Split-Step-Explorer

Accompanying the paper: Bulk-edge correspondence of one-dimensional quantum walks by C. Cedzich, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner, and R. F. Werner.

To operate, move the mouse over the diamond-patterned parameter-space. Clicking on it fixes \(\theta_1\) and \(\theta_2\), while clicking again returns to the mouse-following-mode.

Explanation

System

The famous quantum walk example that is found throughout the literature of topology in quantum walk, the so called split-step-quantum-walk:

\(W=S_\downarrow R(\theta_2) S_\uparrow R(\theta_1)\)

Parameter plane

Here you can set the two angles, \(\theta_1,\ \theta_2 \in [-\pi,\pi]\) by clicking or by moving the crosshairs with the mouse.
The colors indicate the right symmetry index \(\overrightarrow{\mathrm{si}}(W)\) of the walk as defined in the paper.
White lines : here a gap closes (see band structure), so the model is not in the class considered.
The initial point is the so - called Hadamard walk.

Bulk index by
winding number

The ellipse shown is the off - diagonal matrix element of the walk operator with respect to the basis in which the chiral symmetry is diagonal.
The parameter is the quasi-momentum. The winding number of this curve with respect to the origin is shown in the top right corner. By definition, this agrees with the chiral bulk index of the walk and hence coincides with the index given in the parameter plane.

Band structure

The band structure, i.e., quasi-energy over quasi-momentum. Observe that a gap closes at \(0\) or \(\pi\), when a white line is crossed in the parameter plane.

Eigenfunctions of
decoupled walk

These are the eigenfunctions of the walk,
when decoupled by putting a \(\pm\sigma_x\) coin at site \(0\) instead of \(R(\theta_2)\), or
when decoupled by putting a \(\pm i \sigma_y \) coin at site \(0\) instead of \(R(\theta_2)\).
All four decouplings preserve both the chiral and the particle-hole symmetry.
Decoupling with \(\pm i \sigma_y \) is "gentle", since the angle \(\pi/4\) can be changed continuously to the value \(\theta_2\), while satisfying all symmetry-constraints.

Color coding

The color coding for the eigenfunctions is as follows:

\(W\psi=\hphantom{-}\psi\)
\(W\psi=-\psi\)
\(\hspace{.5em}\gamma\psi=-\psi\)
\(\hspace{.5em}\gamma\psi=\hphantom{-}\psi\)

The eigenfunctions are exactly powers: \(\psi(x) = \mu^x \psi_0\). Depending on whether \(|\mu|<1\) (resp. \(|\mu|>1\)) the eigenfunction belongs to the right (resp. left) half chain, provided that the eigenvalue equation is also solved at the boundary.

The symmetry index as defined in the paper can be determined by adding the chiralities (blue minus red) of all eigenfunctions on the right hand side.

An alternative proposal by Asboth and Obuse suggests to get a finer classification by doing this separately for the two eigenvalues, i.e., to either count only solid or only dashed lines. As is evident from comparing the two diagrams, the result then depends on the cutting method. That is, this proposal is not topologically invariant.

The inset table in the figures shows all the indices as defined in the paper in the following format:

\(\mathrm{si_+}\hspace{-.2em}(W_L)\) \(\mathrm{si_+}\hspace{-.2em}(W_R)\) \(\mathrm{si_+}\hspace{-.2em}(W)\)
\(\mathrm{si_-}\hspace{-.2em}(W_L)\) \(\mathrm{si_-}\hspace{-.2em}(W_R)\) \(\mathrm{si_-}\hspace{-.2em}(W)\)
\(\mathrm{si}_\,\hspace{-.2em}(W_L)\) \(\mathrm{si}_\,\hspace{-.2em}(W_R)\) \(\mathrm{si}_\,(W)\)

The underlying theory is now published on the arXiv. It contains the complete classification to fully understand these explanations and its implications.

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