Is there superconductivity in the Hubbard model?

Previously, I considered the tricky problem of Does the doped Hubbard model superconductor?
I mentioned in passing a worrying quantum Monte Carlo study published in PRB in 1999

Correlated wave functions and the absence of long-range order in numerical studies of the Hubbard model
M. Guerrero, G. Ortiz, and J. E. Gubernatis

The graph below shows the distance dependence of the pairing correlation function in the d-wave channel. If superconductivity occurs it should lend to a non-zero value equal to the square of the superconducting order parameter.
It certainly looks like it tends to zero at large distances.

However, careful examination shows that it seems to have a non-zero value of order 0.001.
Perhaps, that is just a finite size effect.
But, we should ask, "How big do we expect the long-range correlations, i.e. the magnitude of the square of the order parameter d, to be?"

A cluster DMFT calculation on the doped Hubbard model (in the PRB below) gives a value of order 0.03 for the order parameter d. This means d^2 ~ 0.001 consistent with the QMC study which claims no superconductivity!

Anomalous superconductivity and its competition with antiferromagnetism in doped Mott insulators
S. S. Kancharla, B. Kyung, D. Sénéchal, M. Civelli, M. Capone, G. Kotliar, and A.-M. S. Tremblay

Similar issues arise when assessing the results of
Absence of Superconductivity in the Half-Filled Band Hubbard Model on the Anisotropic Triangular Lattice
R. T. Clay, H. Li, and S. Mazumdar

If I take the order parameter estimated by a RVB calculation reported in this PRL (by Ben Powell and myself) and square its value it predicts a long-range pairing correlation (~0.001) comparable to the extremely small values found in the numerical study claiming absence of superconductivity.

Clay, Li, and Mazumdar also mentioned the problematic observation that the pairing correlation they calculated did not increase with the Hubbard U. However, my previous post discussed how Scalapino and collaborators argued this is because one needs to factor in the quasi-particle renormalisation Z that also occurs with increasing U. For the half-filled Hubbard model this probably leads to an order of magnitude enhancement of the pairing as U increases towards the Mott insulating phase, since Z decreases from 1 to 0.3 and the renormalised P_d scales with 1/Z^2.

So, I remain to be convinced that superconductivity does not occur in the Hubbard model, both upon doping the Mott insulator or at half-filling near the band-width controlled Mott transition.

Impact factors have no impact on me

There seems to be a common view that on CVs (and grant applications) people should list the Impact Factors for each journal in which they have a paper.
To me this "information" is just noise and clutter.
I do not include it in my own CV or grant applications.
Why?

1. IFs just encode something I know already.
Nature > Science > PRL ~ JACS > Phys. Rev B ~ J. Chem. Phys. > Physica B ~ Int. J. Mod. Phys. B > Proceedings of the Royal Society of Queensland .....

2. There is a large random element in success or failure to get an individual paper published in a high profile journal. e.g., who the referees are.

3. The average citations of a journal is not a good measure of the significance of a specific paper. There is a large variance. What really matters is how much YOUR/MY specific paper in that journal is cited in the long term. Unfortunately, in most cases it is hard to know in less than 3-5 years.

4. Crap papers can get published in Nature and Science. Hendrik Schon published almost 20 papers in Nature and Science. On the other hand, Nobel Prize winning papers are sometimes published in Phys. Rev. B (e.g. giant magnetoresistance).

5. I don't need to know the actual IF of a journal with an impact factor of one or less in order to know that it is a rubbish journal. I already know that because I virtually never read papers in such journals simply because they virtually never contain anything that is significant, interesting, or valid. My "random" meanderings through the literature virtually never lead me there.

6. I remain to be convinced that reporting IFs to more than 2 significant figures and without error bars is meaningful.

I fail to see that alternative metrics such as the Eigenfactor resolve the above objections.

The only value I see in IFs is helping librarians compile draft lists of journals to cancel subscriptions to in order to save money.

I am skeptical that IFs are useful for comparing the research performance of people in different fields (e.g. biology vs. civil engineering vs. psychology vs. chemistry).

And in the end... what really matters is whether the paper contains interesting, significant, and valid results... Actually looking as some of an applicant's papers and critically evaluating them is the best "metric". But that requires effort and thought...

What did Wilson do?

Last week we struggled through chapter 4, "Renormalisation group calculations" of Hewson's book, The Kondo Problem to heavy fermions.

The focus is on Kenneth Wilson's numerical treatment of the Kondo problem, mentioned in his Nobel prize citation. Much of it still remains a mystery to me...
Here are a few key aspects. Please correct me where I am wrong or at least confused...

First, he mapped the three-dimensional Kondo model Hamiltonian into a one dimensional tight binding chain (half-line) with single impurity spin at the boundary. This simplification makes the problem more numerically tractable.

Next, he used a logarithmic discretization (in energy) of the states in the conduction band. This important step is motivated by the logarithmic divergences found by Kondo's perturbative calculation and Anderson's poor man's scaling arguments.

He then numerically diagonalises the Hamiltonian with a discrete set of states for a finite chain. One then rescales the Hamiltonian, truncates the Hilbert space, and adds an extra lattice site.

Eventually, one converges to the strong coupling fixed point and one observes an almost equally spaced excitation spectrum, characteristic of a Fermi liquid.

A surprising thing is that the rescaling parameter Lambda is set to a relatively large value of 2, compared to a value close to one, that one might expect to be needed. Wilson was clever to realise/find that such coarse graining would work so well.

Wilson extracted a large amount of information from his calculations. Here are a few important findings.

1. The impurity specific heat and impurity susceptibility had a Fermi liquid temperature dependence. The latter was given by

[This] shows that there is no residual local moment, and that the impurity spin is fully compensated. The numerical factor 0.4128 is a universal number for the s-d model, and is known as the Wilson number, w. It relates two quite different energy scales for the s-d model, T_K, which is determined from the high temperature perturbative regime, and chi_imp(0), the low temperature susceptibility associated with the strong coupling regime.

2. The Sommerfeld-(Wilson) ratio had a universal value

3. Over an intermediate temperature range (one and half decades) the temperature dependence can be fit to

This Curie- Weiss form corresponds to a reduced moment compared to the free spin form. Thus the impurity moment, even for T ~ T_K, is only of the order of 30% that of the free moment. The residual effects of the screening of the conduction [electrons] persist to very high temperatures because of the logarithmic dependence on T/T_K.

4.  The complete universal dependence with a logarithmic temperature scale is shown below

Writing effective papers

Weston Borden's article 40 years of fruitful chemical collaborations has an significant observation concerning writing effective papers: focus on the physical explanation of the results rather than on the details of the methodology.
He recounts how he he learnt this, while starting out as an Assistant Professor at Harvard, in a collaboration with Lionel Salem. Borden had performed some calculations using the Pariser-Parr-Pople (PPP) model for the electronic structure of conjugated organic molecules [for physicists an extended Hubbard model with long-range Coulomb interactions].

Lionel read my draft, and he promptly rewrote it. Lionel’s revised version, which was the one that we published, focused much more than my draft had on the explanation of the PPP results, rather than on the details of the calculations. This experience taught me a valuable lesson. Although describing the details of calculations and the results obtained from them is certainly important, it is even more important to write a clear, physical explanation of the results. 

This was also the lesson that I learned from the papers that Roald Hoffmann published in the late 1960s and early 1970s. Although it was well-known that the Extended Hückel (EH) method that Roald used was quantitatively unreliable, Roald provided such convincing qualitative explanations of his EH results that it always seemed to me Roald’s EH results must be correct.

I think these observations are just as relevant and important for physicists.

Aside: an earlier post sung the praises of Hoffmann's paper titles.

Borden then makes the important and worrying observation:

Perhaps the tremendous increase in the accuracy of electronic structure calculations during the past 40 years has had the undesirable consequence that computational chemists feel less obliged to provide the kind of detailed physical explanations of their results than Roald routinely furnished 40 years ago.

Transition from a band insulator to a bad metal

Many previous posts have considered how in a metallic phase close to a Mott insulator one can observe a crossover from a Fermi liquid to a bad metal with increasing temperature.

One observes something quite different in FeSi (iron silicide) which has been a subject of debate for several decades. Different paper titles include the following words: Kondo insulator, ferromagnetic semiconductor, unconventional charge gap, strong electron-phonon coupling, Anderson-Mott localization, singlet semiconductor, covalent insulator, correlated band insulator, ferromagnetic metal, ....

At low temperatures FeSi is a semiconductor with a gap of about 50 meV (500 K). Both the spin susceptibility and the resistivity are gapped. However,  around 200 K there is a crossover to a bad metal.
The spin susceptibility has a maximum versus temperature around 400 K and above that can be fitted to a Curie-Weiss form, suggesting the presence of local moments.
The thermopower has a maximum around 50 K with a colossal value of 700 microVolts/Kelvin, making the material attractive for thermoelectric applications. The thermopower changes sign at about 150 K and 200 K.
With increasing temperature the optical conductivity shows redistribution of spectral weight on the electron Volt (eV) scale, an important signature of strong electronic correlations.

There is a really nice paper which provides a compelling theoretical description and explanation of what is going on.
Signatures of electronic correlations in iron silicide
Jan Tomczak, Kristjan Haule, and Gabi Kotliar

The authors perform electronic structure calculations combining Density Functional Theory (DFT) [at the level of Generalised Gradient Approximation (GGA)] with DMFT [Dynamical Mean-Field Theory].
They reproduce the main features of the experimental data.

Here is some of the key physics.
FeSi is a band insulator at low temperatures.
With increasing temperature there is a crossover to incoherence, i.e. the Bloch wavevector is no longer a good quantum number.
Fe is in a mixed valence state with a mean valence (no. of d electrons) of 6.2 and a variance of 0.9.
There is a preponderance of S=1 states, contrary to earlier suggestions that FeSi is a singlet insulator.
The incoherence arises because of fluctuations in the local moment, which is to a large extent non-local.
The results are controlled by the Hund's coupling J rather than the Hubbard U, something also seen recently in other systems with orbital degeneracy [see this two-faced post or discussion of strontium ruthenate or a recent review].

40 years of collaborative quantum chemistry

There is a very nice article in the Journal of Organic Chemistry
With a Little Help from My Friends: Forty Years of Fruitful Chemical Collaborations
by Weston Thatcher Borden

Borden's career is unusual in that he has done both organic synthesis [i.e., actually making molecules] and computational quantum chemistry.

The article is worth reading for several reasons. It describes some
-interesting organic chemistry and shows how quantum chemistry has illuminated it
-characteristics of fruitful collaborations, both between theorists and between theorists and experimentalists
-interesting history and personal vignettes and perspectives

On the latter I found the following throwaway line rather disturbing and disappointing:

When I was an Assistant Professor at Harvard, unlike most of my colleagues in the Chemistry Department, Bill Doering seemed genuinely interested in talking about chemistry with me.

Unfortunately, this happens too often. I would be curious to know why Borden thinks this was. Sometimes it is because people are too "busy" and/or preoccupied with their own little world. The worst reason can be senior scientists actually lose interest in science and get consumed with funding, politics, ...alternatives to struggling to do significant research.

Some of the insights in the article justify a blog post in their own right and so I hope I will post separately about writing up quantum chemistry calculations, tunneling by carbon in organic reactions, symmetry breaking in TMM, and "different electronic states of the same molecule can have different MOs [Molecular Orbitals],..."

A paper of Borden's featured in an earlier post Seeing how degenerate radicals can be

Topological insulators get more interesting

Topological insulators (TIs) are certainly a hot topic. However, there are two things that might make one nervous about all the excitement.

1. All the materials being studied as TIs [e.g. Bi2Se3] actually aren't TIs.
What!? A TI is by definition a bulk insulator with surface metallic states that are topologically protected. However, the actual materials turn out not to be bulk insulators. On a practical level this makes separating out bulk and surface contributions, particularly in transport measurements, tricky. But, also presents an ideological problem: one is not actually studying the phase of matter one wishes one was studying.

2. One could argue that TIs are "just a band structure effect", i.e., they do not involve any quantum many-body physics.

However, these objections are put to rest by a preprint
Discovery of the First True Three-Dimensional Topological Insulator: Samarium Hexaboride
Steven Wolgast, Cagliyan Kurdak, Kai Sun, J. W. Allen, Dae-Jeong Kim, Zachary Fisk

They report electrical transport measurements that show that SmB6 is a bulk insulator with surface metallic states.
This is of particular interest for several reasons

a. The material really is a true topological insulator.
b. The material is a Kondo insulator. [Although strictly the material is in the mixed valence rather than the local moment regime.] The insulating state emerges from strong electronic correlations.
c.  This resolves long standing puzzles about previous transport measurements on this material which did not show activated conductivity at low temperatures. This can now be explained as a sample dependent contribution from metallic surface states.
d. This material was predicted to be a topological Kondo insulator by Dzero, Sun, Coleman, and Galitski.

I also note a recent paper Actinide Topological Insulator Materials with Strong Interaction.

I thank Tony Wright for bringing the preprint to my attention.