Disorder-to-order transition in one-dimensional nonreciprocal Cahn-Hilliard model

We present the phenomenology of the one-dimensional nonreciprocal Cahn-Hilliard model for varying nonreciprocity <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mo>(</a:mo> <a:mi>α</a:mi> <a:mo>)</a:mo> </a:mrow> </a:math> and different boundary conditions. At small <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>α</b:mi> </b:math> , a perturbed uniform state evolves to a defect-laden configuration that lacks global polar order. Defects are the sources and sinks of traveling waves. For a given <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>α</c:mi> </c:math> , defects with a unique wave number that increases monotonically with <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>α</d:mi> </d:math> are selected. A critical threshold <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>α</e:mi> <e:mi>c</e:mi> </e:msub> </e:math> marks the onset of a transition to states with finite global polar order. For periodic boundary conditions, above <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msub> <f:mi>α</f:mi> <f:mi>c</f:mi> </f:msub> </f:math> , the system shows traveling waves that are completely ordered. In contrast, traveling waves are incompatible with the Neumann and Dirichlet boundary conditions. Instead, for <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>α</g:mi> <g:mo>≳</g:mo> <g:msub> <g:mi>α</g:mi> <g:mi>c</g:mi> </g:msub> </g:mrow> </g:math> , we find fluctuating domains that show intermittent polar order, and at large <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:mi>α</h:mi> </h:math> , the system partitions into two domains with opposite polar order.