Extended Hubbard model of high Tc superconducting cuprates

At the moment, I am trying to help out Prof Lee to do a simple numerical calculation for his high Tc model, called it extended Hubbard Model (EHB) [1]. In my opinion, and in many's as well, Prof Lee is among the best physicists in Malaysia. he has very sharp insight in the physics in condensed matter physics, and he prefers more theoretical construct rather than on the computational aspect.

For a 'newbie' physics researcher is looking for a good theoretical topic to work on, Prof Lee's high Tc model is a very attractive research topic. He is a very senior researcher who knows very well the details of the high Tc modelling tools, and he has a very sharp insight on how to proceed with the modelling of the high Tc physics. I am trying to learn from his seniority experience, and also hoping to 'leap off' from his experiences in condensed matter physics, which is not a field of my expertise. The research issue here is to model the behavior of high Tc superconductivity and pseudo gap in cuprate oxides. Many models exist. Prof Lee's pick is the more conventional one based on BCS-like mechanism of phonon-electron interactions, albeit not entirely similar in nature. In his proposal, a non-harmonic terms in the phonon-electron enter the hamiltonian, providing an explanation to the pseudo-gap and Fermi arc seen in high Tc cuprates. From personal point of view, I take my involvement in his the research an opportunity to leap into a real impact research topic by standing on the shoulder of a giant. This is a high impact topic, initiated by our very own local expert to take on a frontier research challenge. 


Hubbard model is the theory that describes Mott insulator very well. In [1], a new term due to non-linear mode between the phonon and electron was added to induce a d-wave symmetry, which is observed in experiments. There are other very attractive features about Prof Lee's model as well. i didn't realise the beauty of it until i myself make a literature search about the high Tc model in cuprate oxide (Nature physics, Vol 2, March 2006). 


Apart from the physics argument justifying the physical origins of the terms in his model, technically, the calculation proceeds along the following way: Build the hamiltonian -> Propose a trial hamiltonian --> Construct the trial free energy --> Use Bogolyubov variantional pricinciple to minimise the trial free energy --> The minimisation results in equations relating various quantities of physical interests, and from here one can determine, say for example, the critical temperature as a function of dopant concentration. The whole machinery is based on the Bogolyubov variational principle, a very powerful theoretical tool used by people in condensed matter physics. The trick here is to be able to guess the appropriate trial hamiltonian, which requires one to have very strong physics insight about the system under investigation. Prof Lai praised highly of Prof Lee for being able to have such a sharp insight to write down the trial hamiltonian for a non-trivial system. The more insight you have on the system, the trial hamilonian constructed would give a more realistic description of the physics of the system. 

How about the exact values of the coupling constants in the Hubbard model? In fact the effective coupling constants in the case of his extended Hubbard model are arbitrary parameters. These are the 't_ij' and 'U' terms. The former represent 'tunnelling' or hopping of an electron from site i to nearest neighbour j (this corresponds to attractive interaction between neighbouring atomic sites), whereas the latter term represent the 'on-site' repulsive interaction when electrons stay on the same ionic site. These are the essential features of the Hubbard model. In the EHB (extended hubbard model) of Prof Lee, these interaction are not known. In my computational calculation I will provide a set of values representing these t_ij and U, then solve the gap equation as a function of temperature and dopant concentration. The gap equation is obtained as a result of the minimisation of the trial free energy using Bogolyubov variation method. What we wish to obtain is the 'superconductivity dome', the critical temperature curve vs dopant concentration, which has been measured experimentally. So if we could some how reproduce such a 'dome' and show it to match with experimental data, we are on the right course towards a full description of high Tc superconductivity in cuprates. 


The modelling of full features of high Tc in cuprates involve many stages, each requires independent pieces of new ingredient. The other features needed are e.g., the pseudogap curve, and magnetic order of high Tc, isotope effect, plus others stuff that i am currently ignorant of. Some of these features could be independent, some could be due to a common physical origin. No one knows for sure at the moment. We can't provide a complete accounts to all the experimental features of high Tc in cuprates in a single shot due to the immense difficulty in the modelling. So we have to go by stages. 

Take the high Tc cuprate as a big zig saw puzzle. we are now assemble the pieces one in a step, with the hope to complete the whole puzzle at the end. The modelling process reminds me of of the construction of any extension of the Standard Model to incorporate neutrino anomalies when writing my PhD thesis.


Reference: 


[1] B. S. Lee, J Supercond Nov Magn (2010) 23: 333-338. Non-linear localised lattice mode coupling mechanism and the pseudogap in high-temperature superconducting cuprates.