Chiral Edges, Interfaces, and Bulk–Boundary Tests

A fractional Hall edge is the boundary realization of the bulk gauge anomaly. Bulk topological data fix its net charge anomaly and chiral central charge, but interactions, disorder, confinement, and reconstruction determine the number, velocities, and equilibration of observable modes. Consequently, two edges of the same bulk phase need not share an identical spectrum.

Required background. Landau-level projection supplies boundary orbitals; fractional Hall fluids supplies the bulk response; K-matrix data supplies Abelian quasiparticles; anomaly inflow supplies the boundary variation.

Helpful background. Two-dimensional conformal symmetry supplies chiral scaling language.

Abelian edge action

For an Abelian $K$ matrix, a general quadratic edge is

$$S_{\partial}=\frac{1}{4\pi}\int dt\,dx \left(K_{IJ}\partial_t\phi_I\partial_x\phi_J -V_{IJ}\partial_x\phi_I\partial_x\phi_J\right) -\frac{e}{2\pi}\int t_I A\,d\phi_I.$$

$K$ fixes equal-time commutators and topology; the positive matrix $V$ contains nonuniversal velocities and interactions. A vertex $e^{il^{\mathsf T}\phi}$ has charge $e\,t^{\mathsf T}K^{-1}l$ and statistical data inherited from the bulk. Gauge variation of the edge cancels that of the bulk Chern–Simons action Wen 1992.

The net electrical anomaly is $\nu=t^{\mathsf T}K^{-1}t$. With full thermal equilibration, the ideal low-temperature thermal Hall coefficient is

$$\frac{\kappa_{xy}}{T}=c_-\frac{\pi^2k_B^2}{3h}, \qquad c_-=\operatorname{signature}K$$

for the minimal Abelian edge. Incomplete equilibration, heat leakage, phonons, and extra integer modes can change the measured value over finite lengths.

Gapping an interface

An interaction $\cos(l_a^{\mathsf T}\phi)$ can pin mutually compatible combinations when

$$l_a^{\mathsf T}K^{-1}l_b=0$$

for all condensed vectors, with $t^{\mathsf T}K^{-1}l_a=0$ when charge is preserved. One also checks locality and primitivity. A complete set of such null vectors can gap a nonchiral interface; a purely chiral edge has no counterpropagating partner and cannot be fully gapped while preserving its anomaly.

Edge reconstruction adds counterpropagating pairs when the confining potential is smooth. Disorder may drive them toward an equilibrated fixed point or localize neutral combinations. Universal tunneling exponents occur only at a specified edge fixed point; electrostatics can make measured exponents nonuniversal.

Bulk–boundary tests

Compare net downstream-minus-upstream charge and heat flow, interface gappability, and anomaly matching—not a single dispersion. State the boundary orientation and which vacuum or phase lies on each side. An interface between two orders is governed by the folded matrix $K_L\oplus(-K_R)$, so identical bulks can gap even when either vacuum edge is chiral.

Exercise

For $K=\operatorname{diag}(1,-1)$ and $t=(1,1)^{\mathsf T}$, show that $l=(1,1)^{\mathsf T}$ is a charge-neutral null vector.

Solution

$K^{-1}=K$. Thus $l^{\mathsf T}K^{-1}l=1-1=0$, and $t^{\mathsf T}K^{-1}l=1-1=0$. A local cosine built from this vector can gap the counterpropagating pair without breaking charge conservation, subject to the locality and primitivity checks.

References

• Xiao-Gang Wen, “Theory of the Edge States in Fractional Quantum Hall Effects,” International Journal of Modern Physics B 6 (1992) 1711–1762, doi:10.1142/S0217979292000840.