Are the definitions of chirality in continuum QFT and the Nielsen-Ninomiya theorem equivalent?

Question

I have seen two definitions of chirality in quantum field theory:

1. According to the Wikipedia article, chirality is defined as whether a particle transforms under a left- or right-handed representation of the Poincare group. Chiral states are eigenstates of the chirality operator $\gamma^5$ with eigenvalues $\pm 1$.
2. In Eq. (4.28) of these notes, chirality of a Hamiltonian $H(\mathbf{k}) = v_i(\mathbf{k}) \sigma^i + \epsilon(k)\mathbf{1} $ is given by $$ \chi=\mathrm{sgn}\ \mathrm{det}(v_{ij}(\mathbf{k}_\alpha)), \quad v_{ij}=\frac{\partial v_i}{\partial k^j}$$ where $\mathbf{k}_\alpha$ are the points at which $v_i(\mathbf{k}_\alpha) = 0$ for all $i$.

At first glance, these two definitons seem completely different. The first definition is concerned with representations of the Poincare group where chirality is a property of states, whilst the second definition defines chirality as a property of the Hamiltonian and not the states.

These two definitions must be related somehow as the latter is used when discussing quantising a fermion on a lattice (the Nielsen-Ninomiya theorem), however I cannot see how these are related. If these two definitions are indeed equivalent, how does one show it?

Answer 1

Of course these two definitions are not equivalent - one is for a continuum QFT and the other for a lattice theory. A lattice breaks Poincaré invariance, so there's no representations of the Lorentz group to look at for chirality - the notion of chirality doesn't really make sense for a lattice at first glance.

However, the notes by Tong you link in the question are trying to make an argument for why the latter is the correct notion of "chirality on the lattice". Note that Tong is looking at the Hamiltonian of a single fermion. Just prior to eq. (4.28), he discussed the "naive" discretized version of a Weyl fermion in 3+1 dimensions, and shows that the single-particle Hamiltonian is given by $$ H(k)\approx \pm (k-k_0)\cdot \sigma,\tag{1}$$ around points $k_0$ in the Brillouin zone where $\sin(k a) = 0$, where the sign $\pm$ corresponds to the chirality of the continuum fermion prior to discretization.

The generalized fermion on the lattice is really just a straightforward generalization: Instead of the $\sin$, we have arbitrary functions $v_i$, and expanding about a point with $v_i(k_\alpha) = 0$, the first non-vanishing order is $$ H(k) \approx (k-k_\alpha)(\mathrm{D}v(k_\alpha))\sigma,\tag{2}$$ where $\mathrm{D}v$ is the Jacobian matrix of the $v_i$, and by comparing eq. (2) to eq. (1) you see that the sign of $\mathrm{det}(\mathrm{D}v(k_\alpha))$ in eq. (2) corresponds to the $\pm$ in eq. (1), hence is related to the chirality of the fermion.