Friday, February 11, 2011

What's new at the fermion zoo?

If there is anything more typical of the landscape of lattice QCD than collaboration acronyms that mean something very different (like a car manufacturer, a color model, or an old DOS command), to people from outside the lattice community, it has to be the fact that each of the aforementioned collaborations uses a fermion action that is in some way different from those of all other collaborations. For gauge actions, there isn't all that much variety (Wilson, tree-level Symanzik, Lüscher-Weisz with or without O(Nfαsa2) corrections, and Iwasaki), but for fermions there is a veritable zoo.

Of course, for every zoo, there is a Linnean system establishing a taxonomy, so the fermion zoo can be ordered by grouping the fermion actions into different classes:

  • Wilson fermions get rid of the doublers by adding a term (the Wilson term) to the action that explicitly breaks chiral symmetry and thus lifts the degeneracy of the doublers, giving them masses of the order of the cut-off. Wilson fermions can be subdivided further firstly into straight Wilson fermions (which have O(a) discretisation effects and hence are rarely used) and O(a)-improved Wilson fermions, which add another term, the Sheikholeslami-Wohlert term, to reduce the lattice actifacts to be O(a2). The numerous individual actions being used then differ mainly by the kind of links that go into the discretised derivatives (and possibly into the SW term), whether they are thin links for rigorous locality and positivity properties, or different kinds of smeared links for empirically better statistical behaviour of various observables.

  • twisted-mass fermions are close relatives of Wilson fermions, consisting of a doublet of unimproved Wilson fermions with a twisted mass term of the form τ3γ5; the doublet is interpreted as the up/down isospin doublet. One of the attractive features of twisted fermions is that spectral observables are automatically O(a)-improved. On the other hand, isospin and parity are violated by cut-off effects, which leads to potentially undesirable features such as a neutral pion with the quantum numbers of the vacuum.

  • staggered fermions reduce the number of doublers to four by redistributing the degrees of freedom between sites. Also here, improvement by adding an additional three-link term (the Naik term) is commonly employed. Significant use is made of smearing to reduce the impact of high-momentum gluons whose exchange results in interactions mixing the different "tastes" of remaining doublers. An advantage of the staggered formalism is the preservation of a residual chiral symmetry; a disadvantage is the need to take the root of the determinant of the Dirac operator (unless one wants to simulate with Nf=4 degenerate flavours), and issue that has been surrounded by some controversy. The actions in current use are the asqtad and HISQ actions.

  • overlap fermions are constructed as an exact solution to the Ginsparg-Wilson relation by means of the overlap operator, which is essentially the matrix sign function of the Wilson Dirac operator. While having the obvious theoretical advantage of exact chiral symmetry at finite lattice spacing, overlap fermions are very expensive to simulate, and thus are not in widespread use yet.

  • domain-wall fermions use a fictitious fifth dimension to realise chiral symmetry by localising the opposite chiralities on different "branes" or domain walls in the fifth direction. They are likewise rather expensive to simulate.


Of course, life being incredibly diverse, every taxonomist will sooner or later run into a creature which defies the existing taxonomic scheme. The past year has, I think, been such an occasion for the fermion zoo, which was increased by the addition of what may become two new families of fermions that straddle the boundaries between the classes outlined above.

One is the family of minimally doubled fermions, which are being championed by Mike Creutz and by people here at Mainz. The idea is to find an action which has the minimal number of doublers permitted for a chirally symmetric Dirac operator by the Nielsen-Ninomiya theorem, i.e. a doublet of fermions that can then be interpreted as the up/down doublet. There are two realisations of this idea, now known as Karsten-Wilczek and Creutz-Borici fermions, respectively, both of which rely on the addition of a Wilson-like term to the action. In a way, this puts them somewhere between Wilson and staggered fermions, the latter because of the existence of taste-changing interactions; of course, no rooting is required to simulate an Nf=2 theory with minimally doubled fermions. The price paid is that, because the line connecting the two poles in momentum space defines a preferred direction, at least one of the discrete spatiotemporal symmetries must be broken; this leads to the possibility of generating additional (relevant in the RG sense) dimension-3 operators in the action, which have to be fine-tuned away. Simulations with minimally doubled fermions are in preparation and will have to deal with these questions; it remains to be seen if this formulation will have practical relevance beyond its obvious theoretical impact.

The other new fermion family are the staggered overlap fermions introduced at this year's lattice conference by David Adams, and which as suggested by the name close the gap between staggered and overlap fermions. The idea here is to perform a similar construction to that used to obtain the overlap operator from the Wilson Dirac operator, but taking the staggered Dirac operator as the starting point. As it turns out, this results naturally in a theory with two fermion flavours, so again no rooting is required to simulate an up/down doublet in this fashion.

Like all taxonomy-defying creatures, these new fermion actions hold the potential to reveal hitherto unknown connections between previously unconnected classes of entities, in this case perhaps by establishing new connections between the number of flavours, chiral symmetry, doubling and the staggered formalism.