Nielsen-Ninomiya theorem

Question

The Nielsen-Ninomiya theorem states that for a local, Hermitian, translationally invariant lattice fermion theory in even-dimensional spacetime, the number of left-handed Weyl fermions is equal to the number of right-handed Weyl fermions.

I don't understand dimensions in this concept. This theorem is stated in some sources to be true for even dimensional systems, and in some sources to be true for odd dimensional systems. For example, it is stated to be true for 1+1 or 3+1 dimensional systems. However, its proof via differential topology is related to the Brillouin torus, which corresponds to 2+1 dimensions.

I'm confused, please help me.

Answer 1

The Nielsen-Ninomiya theorem applies to lattice fermion theories with even spacetime dimension $D$, not the spatial dimension $d$. The confusion arises because the proof involves the Brillouin zone, which has dimension $d$. Spacetime dimension $D$ refers to the total dimensions, including time (e.g., $D=2$ for 1+1 dimensions). Spatial dimension $d$ refers to the number of spatial dimensions only (e.g., $d=1$ for 1+1 dimensions). For a local, Hermitian, translationally invariant lattice fermion theory in even spacetime dimension $D$, the number of left-handed ($N_L$) and right-handed ($N_R$) Weyl fermions are equal:

$$N_L = N_R$$

The theorem concerns the spacetime dimension $D$. The Brillouin zone, used in the proof, is a $d$-dimensional torus. The theorem holds for even $D$ (e.g., 1+1 or 3+1 systems) regardless of the value of $d$. The winding number $W$ relates to the difference in fermion numbers:

$$W = N_L - N_R$$

The theorem asserts that $W = 0$ for even $D$. The core issue is the evenness of $D$, not $d$. The confusion stems from conflating spacetime and spatial dimensions.