Engineering unique localization transition with coupled Hatano-Nelson chains

The paradigmatic Hatano-Nelson (HN) Hamiltonian induces a delocalization-localization (DL) transition in a one-dimensional (1D) lattice with random disorder, in striking contrast to its Hermitian counterpart. The DL transition also persists in the presence of a quasiperiodic potential separating completely delocalized and localized eigenstates. In this study, we reveal that coupling two 1D quasiperiodic Hatano-Nelson (QHN) lattices significantly alters the nature of the DL transition and identify two critical points, V_{c1} &lt; V_{c2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>&lt;</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> , when the nearest neighbors of the two 1D QHN lattices are cross-coupled with strong hopping amplitudes under periodic boundary conditions (PBC). Complete delocalization occurs below V_{c1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> and the states are completely localized above V_{c2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> , while two mobility edges symmetrically emerge about Re[E] = 0 between V_{c1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> and V_{c2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> . Notably, under specific asymmetric cross-hopping amplitudes, V_{c1} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>V</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> approaches zero, resulting in localized states even for an infinitesimally weak potential. Remarkably, we also find that the mobility edges precisely divide the delocalized and localized states in equal proportions. We demonstrate a possible implementation of these findings in a coupled waveguided array which can be exploited to control and manipulate the light localization depending upon the hopping amplitude in the two QHN chains.