Continuous topological phase transition between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> topologically ordered phases

Topological phase transitions beyond anyon condensation remain poorly understood. A notable example is the transition between the toric code (TC) and double semion (DS) phases, which has two distinct <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> topological orders in (2+1) dimensions [ <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mo>(</c:mo> <c:mn>2</c:mn> <c:mo>+</c:mo> <c:mn>1</c:mn> <c:mo>)</c:mo> <c:mi mathvariant="normal">D</c:mi> </c:mrow> </c:math> ]. Previous studies reveal that the transition between them can be either first order or via an intermediate phase, thus the existence of a direct continuous transition between them remains a long-standing problem. Motivated by the fact that both phases can arise from condensing distinct anyons in the <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 order, we introduce a perturbed <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:msub> <g:mi mathvariant="double-struck">Z</g:mi> <g:mn>4</g:mn> </g:msub> </g:math> quantum double (QD) model to study the TC-DS transition. We confirm the existence of a continuous <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mrow> <i:mo>(</i:mo> <i:mn>2</i:mn> <i:mo>+</i:mo> <i:mn>1</i:mn> <i:mo>)</i:mo> <i:mi mathvariant="normal">D</i:mi> </i:mrow> </i:math> XY* transition between the TC and DS phases by mapping it to a two-coupled quantum Ising model. Importantly, using the condensation order parameters and the area law coefficients of the Wilson loops, we further reveal that <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi mathvariant="double-struck">Z</k:mi> <k:mn>4</k:mn> </k:msub> </k:math> anyons, fractionalized from the <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> topological orders, become deconfined at the transition between <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"> <o:msub> <o:mi mathvariant="double-struck">Z</o:mi> <o:mn>2</o:mn> </o:msub> </o:math> topologically ordered phases. Our results open a path toward developing a theoretical framework for topological phase transitions beyond anyon condensation.