Quantum transport with Kwant: from toy models to high performance computing

Transcription

1 Quantum transport with Kwant: from to models to high performance computing Christoph Groth, Michael Wimmer, Anton Akhmerov, Xavier Waintal Kwant workshop, Villard de Lans, 1 5 June 2015

2 Numerical quantum transport is eas scattering region Σlead, GR, G<,... lead 2 lead 0 S, ψ,... lead 1 H= X Hij ii hj i,j... H = VL VL HL VL VL HL VLS VLS HS

3 Reall eas? Construction The construction of a tight binding Hamiltonian matri is often tedious without dedicated support.

4 Reall eas? Lead modes VL VL VL HL HL HL HL HL VL VL VL φn (j) = (λn )j χn (HL + VL λ 1 n + VL λn )χn = Eχn A straightforward eigenvalue problem in principle, but highl non trivial to solve in a robust and general wa.

5 Reall eas? Sstems of linear equations Consider the Hamiltonian and LU decomposition of a simple sstem. Nested dissection Recursive Green s functions Performance comparison Computation time (s) Gaussian elimination kwant - construction kwant - solving RGF - solving 2 L (sites) 3

6 We need a proper code Desirable attributes: Careful software design provide appropriate abstractions user-friendl, general, powerful, etensible Best available algorithms efficient Readil-packaged, well-documented, using a popular language Infeasible as a one-man-project (too big, different areas of competence). While at it, let s make it free ( open source ) for Availabilit Collaboration Reproducible research

7 Introducing Kwant ( project.org/) Full general software package for quantum transport with tight-binding models (an dimension, an geometr, an number of leads, an number of orbitals) Free software (BSD), eas installation on Linu, Mac, and Windows Pthon librar (no input files), optimized with C/C++ and Fortran, etensible, modular Idiomatic ( pthonic ) interface based on phsics concepts (Hamiltonians, sites, orbitals, lattices, smmetries, modes) Faster and more stable than RGF thanks to better algorithms (nested dissection MUMPS, modes) Tutorial, concepts paper, complete reference documentation, discussion list Ideas for a less ugl logo are welcome!

8 A glimpse at Kwant in action import matplotlib.pplot import kwant 5 lat = kwant.lattice.square() ss = kwant.builder() def disk(pos):, = pos return **2 + **2 < 13**2 ss[lat.shape(disk, (0, 0))] = 0.5 ss[lat.neighbors(1)] = 1 kwant.plot(ss)

9 Different shape, different lattice lat = kwant.lattice.honecomb() ss = kwant.builder() def bean(pos):, = pos rr = **2 + **2 return rr**2 < 15 * * rr + **2 * **2 ss[lat.shape(bean, (0, 1))] = 0.5 ss[lat.neighbors(1)] = 1 kwant.plot(ss) import matplotlib.pplot import kwant

10 Just for fun: Kagome supercell with second nearest neighbor hoppings import matplotlib.pplot import kwant lat = kwant.lattice.kagome() sm = kwant.translationalsmmetr(lat.vec((0, 3)), lat.vec((2, 0))) ss = kwant.builder(sm) ss[lat.shape(lambda pos: True, (0, 1))] = 0.5 ss[lat.neighbors(2)] = kwant.plot(ss, unit=0.5)

11 Application/Etension: Time resolved quantum electronics mostl b: B. Gaur, J. Weston, X. Waintal Man interesting applications: Interferometers AC-Josephson effect Majorana fermions... AC Josephson effect without superconductivit, B. Gaur et al. (2014)

12 Time resolved quantum electronics How does it work? H (t) = X Hij (t)c i cj, H(t) = H0 + (t) ij ΨE (~r, t) = Ψst r)e iet + Ψ E (~r, t) E (~ i t Ψ E (~r, t) = H(t)Ψ E (~r, t) + (t)ψst r)e iet + iσ(~r)ψ E (~r, t) E (~ Schrödinger Source Sink Efficient: CPU tn, MEM N Highl parallel Stable differential equation, starts from stationar solution Etension to Kwant

13 Stopping and releasing electrons Stopping electrons with radio frequenc pulses in the quantum Hall regime, B. Gaur et al. (2014)

14 Numerical eperiments with Kwant: Fling qubit Kwant has been envisioned as a tool for numericall-assisted theor, but it works also well for numerical eperiments. Quantitative simulation of a realistic device (in SI units) Computer-aided optimization of designs saves time Theoretical, numerical, and eperimental stud of a fling qubit electronic interferometer. T. Bautze et al. (2014)

15 Other neat applications Superconducting artificial atom based on two inductivel coupled transmons (E. Dumur et al., arxiv: ) Mechanical topological insulators (R. Su sstrunk and S. D. Huber, arxiv: ) Distribution of supercurrent in a disordered Josephson junction (D. Nurgaliev, unpublished) a c b

16 Plans for Kwant Diagrammatic quantum Monte Carlo solver for interacting sstems Kwant 2 More general smmetries (n-d bandstructure, eas PBCs) Better support for Hamiltonians with man parameters Improved performance in special cases (e.g. small sstems) Scalable electrostatics for quantum transport Multi-dimensional scattering a) b) magnetic atoms adatom STM tip superconductor We are looking for people with a strong background in both phsics and computing. Hiring now in Grenoble (2 PhDs) and in Delft (PhD, postdoc).

17 Let us work together on tomorrow s scientific software! In general, please consider sharing useful scientific software, publishing our research in a reproducible wa. As for Kwant, please send ideas, suggestions, questions, and bug reports to kwant-discuss@kwant-project.org, fill in the Kwant user surve at share Kwant-related code. Contributions to Kwant are welcome, and will be attributed prominentl.