Interplay of superconductivity and charge-density-wave order in kagome materials

In the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>A</a:mi><a:msub><a:mi mathvariant="normal">V</a:mi><a:mn>3</a:mn></a:msub><a:msub><a:mi>Sb</a:mi><a:mn>5</a:mn></a:msub></a:mrow></a:math> (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>A</c:mi></c:mrow><c:mo> </c:mo><c:mo>=</c:mo></c:math> K, Rb, Cs) kagome materials, superconductivity coexists with a charge density wave (CDW), constituting a new platform to study the interplay of these two orders. Despite extensive research, the symmetry of the superconducting order parameter remains disputed, with experiments seemingly supporting different conclusions. As key aspects of the physics might lie in the intertwining of electronic orders, a better understanding of the impact of the CDW on superconductivity is crucial. In this paper, we develop a phenomenological framework to study the interplay of superconductivity and CDW order. In particular, we derive a Ginzburg-Landau free energy for both superconducting and CDW order parameters. Given the unclear nature of the superconducting state, we discuss general pairing symmetries with a focus on <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mi>s</d:mi></d:math>-wave, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>d</e:mi></e:math>-wave, and pair-density-wave order parameters. Motivated by experiments, we consider the additional breaking of time-reversal or point-group symmetries of the CDW and determine in detail the consequences for the superconducting state. Our results show how the superconducting state mimics the broken symmetries of the CDW and can guide future microscopic calculations, as well as the experimental identification of the superconducting state in the <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>A</f:mi><f:msub><f:mi mathvariant="normal">V</f:mi><f:mn>3</f:mn></f:msub><f:msub><f:mi>Sb</f:mi><f:mn>5</f:mn></f:msub></f:mrow></f:math> compounds.