Network model for magnetic higher-order topological phases

We propose a network-model realization of magnetic higher-order topological phases (HOTPs) in the presence of the combined space-time symmetry <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msub><a:mi>C</a:mi><a:mn>4</a:mn></a:msub><a:mi mathvariant="script">T</a:mi></a:mrow></a:math>—the product of a fourfold rotation and time-reversal symmetry. We show that the system possesses two types of HOTPs. The first type, analogous to Floquet topology, generates a total of eight corner modes at 0 or <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mi>π</c:mi></c:math> eigenphase, while the second type, hidden behind a weak topological phase, yields a unique phase with eight corner modes at <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>±</d:mo><d:mi>π</d:mi><d:mo>/</d:mo><d:mn>2</d:mn></d:mrow></d:math> eigenphase (after gapping out the counterpropagating edge states), arising from the product of particle-hole and phase-rotation symmetry. By using a bulk <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:msub><e:mi mathvariant="double-struck">Z</e:mi><e:mn>4</e:mn></e:msub></e:math> topological index <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mo>(</g:mo><g:mi>Q</g:mi><g:mo>)</g:mo></g:math>, we found both HOTPs have <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:mrow><h:mi>Q</h:mi><h:mo>=</h:mo><h:mn>2</h:mn></h:mrow></h:math>, whereas <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:mi>Q</i:mi><i:mo>=</i:mo><i:mn>0</i:mn></i:mrow></i:math> for the trivial and the conventional weak topological phase. Together with a <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:msub><j:mi mathvariant="double-struck">Z</j:mi><j:mn>2</j:mn></j:msub></j:math> topological index associated with the reflection matrix, we are able to fully distinguish all phases. Our work motivates further studies on magnetic topological phases and symmetry-protected <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mn>2</l:mn><l:mi>π</l:mi><l:mo>/</l:mo><l:mi>n</l:mi></l:mrow></l:math> boundary modes, as well as suggesting that such phases may find their experimental realization in coupled-ring-resonator networks. Published by the American Physical Society 2024