Scientific fads and fashions

Fashions currently play an essential role in the sociology - and in the funding - of physics and other sciences (mathematics being relatively spared). A specialised subject (such as chaos, string theory, or high temperature superconductors) comes into fashion for a few years and then is dumped. In the meantime the field has been invaded by swarms of people who are attracted by success, rather than the ideas involved. And this changes the intellectual atmosphere for the worse.

David Ruelle, Chance and Chaos (1991), page 71

This issue of fashions in condensed matter physics was also discussed by Sokrates Pantelides in a 1992 Physics Today opinion piece.

Negotiating job offers

I previously posted about considering the relative merits of job offers (if you have more than one). Here are a few random thoughts about negotiating.

First, until you have an offer in writing you do not have an offer. A verbal offer or a promise or a hint of "an almost certain" offer is not an offer. Things can go wrong. Deans can veto decisions of departments. A budget crisis emerges suddenly. Someone in the decision chain dies. The US government defaults on its loans.....

Second, once you have the written offer the dynamics and "power balance" of the relationship between the employer and the potential employee completely change. Before you had to tread lightly and be careful not to offend. Now you can ask for anything. I am not suggesting that you do (see the next point). They can always say no. However, it is quite reasonable to ask them to fly you (and possibly even your family) for a second (or first) look. This is when you really need to find out the "dirt" on who you may be working with or for. Honest and open discussions about expectations and resources are crucial. It is particularly important that you talk to some recent hires for similar positions as yours. For example, if you are being hired as an assistant professor I would try and talk in person to others in the department. Are they happy? Do they have any advice for you? If you are being hired as a postdoc in a specific group then talk to current or recent postdocs from the same group. These conversations can save you a lot of pain and frustration or give an sense of assurance about what you can expect. Experimentalists need to be be certain that the equipment that they need is available (and functioning) or is certain to become available (e.g. through adequate start-up funding) in a timely manner.

Third, be careful that your requests and negotiating points are reasonable. Making unreasonable requests/demands may have a negative effect on your relationship with your future boss and colleagues. Focussing a lot on minor salary details will communicate something about your priorities. Bear in mind that sometimes details of these "confidential" negotiations become more widely known and may affect your reputation in the broader community. [e.g., the Assistant Professor who requested a separate coffee room attached to his lab.]. What is reasonable? That really depends on how much they want you, how many resources are available, and what is "normal". Running possible requests by a more experience colleague before you make them is a good idea.

Fourth, get everything promised in writing. If you don't it is quite likely you won't get what you are promised. Even if you do have it in writing that will not necessarily guarantee it.

Cartoon by Kerry Soper is from here.

The LHC makes me yawn

This post may just reflect my ill informed prejudices. Please correct me.
Sorry, but I cannot get excited about the Large Hadron Collider (LHC). I find it hard to see how it is going to reveal the secrets to the universe, or even to produce new scientific insights comparable to the expense and effort involved.

First, I find it hard to believe that the Higgs boson will not be found. My limited knowledge of particle physics and the standard model is that it just has to be there. Spontaneous symmetry breaking and the associated dynamical mass generation is well established in other areas of physics (e.g. the Meissner effect in superconductors can be viewed as photons acquiring mass, as first emphasized by Anderson). Hence, I won't be surprised if it also works / is present in the Standard model via the Higgs boson.

On the one hand, the LHC (search for the Higgs boson) is an experiment that should certainly be done, in a world in with almost unlimited resources for science.
On the other hand, I remain to be convinced that the LHC is a much better investment than "tabletop science." The latter has produced Graphene, violations of Bell inequalities, spin ice, fractional statistics in the quantum Hall effect, and cuprate superconductors. To me anyone of those is of comparable importance to confirming the Higgs boson is really there.

An earlier post considered the second of Two questions physics can no longer avoid, asked by Martin Gutzwiller in a Letter to Physics Today, published in August 1994.

His first question was, "Has particle physics fulfilled its promise?"

Not all densities are equal

One of the great challenges of describing and understanding strongly correlated metals is that the notion of "charge carrier density" becomes ambiguous. In a simple Fermi liquid metal all the different densities listed below are essentially identical (modulo some factors involving fundamental constants).

• Number of charge carriers (from simple chemical counting).
• The conduction band filling.
• Volume (area) of the Fermi surface
• Drude weight in the optical conductivity
• Hall coefficient (or number)
• Superfluid density of superconducting state

Some of these issues are explored in this PRB by Haerter and Shastry.

Docking station for MacBook Pro

There was one thing I really missed after switching from a PC to a Mac was having a docking station in my office for my laptop. For a Dell PC there was a simple docking station one just dropped the laptop into.
I went without a docking station for the past 18 months with a MacBook Pro. Every morning I would just reconnect the 5 or so connectors needed [large monitor, key board, power, ethernet, printer, ...]
However, I just purchased a Henge Docks. I read a review that was helpful before I bought. I found it hard to believe that it would take about an hour to install. But, it does.
Overall, I agree it is clumsy. But, it is functional and so I am glad I got it.

The hydrogen bonding puzzle

Chemistry is local. For example, the energy and length of a C-C or O-H bond varies little between molecules. It does not matter whether the C-C bond is in ethanol or in a huge protein. (See also an earlier post on transition metal complexes within proteins). One of the great strengths of valence bond theory is that (unlike molecular orbital theory and standard implementations of density functional theory) it captures this local perspective, as emphasized by Sason Shaik.

However, a significant exception to this purely local picture for chemical bonding is provided by hydrogen bonding. We can denote a hydrogen bond by D-H...A where D and A are the donor and acceptor, respectively. The H-bond puzzle is discussed extensively in the 2009 book, The Nature of the Hydrogen Bond by Gastone Gilli and Paola Gilli  (and also in an article from 2000). Here is a paraphrase from page 62 of the book:

"the unique feature of the H-bond is that bonds made by the same donor-acceptor pair may display an extremely wide range of energies and geometries. This extreme variability is well represented by the R1-O-H....:O-R2 bonds that, according to the choice of R1 and R2, can span d(O...O) values from 2.38 to 3.00 Angstroms and d(H...O) ones from 1.20 to 2.00 A, while the bonding energy collapses from 25-30 kcal/mol (~1.0 eV) to less than 1 kcal/mol (~0.04 eV).

The elephant in the lecture theatre

There is an interesting (but long) article in The Australian Higher Education section Hitting the books, or booking the kegs by Glyn Davis. 
(Aside: He is Vice-Chancellor of Melbourne University, where he pioneered a major reconstruction of the undergraduate program, known as the Melbourne Model, inspired by the Bologna Process. I believe this was a virtually unprecedented large scale change in an Australian university, reducing the number of degree options from 96 to 6!)
Much of the article refers to the book, Academically Adrift by Arum and Roksa, which I mentioned in an earlier post. I just mention here a particularly interesting quote from Davis' article:

Like many critics of the American college model, Arum and Roksa argue that incentives are skewed towards research to the detriment of student learning.

This is a situation, suggest Arum and Roksa, that nobody on campus is keen to challenge. American students want to acquire a degree with minimum effort, academics want to devote as much time as possible to research and administrators want to focus on the institutional rankings, which take no account of learning quality. James Wilkinson, director of the Derek Bok Centre for Teaching and Learning at Harvard, reflected this concern in a 2006 Menzies Oration. Said Wilkinson: "It used to be said in the former Soviet Union, where the state was the sole employer, that 'they pretend to pay us, and we pretend to work'. Today, at least in the US, we could emend this cynical observation to read: 'they pretend to teach us, and we pretend to learn'."

Watching damage control in real time

Today I read a nice Nature paper Dynamics and mechanism of repair of ultraviolet-induced (6–4) photoproduct by photolyase from Dongping Zhong's group.

One way that UV can damage DNA (and cause mutations, kill cells, and cause skin cancer) is by fusing together two adjacent pyrimidine rings to produce a molecule 6-4PP.
The figure to the right (taken from here) shows the associated rearrangement of atoms.

What is amazing is that nature has a way to repair this mutation involving a flavoenzyme called photolyase. This paper elucidates (using ultrafast laser spectroscopy) the repair mechanism which is summarised in the schematic below. It contains steps involving electron transfer and proton transfer.

Thermoelectric power in strongly correlated Fermi liquids

Finding universal dimensionless ratios has proven key in understanding both elemental metals and strongly correlated Fermi liquids. Examples of important ratios include those associated with the names Sommerfeld-Wilson, Korringa,  Lorenz, and Kadowaki-Woods. Here is another one...

Today I read I nice article On the thermoelectricity of correlated electrons in the zero-temperature limit by Kamran Behnia, Didier Jaccard and Jacques Flouquet. The graph below shows evidence for a universal ratio for a wide range of materials. The ratio of the slope of the thermopower S(T) versus temperature to the specific heat coefficient gamma equals +/-1/eNA where NA is Avagadro's number and the sign depends on whether the charge transport is via electrons or holes.

Note the log-log scale which covers three decades.

This is the value of the ratio expected for a non-interacting fermion gas. It would be slightly different in Mott's formula for the thermopower with an energy dependent scattering rate [such subtleties are probably lost on the log-log scale]. 

A universal ratio for an Anderson impurity model was predicted by Houghton, Read, and Won [amongst others and discussed in Appendix E of Hewson's book on the Kondo effect].

A discussion of the temperature dependence of the thermopower for a DMFT (Dynamical-Mean-Field-Theory) treatment of the Hubbard model is here.

Being critical about quantum criticality

Recently I read a nice paper by Mike Norman and Andrey Chubukov, High Frequency Behaviour of the infrared conductivity of the cuprates. The paper addresses issue raised by a 2003 Nature paper Quantum critical behaviour in a high-Tc superconductor. The Nature paper gave a detailed analysis of the frequency dependence of the real and imaginary parts of the frequency dependent conductivity of an optimally doped cuprate superconductor. In particular, the data is compared to theoretical work of Anderson and Sachdev. A key figure is the one below. The upper panel shows that the magnitude of the conductivity vs. frequency exhibits a power law dependence over approximately one decade of frequency with a fractional exponent. Anderson has also invoked this power law dependence as evidence for his Hidden Fermi liquid theory.

The lower panel shows that the phase angle of the conductivity is approximately constant and equal to 60 degrees over a large frequency range. This is what was predicted by Anderson.

However, Norman and Chubukov show that there are two alternative explanations of this data, both involving electrons interacting with a broad spectrum of bosons. They work in terms of a frequency dependent (and momentum independent) self energy and no vertex corrections to the conductivity. This formalism was originally developed by Allen in 1971 to describe the effects of the electron-phonon interaction on the optical spectrum of metals. Norman and Chubukov obtain quantitative fits to the data with a self energy which has either a Marginal Fermi liquid form or from a Lorentzian spectrum of overdamped bosons (e.g., spin fluctuations).  What are the implications of this work? Here is my paraphrase of their own conclusions

• the apparent scaling behavior over a wide frequency range may be unrelated to quantum criticality but rather just a consequence of the flattening of the frequency dependence of the scattering rate which is accompanied by a corresponding decrease in the real part of the self energy.
• the behavior of the single particle and optical self-energies is very similar for the marginal Fermi-liquid phenomenology, and the Lorentzian model used in microscopic fermion-boson theories. 
• the data give strong evidence for an upper cutoff of the boson spectrum of around 300 meV. A cutoff of this scale was suggested in the original marginal Fermi-liquid phenomenology. Such a large energy scale would imply that the source of the boson spectrum is collective electronic excitations rather than phonons. Inelastic neutron scattering data show magnetic spectral weight up to this energy scale and, thus, spin fluctuations or possibly other electronic excitations are a natural explanation for the boson spectrum. 

Aside: To me, this all underscores the importance of the method of multiple alternative hypotheses in theory development.

Are students customers?

A colleague at another university recently told me he was aghast his university was now referring to students as "customers".  To see the problem with this terminology, consider the fact that:

• The customer is always right.
• The customer does not do any real work. They just pay for products and services.
• The customer is not really accountable to anyone.

University faculty and administrators do need to do everything they can to create a stimulating environment which gives students opportunities to learn.
Students are not customers. Students are students.

Desperately seeking universal relations for unconventional superconductors

In 2004 Chris Homes et al. published a Nature paper describing a (fairly) universal scaling relation between the superfluid density and the product of Tc the transition temperature and the dc conductivity at T=Tc. This is a generalisation of the Uemura relation which describes underdoped cuprates.

Jan Zaanen wrote a News and Views piece, Why the temperature is high for the Homes article. It features the figure above. Zaanen combined the Homes relation with the Drude formula for the dc conductivity, "Tanner's law" which found that for optimally doped cuprates the superfluid density was approximately one-quarter of the density of "mobile electrons" defined by the low energy spectral weight (Drude weight) in the optical conductivity. This results in a scattering time which is approximately hbar/ 2 pi kB T. He "sexes" this up by calling it "Planckian dissipation".
I wonder if this is really anything more than the Marginal Fermi liquid picture with the dimensionless coupling constant equal to unity.

Aside: Later Frances Pratt and Steve Blundell showed in a PRL that Homes scaling relation does not hold for molecular superconductors. Previously, Ben Powell and I had noted how the organic charge transfer salts deviated significantly from both a simple BCS picture and the phase fluctuation picture proposed by Emery and Kivelson to explain the Uemura relation in the cuprates.

Romantic reductionism

Which is more fundamental, neuroscience or string theory? A debate of this profound question even featured on prime time TV.

Another signature of hydrogen bonding

Previously I posted about how significant redshifts in the frequency of the O-H bond stretch can be a signature of hydrogen bonding. Furthermore, there is a fairly universal relationship between the frequency shift and the bond length and energy.

The graph above shows there is also a fairly universal relationship between the bond energy (horizontal scale) and the change in intensity of the O-H infrared absorption intensity, covering a factor of 200 fold variation in the energy. The details are summarised in this paper. The vertical scale is the bond energy calculated from the empirical relation −ΔH=12.2 ΔA^1/2 where ΔH is the enthalpy in kJ/mol and  ΔA is the change in intensity of the absorption line in units of 10⁻⁴ cm mmol⁻¹.

The scientist who cried "Breakthrough!"

The Boy who cried wolf is one of Aesop's fables. It illustrates the problem of losing credibility due to developing a reputation of make false claims in order to get attention. Yet, the tragedy occurs when a true statement is made; it is ignored because of the lack of credibility of the speaker.
Does it apply in science? I think so. After you have been in science for a while I think there are certain people you just ignore because of previous experience of seeing their claims not bearing the test of time.

The name is Bond, Hydrogen Bond

There is a fascinating (and useful) essay A Bond by Any Other Name by Gautam R. Desiraju in Angewandte Chemie International. It discusses the background behind the recent new definition of a hydrogen bond by IUPAC [International Union of Pure and Applied Chemistry]. [A detailed report is here]. Here is the new definition:

The hydrogen bond is an attractive interaction between a hydrogen atom from a molecule or a molecular fragment XH in which X is more electronegative than H, and an atom or a group of atoms in the same or a different molecule, in which there is evidence of bond formation.

A typical hydrogen bond may be depicted as XH⋅⋅⋅YZ, where the three dots denote the bond. XH represents the hydrogen-bond donor. The acceptor may be an atom or an anion Y, or a fragment or a molecule YZ, where Y is bonded to Z. In specific cases X and Y can be the same with both XH and YH bonds being equal. In any event, the acceptor is an electron-rich region such as, but not limited to, a lone pair in Y or a π-bonded pair in YZ.

Amusingly, one section of the paper is entitled, "The Name is Bond".
More seriously, the paper has a useful section 5.1 List of Criterial useful as evidence (for hydrogen bonding).

The quantum of the gaps

I remember a prominent quantum physicist excitedly describing to me with great passion some unusual experimental observations concerning molecular motors. He said that there was no clear explanation for how they worked and so thought quantum effects must play a role!

I think there is a similar approach (fallacy) with other proposals for "quantum biology", whether it is understanding consciousness (Roger Penrose), evolution (McFadden), smell (Luca Turin), or photosynthesis (a cast of many). 

This reminds me of a phrase, "The God of the gaps" which was coined by one of my heroes, Charles Coulson. Although, best known as one of the founders of quantum chemistry, Coulson also wrote several books about Science and Christianity. Coulson pointed out the fallacy of invoking God to explain what was not currently understood in science.

Entanglement in the 2 impurity Kondo model

At the cake meeting this week we discussed a  nice (short) review of the Kondo effect  by Barbara Jones. [It appeared in the Handbook of Magnetism and Magnetic Materials]. Much of the emphasis is on the two-impurity Kondo model which exhibits competition between the RKKY interaction I of the two impurity spins and the Kondo interaction J of each spin with the conduction electrons.
Of particular interest is the non-Fermi liquid fixed point (quantum critical point) which occurs for I=-2.2TK.

In 2006 Sam Young Cho and I published a paper in Phys. Rev. A which quantified the entanglement between the two impurity spins. A striking thing is that this entanglement vanishes at the quantum critical point. The quantum state of this model also provides a solid state realisation of Werner states [mixed states which have the intriguing property of being entangled but not violating Bell inequalities].

A simple challenge to theorists

[I am] concerned with the spirit that seems to pervade much of theoretical physics nowadays, and with the interests and preferences shown by some of its practitioners. Of course, theoreticians depend on their experimenter colleagues for the bread and butter of their work,  ..... And yet when I try to read some of the current literature or listen to some of my colleagues, I can't help asking: Are they in touch with reality? Are they seriously interested in physics? As an example let me cite antiferromagnetism, which is again a popular topic since the discovery of the superconducting copper oxides. Very sophisticated models of antiferromagnetism have been proposed and worked out in great detail by people who have neither heard of, nor do they care to learn about, hematite, or what everybody else knows as a rusty nail....

... Even among theoreticians the natural order of things should be electron-spin resonance in hemoglobin first, anti-unitary operators and quaternions second....

As a referee for scientific journals I have led a losing battle with authors who don't want to (or can't) explain their general ideas in terms of simple experimental systems, or even in terms of high school geometry as the best in our trade did, from Einstein to Feynman.

Martin Gutzwiller, Two questions physics can no longer avoid, Letter to Physics Today, August 1994.

Quantifying vibronic entanglement

I just finished a paper, Quantum entanglement between electrons and vibrations in molecules, with Laura McKemmish, Noel Hush, and Jeff Reimers.

We consider a simple model Hamiltonian which describes two quantum states interacting with a single vibrational mode [alternatively known as the one mode spin-boson model, the Herzberg-Teller model, or the E x beta Jahn-Teller model].

A couple of things I learnt:
*Realistic model parameter values for six molecules including ammonia, benzene, and semibulvalene.
*The large entanglement which occurs in the strong coupling (adiabatic) limit can be quite "fragile". i.e., it can be destroyed by  a small asymmetry in energy. Compare the top left two boxes in the figure below which shows a colour-shaded plot of the entanglement as a function of the Hamiltonian parameters.

*In contrast, in a regime where all the energy scales are comparable, the entanglement is much more robust.

A curious side anecdote about this paper. Last week we first sent the paper to Physical Review A. However, an editor did not consider it would be "of interest to their readers" and so would not send it out for review. I found that rather disappointing. I wondered if that was because there was a passing reference to Penrose and Hameroff in the conclusion. So, we removed that reference and sent the paper to Journal of Chemical Physics. I would be curious to hear from "Phys. Rev. A readers" whether they think the paper is of interest.

Deconstructing climate change and energy options

Physics Today has a very helpful review of three important books by physicists about climate change and possible future options.

Beyond Smoke and Mirrors: Climate Change and Energy in the 21st Century by Burton Richter

Sustainable Energy - Without the Hot Air by David MacKay

Challenged by Carbon: the Oil Industry and Climate Change by Bryan Lovell.

Fermi liquid transport properties without quasi-particles

Today I encountered the following apparent puzzle. Suppose one has system with a self energy which is the sum of an impurity term and a marginal Fermi liquid self energy. One consequence is that the real part of the self energy is logarithmically divergent at low temperatures. Consequently, the quasi-particle weight vanishes for energies at the chemical potential.
If transport properties (such as the dc conductivity and thermal conductivity) are calculated from bubble diagrams ignoring vertex corrections then it seems the resulting expression only depends on the imaginary part (and not the real part) of the self energy. Consequently, at low temperatures the transport is dominated by impurity scattering and universal Fermi liquid properties such the Wiedemann-Franz law (and the Lorenz ratio) are obeyed.
Thus it seems one can have traditional Fermi liquid signatures without quasi-particles!

This was all stimulated by reading a nice 2002, PRL Heat Transport in a Strongly Overdoped Cuprate: Fermi liquid and a Pure d-wave BCS Superconductor. They observe that the Lorenz ratio has its universal value (to within about 1 %). I was wondering whether these observations at low temperatures had implications for recent work I did with Jure Kokalj, Consistent description of the metallic phase of overdoped cuprate superconductors as an anisotropic marginal Ferm liquid. My current view is that these experiments cannot be used to rule out a marginal Fermi liquid contribution to the self energy, but I welcome comments.

5 Papers every computational chemistry student should read

I have a dream. That every advisor (supervisor) who gets a student to perform a computational chemistry calculation will have them read the following five papers. The papers are from a range of eras and with different emphasis. But, a common theme is the importance of calculations aiding concept development and being aware of the limitations these calculations.
Reading these papers should be like reading the road rules before you get your drivers license.
I list the papers in chronological order.

Present state of molecular structure calculations
C.A. Coulson (1960)
Quantum chemistry and its unachieved missions
Jean-Paul Malrieu (1998)
Is my chemical universe localized or delocalized? is there a future for chemical concepts?
Sason Shaik (2007)
Predicting Molecules - More realism , please!
Roald Hoffmann, Paul Schleyer, and Fritz Schaefer (2008)
Some Fundamental Issues in Ground-State Density Functional Theory: A Guide for the Perplexed
John P. Perdew, Adrienn Ruzsinszky, Lucian Constantin, Jianwei Sun, and Gabor Csonka (2009)

I welcome alternative suggestions. Later I may write more about the individual papers and why I think they are important.

Seeing how degenerate radicals can be

I have been slowly digesting a really nice combined theoretical and experimental paper The Lowest Singlet and Triplet States of the Oxyallyl Diradical which was featured on the cover of Angewandte Chemie in 2009 [For physicists this is the European counterpart to the prestigious JACS = Journal of the American Chemical Society].
[See also the commentary by Bettinger].

Here are a few interesting things I learnt:

• The ground state is a singlet but only 55 meV in energy below the lowest lying triplet. [In most organic molecules the energy difference is ~ 1-2 eV].
• C-C-C angle bending has a frequency of about 400 cm-1 and couples to the electronic transitions. 
• Another vibrational mode which couples strongly to electronic transitions is the C-O stretch [with a frequency of order 1700 cm-1].
• The singlet state is unstable to "disrotatory ring closure" to form cyclopropanone. 

The relevant valence bond structures are

and provide a natural framework to understand the above observations.

This paper is of particular interest to me because the oxyallyl diradical is a simple example of a methine dye [cf. green fluorescent protein, Malachite green] and a ketocyanine dye whose minimal quantum chemical description requires 4 electrons in 3 orbitals. [To physicists 4 electrons in a 3 site Hubbard model]. The paper though uses the framework of 4 electrons in 4 orbitals due to the analogue with trimethylenemethane (TMM) which actually has a triplet ground state. TMM is one of the simplest non-Kekule molecules.

Molecules of chocolate

The Journal of Chemical Education paper on chocolate based demonstrations discusses three key classes of molecules.
Triglyceride is a major component of cocoa butter. It is hydrophobic.

Serotonin is a major component of cocoa powder. It is is largely hydrophilic and so will dissolve in water.

Lecithin is an emulsifier [just like egg which leads to formation of a stable emulsion of oil and vinegar in a salad], an amphiphilic molecule, which promotes mixing of cocoa solids and cocoa butter.

  

Interlayer magnetoresistance in a pseudogap metal

I have been working through a really nice paper, Fermi surface of the electron-doped cuprate superconductor Nd_2–xCe_xCuO₄ probed by high-field magnetotransport by Mark Kartsovnik and collaborators.

The phase diagram of these electron-doped [in contrast to the more common hole-doped cuprates] materials is shown below. x is the Ce content. PG denotes a pseudogap phase.

(b) Shows the Fermi surface expected for x > 0.16 (e.g. from a tight binding model and DFT based calculations) and confirmed by Shubnikov de Haas (SdH) oscillations.

 (c) shows how this Fermi surface may be re-constructed due to a (pi,pi) superlattice potential (which might exist due to co-existing antiferromagnetic (AF) order.

I found the interlayer magnetoresistance measurements shown below particularly interesting. Each curve shows the interlayer resistivity as a function of magnetic field direction (theta= tilt angle from the normal to the layers) for a fixed magnetic field and temperature.

[Above some large angle the resistance goes to zero because the component of magnetic field perpendicular to the layers becomes less than the upper critical field needed to destroy the superconductivity].

What is interesting about these curves?

• They exhibit significant qualitative differences depending on the doping x, even over the narrow range 0.13 < x < 0.17.
• This is probably because the pseudogap has a big effect on the magnetoresistance.
• For x=0.13, 0.15 the dependence on the azimuthal direction (phi) of the magnetic field is very weak (and opposite in sign) compared to that for x=0.16, 0.17.

The only theory of the interlayer magnetoresistance in the presence of a pseudogap that I am aware of is a paper by Michael Smith and myself in 2009. Most of that paper focuses on the case of a field parallel to the layers and shows how the azimuthal angular dependence may reveal the anisotropy of the pseudogap. We did find that the pseudogap reduces the azimuthal anisotropy compared to the anisotropy seen in the normal phase due to anisotropy of Fermi surface properties. This is basically because the pseudogap suppresses large parts of the Fermi surface from contributing to the interlayer conductivity. That physics may be relevant for understanding the data shown above, but a detailed calculation is desirable.