Boundary symmetry breaking of flocking systems

Abstract We consider a flocking system confined transversally between two infinite reflecting parallel walls separated by a distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mo>⊥</mml:mo> </mml:msub> </mml:mrow> </mml:math> . Infinite or periodic boundary conditions are assumed longitudinally to the direction of collective motion, defining a ring geometry typical of experimental realizations with flocking active colloids. Such a confinement selects a flocking state with its mean direction aligned parallel to the wall, thus breaking explicitly the rotational symmetry locally by a boundary effect. Finite size scaling analysis and numerical simulations show that confinement induces an effective mass term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:mrow> <mml:mo>∼</mml:mo> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mo>⊥</mml:mo> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>ζ</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> (with positive ζ being equal to the dynamical scaling exponent of the free theory) suppressing scale free correlations at small wave-numbers. However, due to the finite system size in the transversal direction, this effect can only be detected for large enough longitudinal system sizes (i.e. narrow ring geometries). Furthermore, in the longitudinal direction, density correlations are characterized by an anomalous effective mass term. The effective mass term also enhances the global scalar order parameter and suppresses fluctuations of the mean flocking direction. These results suggest an equivalence between transversal confinement and driving by an homogeneous external field, which breaks the rotational symmetry at the global level.