Quantum transport theory - analyzing higher order Quantum transport theory - analyzing higher order correlation effects by symbolic computationcorrelation effects by symbolic computation

- the development of SymGF- the development of SymGF

PhD Thesis Defense

Feng, Zimin

Feburary 27th, 2012

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Acknowledgements

Guo, Hong – McGill, Physics

Zhang, Xiangwen – McGill, Mathematics

Lei, Tao – McGill, Mathematics

Sun, Qing-Feng – Institute of Physics, Beijing

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How Physics is Done?

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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

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Experimental systems can be complicated, hard to do theory

D.Schröer, L.Gaudreau,S.Ludwig et al. PRB 76 (2007)075306

M.C.Rogge and R.J.Haug Cond-mat. 0707.2058

S. Amaha et al. nanoPHYS'07, Tokyo, Japan ( 2007).

T.Ihn et al.New Journ. Phys.9(2007)111

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Double quantum dots:

M. Ciorga et al, PRB 61, R16 315, (2000) D.Sprinzak et al. PRL 88, 176805 (2002)J.Elzermann et al. PRB 67,161308 (2003)

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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

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If there are strong interactions and strong correlation physics in Hdev, analytic theory can become extremely complicated. For this reason, quantum transport theory for multiple QD has not been done to satisfactory level.

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How to derive formulas in quantum transport theory? (by Green’s function approach)

Equation of motion

Feynmann Diagrams

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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 → ...

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Extremely complicated

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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 !

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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:

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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.

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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

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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.

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Verification of SymGF:

• Sergueev N et al, Phys.Rev.B 65 165303 (2002).

• Meir Y et al, Phys. Rev. Lett. 66 3048 (1991).

• Trocha P et al, Phys. Rev. B 76 165432 (2007).

• Brown K et al, J. Phys.: Condens. Matter 21 215604 (2009).

• Trocha P et al, Phys. Rev. B 78 075424 (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.

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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

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SymGF reveals interesting correlation physics

S. Sasaki et al PRL 103, 266806 (2009)

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SymGF: higher order virtual processes coherently sum up to Kondo resonance

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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.

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Symbolic Computational PhysicsSymbolic Computational Physics

Perhaps: a branch of Condensed Matter Physics

THANK YOU !