The Third Binding

Dirac fields in two dimensions can trap bound states through two known mechanisms. At a domain wall — where a real mass field changes sign — the Jackiw-Rebbi state sits at the interface, pinned to the sign change. At a vortex — where a complex mass field winds around a singularity — the Jackiw-Rossi zero mode sits at the point defect, pinned to the winding number. Walls and vortices. Lines and points. These are the two mechanisms the field has catalogued.

Zhu, Ma, Wang, Liu, Zhang, Wang, Zhang, and Chong (arXiv:2603.28127, March 2026) find a third. A branch cut — a line along which a complex function's phase jumps discontinuously — also traps guided modes. The phase doesn't change sign (that's the wall). It doesn't wind around a point (that's the vortex). It simply breaks: two regions of smooth phase separated by a curve where the phase is undefined. Along that curve, modes propagate. The branch cut is a waveguide.

The modes obey a one-dimensional relativistic Dirac equation along the cut. Their transverse confinement — how tightly they stick to the cut — is energy-independent when the magnitude of the mass field is constant. This is structurally different from domain-wall modes, where confinement weakens near the band gap edge. The cut holds its modes equally well at all energies within the gap. The binding doesn't weaken.

The acoustic experiment confirms it. Solid pillars arranged in a Kekulé-type modulation pattern, with radii encoding the complex mass field's branch structure, guide sound along the cut line — including curved and spiral paths. The relativistic dispersion and energy-independent confinement match the theory.

The structural observation: between the wall and the vortex, there was always a third option hiding in the complex plane. Branch cuts are elementary features of complex analysis — every student encounters them in the first course. But the correspondence between a mathematical discontinuity in a phase field and a physical waveguide was not obvious until someone built it. The mathematics knew about the third binding. The physics had to catch up.