What simple plotting software would you recommend?

I have a second year undergraduate student working with me on a research project.
He needs to start producing some simple plots of one dimensional graphs, of publication quality. The plots will be of comparable complexity/simplicity to the figure below.

What freeware would you recommend?
There is a long list of options on Wikipedia.

The main criteria are

-free
-can be used on a PC and a Mac
-easy to learn and to use

The figure is from this paper.

Why large N is useful

In Hewson's book The Kondo problem to heavy fermions he devotes two whole chapters (7 and 8) to large N limits.

This is a powerful technique in quantum many-body theory and statistical physics. One expands the continuous symmetry of a specific model (e.g. from SU(2) to SU(N)) and then takes the limit of large N. If coupling constants are scaled appropriately the limit of infinite N can be solved analytically by a mean-field theory. One then considers corrections in powers of 1/N. This idea was first pursued for classical critical phenomena where the order parameter was an N-dimensional vector and the symmetry was O(N).

In the Kondo problem this is a particularly powerful and important technique for several reasons.

First, large N can be physically relevant. For example, cerium impurities have N=6 (N=14) when spin-orbit coupling is (not) taken into account.

Second, for any N there are exact results for thermodynamic properties from the Bethe ansatz solution. These provide something to benchmark approximate results from large N treatments.

Third, one can obtain results for dynamical and transport properties which cannot be calculated with the Bethe ansatz.

Fourth, because the analytics/mathematics is relatively simple in the slave boson and diagrammatic formulations one can gain some insight as to what is going on (perhaps).

Fifth, the large N limit and slave bosons can also be used to study lattice models such as Hubbard and Kondo lattice models.

Finally, it works!, giving results that are better both qualitatively and quantitatively than one might expect. For example, Bickers, Cox, and Wilkins used large N to give a comprehensive description of a whole range of experimental properties.

But, the slave boson mean-field theory is far from perfect, working best below the Kondo temperature. In particular, it produces an artifact: a finite temperature phase transition (Bose condensation) at a temperature comparable to the Kondo temperature.

Open questions about multi-band Hubbard models

There is an interesting review article
Magnetism and its microscopic origin in iron-based high-temperature superconductors
by Pengcheng Dai, Jiangping Hu, Elbio Dagotto

The authors highlight

• the diversity of materials and types of magnetic order in the parent compounds 
• just giving a theoretical description of the parent compounds is a challenge
• that these materials are in an intermediate coupling (moderately? correlated) regime where weak-coupling treatments are inadequate 
• there are local magnetic moments at room temperature, which is well above most of the magnetic ordering temperatures
• how poorly understood the multi-band Hubbard model is, even at half filling

The phase diagram below is based on Hartree-Fock theory (which surely is quite inadequate). J_H is the Hund's rule coupling. The "physical region" is probably the parameter regime of the pnictides.

I am interested to know how this diagram compares to DMFT (dynamical mean-field theory) and DMRG studies on ladders.

Culture and the graduate student

Culture is a set of assumptions that are accepted without question.
Culture determines what is right, valued, important, and normal.

At my church there are many graduate students who are not originally from Australia. Many understandably struggle with the dual challenge of postgraduate study and negotiating a foreign culture. Consequently, I got asked to run a workshop "Thriving or surviving in postgraduate research." It covered a wide range of topics including mental health, managing your supervisor, and publishing. Most of the material has appeared on this blog before.

A significant part of the time was spent by the participants completing this worksheet and then discussing their answers among themselves.
I think this is much more effective and less overwhelming than me just telling them what to do, which I fear may be what happens at the workshops run by the university Graduate School (30+ detailed powerpoint slides in 50 minutes?).

Much of it is just as relevant to Australian students but it seems that non-Westerners particularly struggle with asking for and getting help from authoritarian figures such as their supervisors.

What is Herzberg-Teller coupling?

Is it something to do with breakdown of the Born-Oppenheimer approximation?

In molecular spectroscopy you occasionally hear this term thrown around. Google scholar yields more than 3000 hits. But I have found its precise meaning and the relevant physics hard to pin down. Quantum mechanics in chemistry by Schatz and Ratner is an excellent book, but the discussion on page 204 did not help me. "Herzberg-Teller" never appears in Atkins' Molecular quantum mechanics.

So here is my limited understanding.
Herzberg and Teller wanted to understand why one observed certain vibronic (combined electronic and vibrational) transitions that were not expected, particularly some that were expected to be forbidden on symmetry grounds. "Intensity borrowing" occurred.
Herzberg and Teller pointed out that his could be understood if the dipole transition moment for the electronic transition depended on the nuclear co-ordinate associated with the vibration. In the Franck-Condon approximation one assumes that there is no such dependence.

There is a nice clear discussion of this in Section 2.2 of
Spectroscopic effects of conical intersections of molecular potential energy surfaces
by Domcke, Koppel, and Cederbaum

They start with a simple Hamiltonian involving two diabatic states coupled to two vibrational modes. The diabatic states, by definition, do not depend on the nuclear co-ordinates.

They show how in the adiabatic approximation [which I would equate with Born-Oppenheimer] one neglects the nuclear kinetic energy operator and diagonalises the Hamiltonian to produce adiabatic states. But, the diagonalisation matrix depends on the nuclear co-ordinates. Hence, the adiabatic eigenstates depend on the nuclear co-ordinates. In the crude adiabatic approximation one ignores this dependence.

The photoelectron and optical absorption spectra depend on calculated the dipole transition
elements between electronic eigenstates. These depend on the nuclear co-ordinates via the diagonalisation matrix. In Franck-Condon (FC) one ignores this dependence. This dependence is the origin of the Herzberg-Teller coupling.

The figure below, taken from the paper, shows spectra for a model calculation for the butatriene cation. The curves from top to bottom are for Franck-Condon approximation, adiabatic approximation, and the exact result. (Note: the vertical scales are different). Comparing the top two curves on can clearly see intensity borrowing for the high energy transitions. Comparing to the bottom curve shows the importance of non-adiabatic effects; these are amplified by the presence of a conical intersection in the model.

Hence, it should be stressed that Herzberg-Teller and "intensity borrowing" are NOT non-adiabatic effects, i.e. they do NOT represent a breakdown of Born-Oppenheimer. This point is also stressed by John Stanton in footnote 3 of his paper I discussed in an earlier post.

Fulde on the chemistry-physics divide

When dealing with electronic correlations in solids, one finds that they often resemble those in corresponding molecules or clusters. Hence one would expect quantum chemistry and solid-state theory to be two areas of research with many links and cross fertilization. Regrettably this is not the case. The two fields have diverged to such an extent that it is frequently difficult to find even a common language, something we hope will change in the future. In particular it has become clear that the various methods applied in chemistry and in solid-state theory are simply different approximations to the same set of cumulant equations.

Peter Fulde, Correlated electrons in quantum matter (World Scientific, 2012), pages 3-4.

Why you should love diabatic states

An earlier post gave a brief primer on diabatic states.

There is a nice review article
Diabatic Potential Energy Surfaces for Charge-Transfer Processes
V. Sidis

He has an interesting discussion of the historical renaissance of interest in and use of diabatic states in atomic scattering problems.