Effective model for superconductivity in magic-angle graphene

We carry out large-scale quantum Monte Carlo simulations of a candidate field theory for the onset of superconductivity in magic-angle twisted bilayer graphene. The correlated insulating state at charge neutrality spontaneously breaks U(1) Moiré valley symmetry. Owing to the topological nature of the bands, skyrmion defects of the order parameter carry charge <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>2</a:mn><a:mi>e</a:mi></a:mrow></a:math> and condense upon doping. In our calculations we encode the U(1) symmetry by an internal degree of freedom such that it is not broken upon lattice regularization. Furthermore, the skyrmion carries the same charge. The nature of the doping-induced phase transitions depends on the strength of the easy-plane anisotropy that reduces the SU(2) valley symmetry to <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi mathvariant="normal">U</b:mi><b:mrow><b:mo>(</b:mo><b:mn>1</b:mn><b:mo>)</b:mo></b:mrow><b:mspace width="0.16em"/><b:mo>×</b:mo><b:msub><b:mi mathvariant="double-struck">Z</b:mi><b:mn>2</b:mn></b:msub></b:mrow></b:math>. For large anisotropy, we observe two distinct transitions separated by phase coexistence. While the insulator to superconducting transition is of mean-field character, the U(1) transition is consistent with three-dimensional <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>X</f:mi><f:mi>Y</f:mi></f:mrow></f:math> criticality. Hence, the coupling between the gapless charge excitations of the superconducting phase and the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>X</g:mi><g:mi>Y</g:mi></g:mrow></g:math> order parameter is irrelevant. At small anisotropy, we observe a first-order transition characterized by phase separation.