Scaling plots near the Mott transition

Earlier this year Jure Kokalj brought to my attention an interesting PRL
Quantum Critical Transport near the Mott Transition
by H. Terletska, J. Vučičević, Darko Tanasković, and Vlad Dobrosavljević.

My interest in this paper increased this week when Vlad emailed me to tell me about a recent talk at KITP by Kazushi Kanoda. The right side of the slide below [click on it to see it larger] shows a scaling analysis of the temperature and pressure dependence of the resistivity of the organic charge transfer salt kappa-(ET)2(CN)3 near the pressure driven Mott transition. This scaling analysis is based on the theory in the PRL.

The left side shows the Dynamical Mean-Field Theory (DMFT) results [for a Hubbard model at half filling] in the PRL. The top shows the scaling of the resistivity curves and the bottom the T vs. U phase diagram where the yellow region is the "quantum critical" region above the Mott transition.

It is striking that the experimental curves involve scaling over about 6 orders of magnitude of the resistivity.

There are several things that are rather strange/exotic/interesting about the theory. 

First, there is a "duality" between the resistivity in the metallic and insulating sides of the transition.

Second, the critical resistivity is an order of magnitude larger than the Mott-Ioffe-Regel limit.

Third, connections are suggested with the "holographic duality" picture of in papers such as this one by Subir Sachdev.

Aside: This same organic material is also of considerable interest because the Mott insulating phase seems to exhibit a spin liquid ground state and metallic state becomes superconducting at low temperatures. (See this review).

Public accountability gone amok?

Physics Today has a fascinating review by Naomi Oreskes of the book The Hockey Stick and the Climate Wars by Michael Mann.
In the Print edition the review is entitled, "A call to throw caution to the wind".
I suggest reading the one page review in full with two questions in mind:

1.
How would you personally cope with the level of public scrutiny (and attack) that Mann was subjected to following publication of his 1998 Nature paper?
e.g., having emails and grant applications subject to subpoenas from politicians. The Wikipedia page on Mann describes the many investigations he has been subjected to.

2.
Is it really possible for non-scientists to understand and realistically evaluate the evidence for and against scientific hypotheses concerning complex systems?
Oreskes seems to claim not in this political context and suggests Mann should not have been so cautious with his public pronouncements about ambiguities and subtleties in the data.

If people post their answers to one or both of these questions I will then will try and give mine.

d-wave pairing in superconducting organics

We now have significance experimental evidence which should end any debate about whether the superconductivity in organic charge transfer salts is unconventional.

There is a nice preprint
In-plane superfluid density and microwave conductivity of the organic superconductor κ-(BEDT-TTF)2Cu[N(CN)2]Br: evidence for d-wave pairing
S. Milbradt, A. A. Bardin, C. J. S. Truncik, W. A. Huttema, P. L. Burn, S.-C. Lo, B. J. Powell, D. M. Broun

They provide definitive measurements of the temperature dependence of the superfluid density and of the quasi-particle scattering rate. What is distinctly new and exciting about these very careful and precise measurements is the absolute determination of these quantities.

The results also highlight some of the similarities of the organics with the cuprates: a d-wave superconductor "derived" from an antiferromagnetic Mott insulator and that quasi-particle scattering is largely due to electron-electron interactions. Indeed, the key temperature dependences shown below are essentially the same as what is seen in the cuprates.

There are three clear experimental signatures of d-wave pairing:

• the linear temperature dependence of the superfluid density (see figure above)
• the T^3 dependence of the scattering rate in the superconducting state
• the absence of a BCS "coherence" peak in the conductivity near Tc

I particularly like the figure below. It shows how the quasi-particle scattering rate drops dramatically in the superconducting state because the electronic degrees of freedom  responsible for the scattering are being "gapped out" by the superconductivity.

Is open access comical?

There is a video clearly explaining the case for Open Access to research journals. It is illustrated by Ph.D comics. I thank Tony Wright for bringing it to my attention.



A previous post The insatiable greed of commercial journals discussed some of relevant political background in the USA.

To curtail the influence the role of commercial journals, I think the least we can do is aim to mostly publish in journals run by professional societies such as American Physical Society and the American Chemical Society.

I also think research groups should be putting their "raw" research data sets online so others can analyse them independently.

I welcome comments.

The quality of agreement and comparisons

Many authors like to make statements such as "our calculated value of 1.23 for X is in excellent agreement with the experimental value of 2 for X".

In contrast, this week I read the modest statement "the DMFT value of 0.66 is in mediocre agreement with the Brinkman-Rice value of 0.47" in this review [below eqn. (266)].

I think the quality of agreement in such comparisons is in the eye of the beholder. Different people can have quite different standards. Furthermore, the level of agreement one might hope for depends strongly on the context and many other factors. e.g., how many free parameters there are, how difficult it is to calculate or measure the relevant quantity, and how sensitive the calculated quantity is to the level of theory.

Let me illustrate with two statements a hypothetical author might write in the future:

1.
"my new formulation of renormalisation in quantum electrodynamics (QED) leads to value for the g-factor of the electron that agrees with experiment to one part in a thousand"

2.
"my new quantum field theory of gravity leads to a value of the cosmological constant which differs from the observed value by a factor of one hundred."

To the novice, 1. sounds very impressive and 2. sounds lame.
However, paper 1. would be ignored and paper 2. may lead to a Nobel Prize!
Why?
Existing versions of renormalisation in QED give values of g that agree with experiment to about 15 significant figures!
Existing formulations of the cosmological constant in quantum field theory give values that differ from experiment by about 100 orders of magnitude!
[Weinberg has a nice looking review on this that I hope to read sometime....]

The solution to this problem?
I think we should simply write: "our calculated value of 1.23 for X can be compared with the experimental value of 2 for X".
Let the reader decide for themselves whether they think the "agreement" is excellent, impressive, surprising, disappointing, mediocre, poor, .....

Searching for the nodes in an unconventional superconductor

In the cuprate high-Tc superconductors it is now well established experimentally that the superconductivity has d_x^2-y^2 symmetry with nodes in the energy gap along the Brillouin zone diagonals. This is what is predicted by spin fluctuation and RVB theories.

In the organic charge transfer salts the experimental situation is not as clear. There are clear theoretical predictions of unconventional superconductivity with nodes in the gap.
Hence, I was interested to read the paper:

Location of gap nodes in the organic superconductors kappa-(ET)2Cu(NCS)2 and kappa-(ET)2Cu[N(CN)2]Br determined by magnetocalorimetry
L. Malone, O.J. Taylor, J.A. Schlueter, and A. Carrington

This experiment is difficult [measuring the specific heat at low temperatures while rotating a magnetic field within the layers] and its interpretation is subtle. The conclusion is that the nodes are as shown below. This is where they are predicted to be by both spin fluctuation and RVB theory. (A nice article by Ben Powell discusses the relevant theory).

The authors (helpfully and honestly) point out that these results are inconsistent with some thermal conductivity measurements. 

However, Tony Carrington pointed out to me:

The point is that the results could be consistent if they are in different parts of the B/T phase diagram.  The problem is that in both thermal conductivity and specfic heat measurements the sign of the oscillations changes as a function of B and T.  In the low B, low T limit both give a maximum when B is in the antinodal direction, but this becomes a maximum at higher T and higher B.  For thermal conductivity this occurs at a different temperature compared to the specific heat (according to Vekhter's model).  So both could be consistent with the same nodal structure.

Something that would be particularly interesting would be to do similar measurements on  a different organic charge transfer described in an earlier post because RVB theory predicts the nodes would be in a different location.

How many authors is too many?

Physics Today recently published an excellent commentary Too Many authors, two few creators by Philip Wyatt. It is worth reading in full.

He compares the number of authors per paper in 8 major journals in 1965 and 2011. The number of single authors has plummeted while the number of papers with 4+ authors has dramatically increased.
In contrast, the number of authors of patent applications has remained steady.
Wyatt argues that this reflects a decrease in creativity. This is all caused by the pressure for everyone to have large numbers of publications and citations to survive in science.
He asserts that many authors have not made significant scientific contributions to a paper and so should only be in the acknowledgements section.
He sings the praises of single author papers, as I do.

Why has not the number of authors on patent applications increased?

If a listed inventor, or “innovator,” did not actually contribute to the invention, the issued patent will be void if such deception is ever discovered. The patents most easily challenged in court may well be those with extraordinary numbers of inventors.

Overall I agree with Wyatt's concerns.
Some people are getting credit they don't deserve.
Multiple authors make it very hard to evaluate the quality and quantity of individuals contributions.
Too much time is being wasted producing papers as opposed to doing creative and productive science of lasting value.

The commentary brought a number of interesting responses. I particularly agreed with his  response included the statement:

so much effort seems focused on writing papers just to secure funding or a job that many fundamental building blocks needed as tools to spur creativity in our young scientists are lost in the process.

What is my information diet?

Like most scientists I don't think I have a very balanced or healthy or disciplined diet. It has changed over the years, both with seniority and the internet.

I tend not to work on highly fashionable topics and so don't feel I have to look at the arXiv each day or the Table of Contents of journals. I hope (and expect) that postdocs or collaborators will bring to my attention particularly noteworthy recent papers or articles. I greatly appreciate it when they do. Colleagues also kindly send me some their recent work that they think may be of interest. I appreciate this and unlike some do not begrudge it as shameless self-promotion. On the other hand I don't appreciate being sent stuff which is of marginal or no relevance.

A lot of stuff I need I find with Google Scholar using carefully chosen word searches.

About 15 years ago, I believe I once heard Doug Stone claim that one thing he learnt from Patrick Lee was that he did not need to keep up with journals. Why? If there was some new important development he needed to know about someone would come through his office door and tell him about it.
I don't rate myself in this class. But it is an insight into how some high-profile people operate.

I only go to a couple of conferences per year. I also visit a few departments each year and talk with colleagues individually about what they are working on. These are a good way to catch up.

What is a balanced diet?
I am not sure.
One has to balance Nature/Science and PRL and PRB and others.
One also has to balance theory and experiment.
Furthermore, some of us have to balance chemistry and physics.
One also has to balance papers of broad interest, reviews, foundational references, and latest results.

In general I spend too much time downloading, printing and not reading. I also probably read too much chemistry and experiment, rather than theory. On the other hand, some of my past main successes have been built on that focus.
I also probably read too many papers superficially rather than properly understanding a few.

For the last year, I have been getting the weekly update from Science. But, the main reason is not to find out which condensed matter physics group got a paper in! Rather, I try to read some of the summaries of the biology, ecology, climate change, geology, chemistry, and psychology papers. I usually struggle to understand but it is a nice intellectual challenge. I hope to learn just a couple of basic things...

I thank Ted Sanders for suggesting this post.
If it seems a bit rambling it is because I don't really have a handle on this issue.

I welcome suggestions and ideas, from both junior and senior people, on how they manage this problem.

Significance of the Kondo paradigm

This week we am part of new reading group which aims to work through Hewson's beautiful book The Kondo Problem to Heavy Fermions.
Why the choice of this topic?
Is it only of interest and relevance to people working on the Kondo physics and/or heavy fermion compounds?
I would say definitely no!

The Kondo problem represents a very important paradigm in quantum many-body physics. Perhaps the other main (well established and accepted) paradigms are

• Fermi liquid theory,
• BCS theory of superconductivity,
• Mott insulators, 
• spontaneous symmetry breaking
• quantum antiferromagnets (Heisenberg models and Anderson superexchange),
• Fractional Quantum Hall effect

Asides:
1. Perhaps Mott insulators should not be on the list because there is no well-accepted theory beyond the "zeroth-order approximation".
2. Two other paradigms that I believe will eventually be accepted are Dynamical Mean-Field Theory and the RVB theory for superconductivity in proximity to a Mott insulator.
3. Is the shortness of the list a reflection of our success or failure? Is there a need for only a few paradigms or is it just we have solved so few problems?

Furthermore, the Kondo model/problem exhibits rich physics and key concepts in quantum many-body theory including

• non-perturbative effects
• asymptotic freedom: weak coupling scales to strong coupling as the temperature/energy decreases
• scaling and universality [a single energy scale: the Kondo temperature]
• the only (?) well-established realisation of a non-Fermi liquid fixed point [in multi-channel problems]
• a simple physically transparent variational wave function [Yosida]

It is a benchmark for testing numerical and analytical methods since it can be solved exactly analytically using the Bethe ansatz and numerically using the Numerical Renormalisation Group. Its fixed points can be described by boundary conformal field theory.

Furthermore, Kondo physics is at the heart of the Dynamical Mean-Field Theory (DMFT) treatment of the Mott metal-insulator transition and electronic structure methods (e.g. DMFT+LDA).

Finally, Nozieres says the Kondo effect "is typical of what real theory should be, using tortuous roads towards simple final results".

I welcome additions and subtractions to my lists.

The Higgs boson in social context

A week ago I gave a talk "The Higgs Boson: scientific reality vs. media hype" at the Centre for Science, Religion, and Society at Emmanuel College at UQ. You can see the slides on my other blog.

Why do I keep blogging?

Sometimes I get asked, "Why do you do it?"
"How much time does it take?"

It does require significant time. I estimate I spend an average of 30-60 minutes per post and I try to make one post per weekday.

But, why do it?

1. It is fun.
Why? Probably because I like learning and understanding things. If I have to write something concrete about a paper or an idea then I am forced to understand it better and more carefully think through the basis of my opinions.

2. It saves time.

-On career advice I don't have to keep repeating myself to students and postdocs. I can just refer them to posts on the relevant topics.

-I (or a reader) may detect mistakes in my thinking, that might have gone undetected for a period, wasting time.

-blog=Web log=diary=note book.
It is a good source of notes. Searching old posts I can find information and ideas I have already forgotten about or might be lost otherwise.

-hopefully it is a useful source of information about me (both science and philosophy) for prospective students, postdocs, and collaborators.
This will attract the like-minded and scare off those with different interests and values.

Finally, progress and achievement in science is very slow.
Making a post gives me a tangible sense of achievement for each day.

So even if no one read it I think I would still do it. But, the positive feedback I get is encouragement to keep going. The best encouragement is when a post generates some constructive debate and discussion.

How good metals turn bad

There is a really nice preprint
How bad metals turn good: spectroscopic signatures of resilient quasiparticles
by Xiaoyu Deng, Jernej Mravlje, Rok Zitko, Michel Ferrero, Gabriel Kotliar, and Antoine Georges

They study a doped Hubbard model using Dynamical Mean-Field Theory (DMFT). Although the ground state is a Fermi liquid this is only a good description at very low temperatures. In particular, the quadratic temperature dependence (characteristic of a Fermi liquid) only occurs below a temperature of about 0.05 delta D [where delta=doping and D=band width]. But, well-defined quasi-particles still exist all the way up to the "bad metal" region at which the mean-free path is comparable to the lattice constant.

I found this resilience of quasi-particles somewhat surprising [I am not quite sure why] and interesting. Furthermore, in this intermediate temperature regime there is large entropy and significant local magnetic moments, and Kelvin's formula gives a good description of the temperature dependence of the thermoelectric power.

The strong correlations lead to a significant particle-hole asymmetry (in the self energy and single-particle density of states) via the subtle interplay of the lower Hubbard band with the quasi-particle peak in the density of states.

To me this all illustrates how much (but obviously not all) of the rich physics seen in strongly correlated materials can be captured by DMFT.

Note added. I am told that the axis label on the left figure is in error and it should be 1/Z not Z, i.e. Z actually increases with temperature.

Extracting Berry's phase from experiments on topological insulators

Tony Wright and I just finished a paper
Quantum oscillations and Berry's phase in topological insulator surface states with broken particle-hole symmetry

Quantum oscillations [e.g., Shubnikov de Haas] can be used to determine properties of the Fermi surface of metals by varying the magnitude and orientation of an external magnetic field. Topological insulator surface states are an unusual mix of normal and Dirac fermions. Unlike in graphene and simple metals, Berry's geometric phase in topological insulator surface states is not necessarily quantised. We show that reliably extracting this geometric phase from the phase offset associated with the quantum oscillations is subtle. This is especially so in the presence of a Dirac gap such as that associated with the Zeeman splitting or interlayer tunneling.

We develop a semi-classical theory for general mixed normal-Dirac systems in the presence of a gap, and in doing so clarify the role of topology and broken particle-hole symmetry. We propose a systematic procedure of fitting Landau level index plots at large filling factors to reliably extract the phase offset associated with Berry's phase.

Comments and feedback would be very welcome.

Connecting the pseudogap to superconductivity in organics

For the cuprates an outstanding question concerns whether there is a connection between the pseudogap and superconducting states. The same question arises in organic charge transfer salts, although it has received little attention. For the organics evidence for the pseudogap is rather indirect, mostly coming from NMR, which shows a reduction in the Knight shift and the relaxation rate below about 50 K. There is no direct evidence of nodes in the pseudogap.

A recent PRB presents evidence from ultrasound measurements that there is an intimate connection between the two.
Symmetry-imposed signatures at the pseudogap crossover in κ-(BEDT-TTF)2X organic superconductors
by Mario Poirier, Maxime Dion, and David Fournier

The figure below shows the temperature dependence of different elastic moduli near the temperature at which the pseudogap opens.

A post from last year considered in detail an earlier paper by the same group [plus their theory colleague A.M. Tremblay] that considered the anisotropy in the acoustic anomalies at the superconducting transition.

Note that the anisotropies are the same for both transitions, suggesting that the coupling of the relevant order parameters to the different components of the elastic tensor is the same.

This earlier paper presented a theory for the elastic anomalies near the superconducting transition. This theory would mean that the pseudogap and the superconducting order parameter have the same symmetry.
My earlier post raised some concerns about the theory, including that it predicts a symmetry distinct from the B2g symmetry predicted by RVB and spin fluctuation calculations (and this recent experimental paper).
Hence, I think there is a need for a slightly different theory of the acoustic anomalies. Nevertheless, I don't think that will change the main conclusion of the current paper: that the pseudogap and the superconducting order parameter have the same symmetry.
Indeed, this was the last sentence of a 2005 PRL that Ben Powell and I wrote about an RVB theory of these materials.

Not all iron pnictide superconductors are the same

I am not working on the new iron pnictide superconductors and so am only casually following the field. Here is one recent paper that got my attention.

Nodal versus Nodeless Behaviors of the Order Parameters of LiFeP and LiFeAs Superconductors from Magnetic Penetration-Depth Measurements
K. Hashimoto, S. Kasahara, R. Katsumata, Y. Mizukami, M. Yamashita, H. Ikeda, T. Terashima, A. Carrington, Y. Matsuda, and T. Shibauchi

The temperature dependence of the London penetration depth [which determines the superfluid density] is strikingly different for LiFeP and LiFeAs. This suggests that for the former there are nodes in the superconducting gap.

Furthermore, they correlate the presence of nodes in the superconducting energy gap with  the height of the pnictogen atoms (Pn=As or P) above the plane of the Fe atoms in the crystal structure.

Overselling cold atoms

A few previous posts about BECs in dilute atomic gases show that I am at times skeptical about some of the claims made by members of the cold atom community. Those posts also generated some interesting and worthwhile comments.

Yesterday I endured an irritating seminar about realising spin-orbit coupling, topological superconductors, and Majorana fermions in cold atom systems. It was claimed that all of the problems and ambiguities with observing these effects in condensed matter systems could be solved in cold atom systems. I wish this were true. However, it seemed to me that the complexities and challenges associated with the speakers proposed cold atom realisation was just as great. The speaker made the fundamental mistake, never offer undefendable ground.

I mention this because I have heard several cold atom talks (and reviewed grant applications) like this. Basically, there is a lot of hubris and hype. There is also ignorance of the existence of standard condensed matter techniques (e.g. inelastic neutron scattering and scanning tunneling microscopy) and of history (e.g. the Josephson effect).
But, I realise not everyone in the cold atom community is like this.

Quantum many-body systems are extremely difficult to study, both theoretically and experimentally. Every system and every technique has advantages and significant disadvantages. 

Dilute atomic gases have a greater tuneability than most solids. However, interpreting experiments is subtle because of spatial non-uniformity, non-equilibrium effects, lack of robust thermometry, .... Furthermore, one can access a limited range of densities and the interactions are short-ranged.

A strong case can be made that dilute atomic gases are of interest in their own right and are complementary to other condensed matter systems. I think it is counter-productive to claim much more...

Physics Nobel highlights high value of table-top science

I really like the award of the 2012 Physics Nobel Prize to Serge Haroche and David Wineland. (I predicted entanglement but not these individuals).
Their studies have made theoretical ideas about quantum decoherence and entanglement actually testable in the lab. A nice summary of the scientific background is here.

One thing the award highlights to me is the high value (return on investment) of table-top science. I would guess that Haroche and Wineland's annual research budgets would be less than one million dollars.  This is to be contrasted to How much does it cost to find a Higgs boson? (about 10 billion dollars).

This award also highlights the incredible and enviable track record of NIST, who have received 4 Nobels in the past 15 years (Bill Phillips, 1997; Eric Cornell, 2001; John Hall, 2005; Phillips, 2012). The former 3 recently testified to the US congress that NIST management style was key to their success.

Table top science is relatively cheap. It is the most cost-effective investment for smaller countries and institutions.

An essential state model for the pseudogap state

Understanding the origin of the pseudogap state in cuprate superconductors (and organic charge transfer salts) remains a challenge. It has now been shown that within a doped Hubbard or t-J model one can produce a pseudogap like state when the model is treated at the level of the Dynamical Cluster Approximation. The DCA is a generalisation of Dynamical Mean-Field Theory (DMFT) to small clusters. Although this is a significant advance it is still somewhat at the level of a "black box". One would like to know the essential physics.

Jaime Merino and Olle Gunnarsson have a preprint which shows how a two-site two-orbital model can capture some of the essential physics. This model is motivated by DCA calculations on the smallest four site cluster. The two orbitals correspond to (0,pi) and (pi,0) in the first Brillouin zone. Each orbital couples to an independent bath.
As U increases there is a transition from the cluster orbitals forming a Kondo singlet with the bath states to formation of a non-degenerate bound state on the cluster. The latter corresponds to formation of the pseudogap.
For larger clusters the coupling to the baths are much stronger for the nodal regions than the anti-nodal regions.

The worst stage of an academic career

Last night it was announced that John Gurdon shared the 2012 Nobel Prize in Physiology or Medicine. The Guardian website report had this interesting quote from him.

Probably the worst stage in an academic career is when starting on one's own as a new assistant professor, with a hefty load of new lectures to be prepared, the need to acquire research support for an independent program, the wish to attract students to form a group, and no one except oneself to do the lab work with which to attract students and research support. I was very fortunate to be joined, within two years of starting at this level, by two outstanding students, Christopher Graham and Ron Laskey.

"From Nuclear Transfer to Nuclear Reprogramming: The Reversal of Cell Differentiation" - J.B. Gurdon, Annual Review of Cell and Developmental Biology Vol. 22: 1-22 (Volume publication date November 2006)

A helpful question to your audience

Does anyone have any questions so far?

If I am giving a seminar I try to ask the audience this at several points in the talk, particularly if there have been no questions so far.
Even if no one has a question is shows them you are open to them, and will keep the audience more engaged because they may make more of an effort to think of questions.

Deconstructing the Hall effect in quasi-one-dimensional metals

Understanding the Hall effect in strongly correlated electron materials is a challenge. In simple metals, the Hall coefficient R_H is, to a good approximation, equal to the inverse of the charge carrier density, and so is weakly dependent on temperature.
Furthermore, it has the same sign as the charge carriers.

In contrast, in strongly correlated metals, R_H can vary significantly with temperature, including changing sign, and its magnitude and signh can be significantly different from the charge carrier density estimated from band structure calculations or alternative experimental measurements such as the Drude weight in the optical conductivity.
I have written several previous posts about this issue. A post on the cuprates illustrates the problem.

There is a really nice paper Hall effect in quasi-one-dimensional metals in the presence of anisotropic scattering by Nicholas Wakeham and Nigel Hussey.
They show that due to the highly anisotropic band structure in a quasi-one-dimensional Fermi liquid metal, a variation of the scattering rate of just a few per cent over the Fermi surface, can significantly modify the Hall coefficient.
The figure below compares experimental data for the cuprate PrBa2Cu4O8 to a simple model with a temperature dependent anisotropy of less than 4 per cent.
The measured Hall coefficient is an order of magnitude smaller than the band structure value and changes sign twice as a function temperature.

This relatively simple explanation of complex behaviour should caution against exotic and non-Fermi liquid interpretations of Hall effect data (e.g., this PRL).

Tables are wonderful

Consider including a Table in your next paper.
They can be very useful.

For example, they can compactly summarise previous work (experimental and/or theoretical) relevant to your paper.

To me Tables highlight an important aspect of science that I think is increasingly devalued or ignored: comparison. Everyone whats to talk about their own results but is less inclined to critically compare their results to earlier work and put it in that context.

It may be painful to admit it but the Table may actually be more useful to the community than the main results in your paper.
I believe Jim Brooks once told me that this was an extremely valuable lesson he learnt from his thesis advisor Russell Donnelly.

To make this concrete I include below a Table from one of my papers with Jaime Merino.

A limit to my understanding II

This is a follow up on a previous post which discussed the strange fact that in quantum many-body theory certain limits (e.g., zero frequency, and the long wavelength) do not necessarily commute.

Today I learnt that in 1960 Kohn and Luttinger pointed out that the following definitions of the ground state energy are not necessarily the same:

1. The lowest eigenvalue of the Hamiltonian in the thermodynamic limit.

2. The zero temperature limit of the free energy of an infinite system.

This observation was based on an examination of Goldstone's perturbation theory, which is based on 1.

Ward and Luttinger later showed that for spatially isotropic band structures and interactions the two are equivalent.
Metzner and Vollhardt [where I learned all this] claim that for the Hubbard model, to second order in U, the two are equivalent.

Aside: Kohn was very productive in the early 1960's! Another recent post referred to his seminal paper on insulators.

Two delusions we suffer from

Both things we desperately want to believe. We tell them to our colleagues.
The first may also be told by department chairs to try and persuade someone to teach a course.
The second we may tell prospective students.

1.
Teaching this course for the first time won't really take that much time.
I won't be a perfectionist. I will cut corners. After all, I have someone else's notes (and assignments, exams, ...). It is a really good textbook....

2.
My student can complete their Ph.D in the minimum amount of time [3 years in Australia].
They will work really hard. They are smart. They will cut corners. I will supervise them closely. The project is quite straight-forward. I won't repeat past mistakes...

Face it. Neither is true.
Or do you believe them?
Why or why not?

Breakdown of the Luttinger liquid paradigm

At the Journal Club for Condensed Matter Patrick Lee has a helpful commentary on a recent preprint

Non-Fermi liquid d-wave metal phase of strongly interacting electrons
Hong-Chen Jiang, Matthew S. Block, Ryan V. Mishmash, James R. Garrison, D. N. Sheng, Olexei I. Motrunich, Matthew P. A. Fisher

The preprint presents numerical evidence that a t-J model on a ladder with a particular kind of ring exchange has an exotic ground state.

This state is particularly interesting for two reasons.

First, it falls outside the Luttinger liquid paradigm which is the one-dimensional version of Landau's Fermi liquid theory.

Second, this exotic state is well described by a variational wave function based on the  "parton" construction [a generalisation of slave bosons] where electron operators are replaced by a product of three fermion operators.

The million dollar question remains: is any of this relevant to two dimensions?

The authors state, "estimating the strength of K [ring exchange] .... in real materials such as La2-xSrxCuO4 is an interesting question."

This is particularly important because they require K larger than the hopping t to get the non-Fermi liquid phase.

I thought the ring exchange was the four spin ring exchange occurs in solid 3He and in a weak Mott insulator. However, it is different. As the figure above shows this "ring exchange" involves spatially rotating spin singlets. I thank Matthew Fisher for clarifying my confusion on this point.

For the parent Mott insulator La2CuO4 the magnitude of the 4-spin ring exchange was determined by inelastic neutron scattering a decade ago by Radu Coldea and collaborators. It is smaller than t.

Previously, I posted about nice related work by some of the same authors, on a Heisenberg model with 4 spin-ring exchange on a multi-leg ladder. This model is directly relevant to the possible spin liquid state in organic charge transfer salts.

A highly original model for water

I find it is rare that I read a paper that I think is highly original and creative. One I read recently is
Water Modeled As an Intermediate Element between Carbon and Silicon
by Valeria Molinero and Emily Moore

They model water molecules as single atoms! It is just like a spherical cow. And it works..

The abstract of the paper is beautifully written and very informative and so I reproduce it here.

Water and silicon are chemically dissimilar substances with common physical properties. Their liquids display a temperature of maximum density, increased diffusivity on compression, and they form tetrahedral crystals and tetrahedral amorphous phases. The common feature to water, silicon, and carbon is the formation of tetrahedrally coordinated units. We exploit these similarities to develop a coarse-grained model of water (mW) [monatomic Water] that is essentially an atom with tetrahedrality intermediate between carbon and silicon. mW mimics the hydrogen-bonded structure of water through the introduction of a nonbond angular dependent term that encourages tetrahedral configurations. 

The model departs from the prevailing paradigm in water modeling: the use of long-ranged forces (electrostatics) to produce short-ranged (hydrogen-bonded) structure. 

mW has only short-range interactions yet it reproduces the energetics, density and structure of liquid water, and its anomalies and phase transitions with comparable or better accuracy than the most popular atomistic models of water, at less than 1% of the computational cost. 

We conclude that it is not the nature of the interactions but the connectivity of the molecules that determines the structural and thermodynamic behavior of water. 

The speedup in computing time provided by mW makes it particularly useful for the study of slow processes in deeply supercooled water, the mechanism of ice nucleation, wetting-drying transitions, and as a realistic water model for coarse-grained simulations of biomolecules and complex materials.

The success of the model highlights how the properties of liquid water are emergent.

The authors recently used this model in a Nature paper,

Structural transformation in supercooled water controls the crystallization rate of ice