Master Thesis — Realspace Braiding of Majorana States

Jakob Teuffel

Full thesis can be found here

Numerical simulation and analysis of non-Abelian anyons in 2D chiral topological superconductors, with focus on adiabatic braiding of Majorana zero modes and their potential for fault-tolerant quantum computation.

Abstract

This Master Thesis demonstrates that the non-Abelian exchange of Majorana modes in the 2D chiral topological p-wave superconductor vortices is possible with a non-zero energy splitting. The dependence of the energy splitting on the distance and radius of the vortices is shown, and a further exponential energy splitting in dependence on the distance between the vortex and the bulk edge is demonstrated. The adiabatic evolution of the quasi-particle exchange is visualised in real space. For the non-adiabatic regime, the dependence of the fidelity on the exchange speed is computed numerically for two and four vortices. The adiabatic and diabatic limits are discussed and a theoretical lower limit for the angular velocity for a non-zero energy splitting and two vortices is derived.

Key Results

Eigenvalue Spectrum with +4t Chemical Potential Shift

Spectrum for discrete vortices (zero energy mode highlighted in red), parameters: t=¼, Δ=1, R=Grid/8, as a function of μ₀ with a +4t shift. In this configuration only one side of the vortex is in the topological regime — the topological phase must be on the outside of the vortex for a zero-energy mode to exist. The spectrum is mirrored relative to the −4t case, with the phase transition going from chiral− to trivial instead of chiral+ to trivial.

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Adiabatic Exchange — Filmstrip

Complex plot of u_{x,y} + v_{x,y} for the Majorana mode γ_A at θ=0°, θ=90°, and θ=180° during a clockwise exchange. White arrows on the intermediary state indicate the rotation direction; dotted coloured lines connect identical vortices. The vortices exchange as predicted, with a global π/2 phase. The γ_A mode initially on the right (red/cyan) ends at the position of γ_B, and the colour regions shift in a way that recovers the full exchange statistic τ(σ₁⁻¹).

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Non-Abelian Exchange Map — All Four States

Complex plots of u_{x,y} + v_{x,y} mapping all four states reachable via repeated clockwise exchange. Moving clockwise through the four images corresponds to one additional clockwise vortex exchange each step. Dotted connecting lines indicate which vortex is which, with line colour encoding the sign picked up by the Majorana mode during exchange. The vector in the top-left shows the sign convention (i denoting the γ_B mode). Demonstrates the Z₄ periodicity of the exchange statistics.

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Non-Adiabatic Fidelity — Two Vortices

Propagators |⟨d d†(−π/ω)⟩|² and |⟨d†(−π/ω) d†⟩|² for the exchange of two Gaussian vortices (μ=1, Δ=1, t=−⅔, R=1/35) as a function of rotation speed ω. For ω ≫ 1 the system is diabatic (no exchange); for ω < 10⁻⁴ the adiabatic regime begins. In between, oscillatory behaviour is observed with sharp crossings where the exchange fidelity becomes large — corresponding to quasi-resonant passages through the energy gap.

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Non-Adiabatic Fidelity — Four Vortices

Fidelity and phase of the σ₂ exchange (exchange of two Majorana modes from different edge states) in the four-vortex system (μ=1, Δ=1, t=−⅔, R=1/35). The σ₂ braid group generator of B₄ mixes modes between the two edge states d₁, d₂. The adiabatic expectation values are ⟨d₁ d₁†(−π/ω)⟩ = ½, ⟨d₂ d₁†(−π/ω)⟩ = −i/2, ⟨d₁†(−π/ω) d₁†⟩ = ½, ⟨d₁†(−π/ω) d₂†⟩ = −i/2.

Repository Structure

Master-Thesis/
├── Thesis/          # LaTeX source
├── Simulation/      # C++ simulation (CMake)
├── Evaluation/      # Python analysis scripts
└── Mathematika/     # Mathematica notebooks

Thesis/

LaTeX source for the written thesis. Main entry point: Thesis/Master.tex. Chapters cover BdG formalism, Majorana fermions, non-Abelian anyons, topological invariants, and time evolution / braiding numerics. Build output in Thesis/build/.

Simulation/

C++ simulation of the 2D topological superconductor Hamiltonian. Supports:

• Eigenvalue/eigenvector diagonalization (Eigen + Accelerate/OpenBLAS on macOS, MKL on Linux)
• Adiabatic time evolution of vortex configurations
• Optional GPU acceleration via MAGMA

Build:

cd Simulation
mkdir -p build && cd build
cmake .. \
  -DCMAKE_BUILD_TYPE=Release \
  -DGRID=<N> \
  -DRAD=<R> \
  -DDIST=<D> \
  -DT_RES=<steps> \
  -DT_C=<chemical_potential> \
  -DT_ROT=<rotation_time> \
  -DW_START=<log_omega_start> \
  -DW_END=<log_omega_end> \
  -DW_RES=<omega_steps> \
  -DTHREADS=<mkl_threads> \
  -DTHREADSHL=<omp_threads>
make -j$(nproc)
./sim

Key flags: -DTIME_EVOLUTION=TRUE for braiding runs, -DUSE_MAGMA=TRUE for GPU, -DVERBOSE=TRUE for debug output.

Example — two-vortex adiabatic exchange, time evolution:

cd Simulation
mkdir -p build && cd build
cmake .. \
  -DCMAKE_BUILD_TYPE=Release \
  -DTHREADS=18 \
  -DTHREADSHL=18 \
  -DGRID=36 \
  -DRAD="1.0/16.0" \
  -DDIST=10 \
  -DT_RES=500 \
  -DT_ROT=0.5 \
  -DT_C=-2.0 \
  -DW_START=-4.0 \
  -DW_END=2.0 \
  -DW_RES=4 \
  -DUSE_MAGMA=FALSE \
  -DUSE_GPU=FALSE \
  -DVERBOSE=TRUE \
  -DTIME_EVOLUTION=TRUE
make -j$(nproc)
./sim

Parameter	Value	Meaning
CMAKE_BUILD_TYPE	Release	Compiler optimizations on (-O3, architecture flags). Use RelWithDebInfo for profiling.
THREADS	18	Thread count for the BLAS/LAPACK backend (MKL on Linux, vecLib on macOS). Controls low-level parallelism inside each matrix diagonalization.
THREADSHL	18	OpenMP thread count for the high-level outer loop (iterating over time steps or ω values). Effective total concurrency = THREADS × THREADSHL.
GRID	36	Lattice size N. The BdG Hamiltonian is 2N² × 2N² (here 2592 × 2592). Larger grids reduce finite-size effects at quadratic cost.
RAD	"1.0/16.0"	Vortex radius in lattice units (≈ 0.0625). In discrete mode: vortex core is the set of sites within this radius. In Gaussian mode (-DGAUSS): half-width of the Gaussian chemical-potential modulation. Passed as a C++ expression, evaluated at compile time.
DIST	10	Vortex separation scale factor. The vortex centre is placed at DISTANCE = GRID/4 × DIST lattice sites from the grid centre.
T_RES	500	Number of time steps for the ODE integrator during time evolution. Higher values give finer resolution along the exchange path.
T_ROT	0.5	Number of full rotations in the exchange path. T_END = −2π × T_ROT, so 0.5 = half-rotation = one vortex exchange τ(σ₁).
T_C	-2.0	Hopping amplitude t in the BdG Hamiltonian (nearest-neighbour tunnelling energy). Sets the bandwidth and together with μ determines whether the system is in the topological phase.
W_START	-4.0	Log₁₀ of the minimum angular velocity for the fidelity sweep: ω_min = 10⁻⁴ (deep adiabatic regime).
W_END	2.0	Log₁₀ of the maximum angular velocity: ω_max = 10² (deep diabatic regime).
W_RES	4	Base resolution of the ω sweep. Actual number of ω values compiled in = W_RES × THREADS = 72, distributed evenly in log-space between W_START and W_END.
USE_MAGMA	FALSE	Disable GPU-accelerated diagonalization via MAGMA. Set TRUE if a CUDA-capable GPU and MAGMA are available.
USE_GPU	FALSE	Disable GPU offload for matrix operations. Required alongside USE_MAGMA=TRUE for full GPU path.
VERBOSE	TRUE	Enable debug output (DEBUG_ACTIVE defined). Prints progress, thread counts, and intermediate results to stdout.
TIME_EVOLUTION	TRUE	Run the adiabatic time evolution / braiding simulation. When FALSE, only the static eigenvalue sweep (μ or Δ scan) is performed.

Output: CSV files written to Simulation/build/ (eigenvalue spectra, eigenvector data, fidelity decay).

Evaluation/

Python scripts for post-processing simulation output. Scripts read CSV files from Simulation/build/ and produce plots. Dependencies: numpy, matplotlib, scipy.

Script	Purpose
Eval_EigenValues.py	Plot eigenvalue spectra
Eval_EigenVectors_1D.py	1D eigenvector visualization
Eval_EigenVectors_2D.py	2D eigenvector color maps
Eval_EigenVectors_2D_3D-Scatter.py	3D scatter of 2D eigenvectors
Eval_Matrix.py	Hamiltonian matrix visualization
Kitaev_Continuety.py	Kitaev chain continuity analysis
JobFactory.py	Batch job generation for cluster

Mathematika/

Mathematica notebooks for analytical cross-checks (Calcs.nb, Majoranaize.nb).

Dependencies

Library	Purpose
Eigen ≥ 3.3.9	Linear algebra
Accelerate (macOS) / MKL (Linux)	BLAS/LAPACK
OpenBLAS (macOS)	LAPACKE C interface
OpenMP	Parallelism
Boost ≥ 1.70	Utilities
MAGMA (optional)	GPU diagonalization