Quantum transport theory - analyzing higher order correlation effects by symbolic computation - the development of SymGF PhD Thesis Defense Feng, Zimin Feburary 27th, 2012

Symbolic Computational Physics - Feng2 Acknowledgements Guo, Hong – McGill, Physics Zhang, Xiangwen – McGill, Mathematics Lei, Tao – McGill, Mathematics Sun, Qing-Feng – Institute of Physics, Beijing

Symbolic Computational Physics - Feng3 How Physics is Done?

Symbolic Computational Physics - Feng4 We wish to understand the microscopic physical process Fit experimental data with theoretical model and curves If no theory properly describes data, come up with a new model. Ex: Kondo effect in quantum dot transport

Symbolic Computational Physics - Feng5 Experimental systems can be complicated, hard to do theory D.Schröer, L.Gaudreau,S.Ludwig et al. PRB 76 (2007) M.C.Rogge and R.J.Haug Cond-mat S. Amaha et al. nanoPHYS'07, Tokyo, Japan ( 2007). T.Ihn et al.New Journ. Phys.9(2007)111

Symbolic Computational Physics - Feng6 Double quantum dots: M. Ciorga et al, PRB 61, R16 315, (2000) D.Sprinzak et al. PRL 88, (2002) J.Elzermann et al. PRB 67, (2003)

Symbolic Computational Physics - Feng7 How quantum transport theory is done? The model: Lead-Device-Lead Non-interacting leads Current proportional to the rate of change of electrons in a lead If there are strong interactions and strong correlation physics in H dev, analytic theory can become extremely complicated. For this reason, quantum transport theory for multiple QD has not been done to satisfactory level.

Symbolic Computational Physics - Feng8 How to derive formulas in quantum transport theory? (by Green’s function approach) Equation of motion Feynmann Diagrams

Symbolic Computational Physics - Feng9 When quantum-dots contain strong interactions... Suppose a Hamiltonian has on-site interaction U and we need to calculate its Green's function: 2-particle GF → 3-particle GF → 4-particle GF →... Extremely complicated

Symbolic Computational Physics - Feng10 New idea – SymGF: symbolic tool for deriving high-order formulas H→SymGF→G Automatically and symbolically derives the Green's function of a given Hamiltonian by a computer: complicated problems can now be solved. Results are given analytically. Order of expansion is controllable. Developed with Mathematica Widely tested for its reliability Using SymGF, investigating higher-order processes and complicated device configurations become possible !

Symbolic Computational Physics - Feng11 Why not done earlier ? Computer Algebra System (CAS) started in 1960's Widely used in scientific research Has established packages in high- energy physics Condensed matter physics is quite versatile; Each problem has its own Hamiltonian and its own methodology: developing a symbolic tool for each problem is not viable. Exception: quantum transport theory

Main features of SymGF: Symbolic Computational Physics - Feng12 3 sets of inputs to SymGF: Output of SymGF : 1.Hamiltonian in second quantized form; 2.anti-commutation relations of the operators that appeared in the Hamiltonian; 3.Truncation rules. - this determines the order of expansion The desired Green's function of the given Hamiltonian at given order of expansion.

What is in SymGF? Methods implemented in SymGF of solving the equations of motion: Gaussian Elimination Preconditioned Iteration Graph-Aided Solution Direct Iteration Self-energies are automatically defined during the solution Automatic derivation of all required equations of motion Automatic recognition of applicability of truncation rules Keeping specific equal- time correlators at user's mandate Symbolic Computational Physics - Feng13

Symbolic Computational Physics - Feng14 Demonstration An example run of SymGF to reproduce the analytical derivation of PRL 66, 3048 (1991). Single quantum dot transport problem with on- site interaction.

Symbolic Computational Physics - Feng15 Verification of SymGF: Sergueev N et al, Phys.Rev.B (2002). Meir Y et al, Phys. Rev. Lett (1991). Trocha P et al, Phys. Rev. B (2007). Brown K et al, J. Phys.: Condens. Matter (2009). Trocha P et al, Phys. Rev. B (2008). It took SymGF less than two minutes to derive the analytical formula for these different problems, and the results are exactly the same as derived by hand.

Symbolic Computational Physics - Feng16 Application - side-coupled double QD: extremely difficult if not impossible to derive higher order formulas by hand

The model for the side-coupled double quantum dot system

Symbolic Computational Physics - Feng18 SymGF reveals interesting correlation physics S. Sasaki et al PRL 103, (2009)

Symbolic Computational Physics - Feng19 SymGF: higher order virtual processes coherently sum up to Kondo resonance

Symbolic Computational Physics - Feng20 Outlook for SymGF: going beyond existing theory! Long range potential: going beyong random phase approximation? Long range potential: include more than just the most diverging terms? Include dynamic dipole-dipole interaction? Perhaps quadripole interaction as well? (computing van der Waals interaction from 1st principles) The idea of SymGF opened new doors for theoretical condensed matter physics.

Symbolic Computational Physics - Feng21 Symbolic Computational Physics Perhaps: a branch of Condensed Matter Physics THANK YOU !