to the questions. Do not cut off the questioner, even if you think they are mistaken. Let them finish. You can learn a lot from the questions and criticism and skepticism of your audience, even if you disagree. Answering, "I don't know. I will try and find an answer for you" is preferable to waffling on about things you do not understand or making claims you cannot back up. The latter reduces your credibility considerably more than the former. Caveat: it is easier to give advice than to actually put it into practice. Like all advice I give here I do not claim that I always do it myself. But, it is what I want to aim for.

The hierarchy of objects and descriptions associated with theories of electronic properties of solids. [n.b. the arrows should point up not down! I am graphically challenged] At the level of quantum chemistry one can describe the electronic states of single (and pairs of) molecules in terms of molecular orbitals which are linear combinations of atomic orbitals. [This is Laughlin and Pines, Theory of Everything! ] Just a few of these orbitals interact significantly with neigbouring units in the solid. Low-lying electronic states can be described in terms of itinerant fermions on a lattice and an effective Hamiltonian such as a Hubbard model. In the Mott insulating phase the electrons are localised on single lattice sites and can described by a Heisenberg spin model. The low-lying excitations of these lattice Hamiltonians may have a natural description in terms of quasi-particles which can be described by a continuum field theory such as a non-linear sigma model.

Last week I briefly discussed the notion of quasi-particle weight and recommended P.W. Anderson, Concepts in Solids: Lectures on the Theory of Solids, Chapter 3. This week we looked at Chapter 4, in Advanced Solid State Physics, The Hartree-Fock approximation. This is can be viewed as a mean-field theory and/or a variational approximation to the ground state energy. One assumes many-particle ground state wavefunction can be written as a single Slater determinant. One way to understand the Hartree-Fock equations (4.23) as a mean-field theory is the following. Consider the Coulomb interaction energy between an electron in one orbital (denoted nu) [which has an associated charge density] and the charge density associated with all the other occupied orbitals. The one electron energies are actually the Lagrange multipliers introduced to care of the normalisation constrain for the one electron orbital wave functions. The exchange interaction (between electrons in different orbitals) arises

In the latest issue of Nature, Leon Balents has a nice comprehensive review article, Spin liquids in Frustrated Magnets . In a Box at the side Leon considers the problem of quantifying the amount of frustration in an antiferromagnetic material (or model) and considers a measure (emphasized by Ramirez in a classic review ) f=T_CW/T_N, the ratio of the Curie-Weiss temperature to the Neel temperature, at which three-di mensional ordering occurs. Although, indicative of frustration I think we should come up with a more precise measure. One limitation of this measure is it does not separate out the effects of fluctuations (both quantum and thermal), dimensionality, and frustration. For strictly one or two dimensional systems, T_N is zero. For quasi-two-dimensional systems the interlayer coupling determines T_N. Thus, f would be larger for a set of weakly coupled unfrustrated chains than for a layered triangular lattice in which the layers are moderately coupled together. In 2005, Weihong

As a physicist trying to understand what makes a good organic LED material it is easy to get lost in all the chemical details. Today I came across a review, Light-emitting iridium complexes with tridentate ligands , by W^3 (Williams, Wilkinson, and Whittle!) which has the cute and useful abstract picture below If you wade through the paper Figure 17 (below) suggests how to understand differences between different ligands in terms of frontier orbitals. They also discuss how the character of the emitting triplet state changes for the different complexes.

I now discuss a second of the four articles I introduced in an earlier post . This article is one I know that Phil Anderson likes to rant about. Brian Pippard , The Cat and the Cream , 1961. A banquet speech given at a conference on superconductivity held at IBM's new lab at Yorktown Heights, and later published in Physics Today . Here are a few choice quotes: the era of the great breakthrough is over..... I have found that when I suggest to senior physicists that the end of physics as we know it is in sight, they tell me, "That's just what everybody was saying in 1900". Now this may be a justification for optimism, but let's first ask whether the historical parallel is sound. I think in many ways it is. ....If you don't believe me, ask yourselves this question: Apart from the field of fundamental particles, what is the most recent discovery in physics that still remains in essence a mystery? I think I might remark that in low-temperature physics the disappe

Yesterday there was an interesting colloquium, Simulating star cluster evolution on high-end graphics cards, by a new UQ staff member, Holger Baumgardt. The fact I am writing a blog post about it should be taken as a compliment and the comments below need to be taken with a large grain of salt, since the subject goes way beyond my expertise. But they may be interesting in terms of how an outsider sees things... First, a few things I learnt. Globular clusters are 10-12 billion years old, much older than typical galaxies. Black holes may be at the centre of most galaxies. UCDs are ultra-compact dwarf galaxies somewhere between star clusters and galaxies. The figure below shows a plot of the radii of different objects versus their mass. It is taken from this paper. Scaling relations for low-mass, hot stellar systems: half-light radius plotted against total mass.The dashed line shows the fitted relation for elliptical galaxies, while the solid lines indicates the median for Galactic glob

John R. Platt has featured on this blog before because of his paper in Science about "Strong Inference" and the "method of multiple alternative hypotheses". This stimulated my New Year's resolution. Platt also wrote a really nice theory paper about organic dyes that will feature soon because Seth Olsen has a paper about to appear in Journal of Chemical Theory and Computation which gives a rigorous quantum chemical basis for Platt's theory. I was wondering what happened to Platt and found an interesting 1992 Obituary in the New York Times. I also found that Platt was on a panel discussion with Michael Polanyi about the interplay of reductionism and emergence in biology, physics, and chemistry.

This weeks reading from Advanced Solid State Physics by Philip Phillips is Chapter 3, Second Quantization. Some key ideas: Quasi-particles in quantum many-body theory are either fermions or bosons. The difference is seen in whether their creation and annihilation operators obey anti-commutation or commutation relations. The second quantisation formalism provides a convenient and powerful book-keeping device to keep track of the fermionic (or bosonic) character of many-particle states. A key result to become familiar with is writing a many-body Hamiltonian operator in second-quantised form, as in equation (3.33). This is the starting point of Chapter 4, The Hartree-Fock approximation. In the next chapter we will see how the anti-symmetric (fermionic) character of many-electron quantum states has an important physical consequence: the exchange interaction which leads to an energy splitting of singlet and triplet excited states. [A clear discussion of this is in Chapter 18, The Helium Ato

In Mike Norman's nice review of the theory of superconductivity in the cuprates he states: The RVB spin gap was probably the first prediction for the subsequently observed pseudogap phase. In RVB theory, the pseudogap phase corresponds to a spin singlet state (with its resulting spin gap) but no phase coherence in the charge degrees of freedom. One of the interesting ideas to emerge from this was an explanation for transport in this phase, which reveals a metallic behavior for in-plane conduction, but an insulating behavior for conduction between the planes. In the RVB picture, the metallic behavior is due to the fact that the holons can freely propagate. But to tunnel between the planes, the holons and spinons must recombine to form physical electrons, and this costs the spin gap energy, thus one obtains insulating like behavior for the c-axis conduction (Lee, Nagaosa, Wen, 2006). This “gap” has now been directly seen in c-axis infrared conductivity data (Homes et al., 1993). Ho

I have been reading through Bruce Normand's review article, Frontiers in frustrated magnetism article in Contemporary Physics. The focus is on spin liquids: how to define them and whether they exist in real materials and/or whether they are the ground state of physically reasonable quantum lattice Hamiltonians. Bruce considers three different classes of spin liquids, each being defined by their excitation spectrum. In the figure below, singlets are red and triplets are blue. (b) Type I. There is gap between the singlet ground state and both the lowest-lying triplet state and the first excited singlet state. (c) Type II. There is gap between the singlet ground state and the lowest-lying triplet state. There is no gap to a continuum of low-lying singlet states. (d) Algebraic spin liquids. There is no gap to either the lowest-lying triplet state or the first excited singlet state. An important question is how to distinguish these different states experimentally. One basic question I

I liked this xkcd cartoon since so many seem to feel they are well qualified to pass judgement on the evidence for and against climate change, biological evolution, big bang cosmology, consciousness, Schrodingers cat... I thank Nathan Campbell for bringing it to my attention.

This is the first post about one of the four articles mentioned in How wrong about the future can you be? Serge Haroche and Jean-Michel Raimond, Quantum Computing: Dream or Nightmare? Physics Today, 1996. Actually, I think these authors were essentially correct 14 years ago when they concluded: Even if quantum computing remains a dream, the physics of quantum information processing at the level of a few qubits is fascinating. Experiments on entangled particles with ions in a trap or atoms in a cavity will help us understand the fundamental aspects of quantum measurement theory, and they may lead to major improvements in the precision spectroscopy of simple quantum systems. .... Testing quantum decoherence in conceptually simple experiments is also an important and challenging task. Rather than teaching us how to build a large quantum computer, such experiments are more likely to teach us about the processes that would ultimately make the undertaking fail. It is important to adve

Fritz Schaefer is one of world's leading computational chemists. It is interesting read his list of minimum requirements for a first draft of a paper from his group. Two things to note: the emphasis that computational chemistry should be motivated by and ultimately connect with experiment the figures are key

Considering natural orbitals in quantum chemistry this week reminded me of a key property of Fermi liquids: the existence of a discontinuity in the wavevector dependence of the Fourier transform of the one-electron reduced density matrix n(k). The figure below is taken from a PRB paper by Paola Gori-Giorgi and Paul Ziesche. It shows the wavevector dependence of n(k) for a uniform electron liquid (at density corresponding to an interparticle spacing of rs=5, comparable to a typical metal). k=1 corresponds to the Fermi wavevector. For a non-interacting fermion system n(k)=1 for k less than 1 and 0 for k >1. The figure below shows the magnitude of zF, the jump in n(k) at k=1, as a function of rs. Increasing rs corresponds to decreasing electron density and increasing correlation effects. This quantity zF can be related to the quasi-particle weight associated with the pole in the one-electron Greens function. The smaller zF the larger the effective mass of the quasi-particles. In a fe

Distinguished scientists trying to predict the future direction of a field of study is a dangerous (but fascinating) exercise. I have on my desk four such articles. I hope to comment on each in turn, but it is interesting to compare them since they are all very different, have had different impacts, and in hindsight some were pre-scient and others off track. Here they are in chronological order: Niels Bohr, Light and Life, given at the International Congress of Light Therapy in 1932, and published in two parts in Nature in 1933 ( Part 1 and Part 2 ). Charles Coulson, Present State of Molecular Structure Calculations, Given as an "after-dinner" speech at a quantum chemistry conference in 1959. Brian Pippard, The Cat and the Cream, 1961 A conference banquet speech at a conference held at IBMs new lab at Yorktown Heights, and later published in Physics Today . Serge Haroche and Jean-Michel Raimond, Quantum Computing: Dream or Nightmare? Physics Today, 1996.

Today I gave a talk at the weekly UQ Condensed Matter Theory group meeting (cake meeting because someone always brings a cake!) on the possible role of entanglement measures in quantum chemistry. I only got through the first half of these notes and so will continue next week. Any comments welcome.

A fundamental question concerning dye-sensitized solar cells is what determines the speed and efficiency with which charge is injected from the dye to the semiconductor (such as titanium oxide) on which it is absorbed? A nice review by Liu and Andersen contains the summary diagram. One important point I learnt is that (at least in some dyes) the ultrafast injection (~tens fsec) from the singlet excited state of the organometallic dye complex (such as those shown above) competes with intersystem crossing to the triplet excited state, from which injection can also occur but not as quickly.

This weeks reading from Advanced Solid State Physics by Philip Phillips is Chapter 2, The Born-Oppenheimer [BO] Approximation. This reflects a key idea in quantum many-body theory: when there is a clear separation of energy scales one can often simplify the theory greatly and identify distinct quasi-particles [e.g., phonons and electronic quasi-particles] corresponding to the different energy scales. Without BO there would be no quantum chemistry and no quantum theory of electronic properties of crystals. BO is a consequence of the fact that electrons are much lighter than nuclei. Consequently, electrons in a molecule or solid have average velocities that are typically orders of magnitude faster than the nuclei. Equation (2.1) is the Hamiltonian The symbols Z α and M α are the atomic number and mass of the α th nucleus, R α is the location of this nucleus, e and m are the electron charge and mass, r j is the location of the j th electron, and ℏ is Planck's constant. If we

In a 1987 Science paper Phil Anderson made a radical proposal, stimulated by the discovery of high-Tc superconductivity in layered copper oxides. [A meausure of the influence of Anderson's paper is that it has been cited more than 4000 times]. My version of Anderson's proposal is: The fluctuating spin singlet pairs [produced by the exchange interaction] in the [Mott] insulating state become charged superconducting pairs when the insulating state is destroyed by doping, frustration or reduced correlations. These fluctuations are enhanced by spin frustration and low dimensionality. To me the idea is radical and rather counter-intuitive because in one sense insulating and superconducting states are so different. One has infinite conductivity and the other zero. Ben Powell and I have been invited to write a review article for Reports in Progress in Physics concerning organic charge transfer salts which can be described by a Hubbard model on the anisotropic triangular lattice. [An

I was going to write a post about email and then remembered I wrote a post Think twice before you send that email in the early days of this blog. It says much of what I wanted to say again. Think twice (or cool off) before you hit Reply or Send or Forward. It is also worth reading some of the lists of "email etiquette" on the web. Here is one that I thought was pretty could. Some of this may seem obvious and common sense. But to quote Steven Covey, "Common sense is not necessarily common practice."