What is the physical significance of links between sub lattices in the staggered fermion formulation of lattice gauge theories?

Question

I am currently learning about Lattice Gauge theories and have come across the Lattice formulation in terms of staggered fermions (see e.g. Susskind 1977, Kogut & Susskind 1975).

As I understand it, we take the components of e.g. a 2-component fermion spinor and spread them out on two sub lattices, which are then endowed with the staggered sign to beat the Fermion Doubling problem. Now the way the gauge field enters is through parallel transport operators on links between lattice sites. They represent the degrees of freedom of the gauge field between the different lattice sites.

What is now unclear to me is what the physical significance / interpretation of links between sites of the sub lattices are? The links between physical lattice sites are clear to me: Since my gauge transformation affects every point in spacetime differently, I need to include parallel transporters when I wish to compare one spacetime point to another. But as I understand it, each site has a corresponing site on the second sub lattice (this is where the second spinor component, the antiparticle goes), but these two points would in fact correspond the the exact same location in spacetime. So why then is there a link between these points and what is the significance of this link?

Answer 1

This is a nice question; staggered fermions can be tricksy and it's possible to describe them in various ways. In the formalism I'm used to, all sites are equally (un)physical. They are a finite distance apart and only correspond to the same location in spacetime in the $a \to 0$ continuum limit that removes the regulator $\propto 1 / a$ and provides a physical quantum field theory. It's only in this limit that the spinor components formally recombine into the Dirac or Majorana fermions of interest.

Rather than thinking about 'spreading out' the components of a single fermion, you can construct staggered fermions by taking naive lattice fermions (with $2^d$-fold doubling), choosing a spin-diagonal basis for them, and then just discarding all but one spinor component on each site. For $d = 4$, this reduces the number of Dirac fermions in the continuum limit from 16 to 4.

Note that there are still 'particles' and 'anti-particles' at every site. (The "second spinor component" isn't exactly "the antiparticle".) You can further halve the doubling by discarding the 'particles' on all odd sites and the 'anti-particles' on all even sites. This is known as the "reduced staggered" fermion formulation and is not very widely used, for two reasons. 1) It forbids mass terms of the form $m\overline\chi_n\chi_n$ at lattice site $n$. 2) For fermions in complex irreps of the gauge group (including the fundamental rep of SU($N$) with $N \geq 3$) it introduces a sign problem.

Anyway, depending on your background you might recognize the distribution of spinor components around the unit hypercube as corresponding to a "twisting" of the Lorentz and flavor symmetries.

I'd suggest consulting modern textbooks or lecture notes such as hep-lat/9807028 rather than relying on the original literature from the '70s.