Integer Quantum Hall Matter and Chern Insulators

An integer quantum Hall or Chern insulator is a two-dimensional charge-conserving gapped phase whose occupied projector has nonzero first Chern number. That integer fixes the Hall coefficient and the net chirality at an interface, provided orientation, charge sign, chemical potential, and gap hypotheses are held fixed.

Required background. Berry geometry supplies the occupied-projector curvature; Kubo response supplies the transport limit.

Helpful background. Background responses supplies the Chern–Simons interpretation.

Chern number and Hall sign

For positive Brillouin-zone orientation $dk_x\wedge dk_y>0$ and $\mathcal A_i=i\langle u|\partial_i u\rangle$, define

$$C=\frac{1}{2\pi}\sum_{n\in\mathrm{occ}} \int_{\rm BZ}\Omega^{(n)}_{xy}(\mathbf k)\,d^2k\in\mathbb Z.$$

For electrons of charge $-e$ and $j_x=\sigma_{xy}E_y$, the dc Hall conductivity in this convention is

$$\sigma_{xy}=-C\frac{e^2}{h}.$$

The minus sign is not universal notation: using $-i\langle u|\partial u\rangle$, reversing $(k_x,k_y)$, or reporting carrier rather than electron charge changes the printed relation. The invariant comparison translates all conventions together. The Kubo derivation requires the transport limit and a Fermi level in a bulk spectral gap; with disorder, a mobility gap suffices because localized states do not carry dc Hall current Thouless et al. 1982.

A lattice two-band model

Consider

$$H(\mathbf k)=\sin k_x\,\sigma_x+\sin k_y\,\sigma_y +(m+\cos k_x+\cos k_y)\sigma_z.$$

The gap closes at $m=-2,0,2$. For the lower band and the conventions above,

$$C=\frac12\big[\operatorname{sgn}(m+2)+\operatorname{sgn}(m-2)-2\operatorname{sgn}m\big],$$

so $C=1$ for $-2<m<0$, $C=-1$ for $0<m<2$, and $C=0$ outside. Each massive Dirac point contributes a half-integer local value, but the lattice supplies all cones and makes the total integer. A continuum Dirac Hamiltonian without its ultraviolet regularization cannot determine the absolute lattice Chern number.

Bulk–edge statement

At an interface between insulators with Chern numbers $C_L$ and $C_R$, spectral flow gives net right-minus-left-moving electron channels equal to $-(C_R-C_L)$ under the same edge-orientation convention. Edge reconstruction may add counterpropagating pairs, but it cannot change the net chirality without closing the bulk mobility gap or violating charge conservation. This is stronger than observing one edge-shaped dispersion: the count must be compared across a specified interface Hatsugai 1993.

Weak interactions preserve the integer phase while the many-body gap and charge conservation survive. Intrinsic fractional Hall order can have a fractional Hall coefficient and anyons and is not classified by an occupied-band Chern number. A local Berry-curvature hotspot, a band inversion, or an edge-like state alone is insufficient.

Exercise

Evaluate the Chern number of the lattice model at $m=-1$ and give $\sigma_{xy}$ in the stated convention.

Solution

The signs are $\operatorname{sgn}(m+2)=1$, $\operatorname{sgn}(m-2)=-1$, and $\operatorname{sgn}m=-1$. Therefore $C=(1-1+2)/2=1$ and $\sigma_{xy}=-e^2/h$.

References

• Yasuhiro Hatsugai, “Chern Number and Edge States in the Integer Quantum Hall Effect,” Physical Review Letters 71 (1993) 3697–3700, doi:10.1103/PhysRevLett.71.3697.
• David J. Thouless, Mahito Kohmoto, M. Peter Nightingale, and Marcel den Nijs, “Quantized Hall Conductance in a Two-Dimensional Periodic Potential,” Physical Review Letters 49 (1982) 405–408, doi:10.1103/PhysRevLett.49.405.