Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a nontrivial <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn></a:msub></a:math> index in a spinful two-dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:msub><c:mi mathvariant="script">C</c:mi><c:mi>s</c:mi></c:msub><c:mo>=</c:mo><c:mrow><c:mo>(</c:mo><c:msub><c:mi mathvariant="script">C</c:mi><c:mo>↑</c:mo></c:msub><c:mo>−</c:mo><c:msub><c:mi mathvariant="script">C</c:mi><c:mo>↓</c:mo></c:msub><c:mo>)</c:mo></c:mrow><c:mo>/</c:mo><c:mn>2</c:mn></c:mrow></c:math>. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi mathvariant="double-struck">Z</g:mi><g:mn>2</g:mn></g:msub></g:math> index vanishes. Identification of solid-state systems hosting an odd, quantized <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:msub><i:mi mathvariant="script">C</i:mi><i:mi>s</i:mi></i:msub></i:math> and trivial <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:msub><k:mi mathvariant="double-struck">Z</k:mi><k:mn>2</k:mn></k:msub></k:math> index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi mathvariant="double-struck">Z</m:mi><m:mn>2</m:mn></m:msub></m:math> index and odd <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mi mathvariant="script">C</o:mi><o:mi>s</o:mi></o:msub></o:math>. We demonstrate the potential of one such system, <q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mrow><q:msub><q:mi>Ti</q:mi><q:mn>2</q:mn></q:msub><q:msub><q:mi>CO</q:mi><q:mn>2</q:mn></q:msub></q:mrow></q:math>, to support Majorana corner modes via the superconducting proximity effect. Published by the American Physical Society 2024