can a gapless system be a topological state?

For a gapless system without boundary (i.e. in the bulk there is gapless excitation while no clear meaning of boundary excitations like QFT), can it be a topological state? What is the property of EE in these systems? Thanks!

One first needs to define what it means for a state to be topological when the system is gapless. This is not an easy question! I don't think there is a unified answer at the moment.

I'll give some examples of exotic gapless states which are "proximate" to gapped topological states. Here I take "topological" to mean exotic low-energy excitations (much like anyons). But of course there are other notions of topological gapless states, such as Weyl semimetals in 3D which can be characterized by exotic boundary states.

1. Emergent Fermi liquid. By this I mean the following: starting from a gapped topological phase, which contains emergent fermionic quasiparticles (not the electrons). Now imagine the system is tuned in such a way that the gap to this fermionic excitation is reduced, while the gap to all other types of quasiparticles do not close. Eventually, the gap to the fermionic quasiparticles close, and we imagine those fermions form some kind of Fermi liquid. A well-known example of this type is Kitaev's honeycomb lattice model, at the isotropic point. The gapless emergent fermions are still coupled to a $\mathbb{Z}_2$ gauge field. A related example is the $\nu=1/2$ composite Fermi liquid in FQH, but it has a different gauge structure ($\mathrm{U}(1)$ gauge field).

2. Fluctuating order parameters. Some topological states, especially those that occur in free fermion band systems, are sometimes associated with symmetry breaking. For example, topological superconductors break $\mathrm{U}(1)$ symmetry. One can ask what if the superconducting order parameter has phase fluctuations (gapless Goldstone mode). An example that is studied very thoroughly is 1D class D superconductor, with such phase fluctuations, see http://arxiv.org/abs/1106.2598. The phase fluctuations change the exponential splitting between the degenerate ground states to power law.

I think in both cases people would have no issue calling the gapless states "topological", since they still inherit some of the nontrivial low-energy excitations from the nearby topological states. But a general characterization of gapless states does not exist yet.

I recommend the very inspiring paper http://arxiv.org/abs/1212.6395, where the authors defined "quasi-topological phases".

• Thanks for your helpful comment! More naive questions: Is Weyl semi-metal a topological system with gapless excitations? Is it possible that for such system (e.g. translational invariant ideal WSM in some strongly coupled case) there is no Fermi arc like boundary excitations? Thanks! – user9510 Dec 11 '15 at 20:56
• Ha, I forgot to mention this most popular example. Weyl semi-metal is topological in the sense that the Weyl point is a monopole in the momentum space, although they are not stable to translation-breaking perturbations. Although I don't know what "strong coupled case" you have in mind, it is unlikely that the interactions can get rid of Fermi arc without a phase transition in the bulk. – Meng Cheng Dec 11 '15 at 21:13
• I was curious about the role of boundary excitations in WSM. Is it possible to realise a similar topological system without any boundary excitations? Maybe I am wrong, since the effective electromagnetic interaction strength in WSM systems is or order 1, WsM should be a strongly coupled system. The free Weyl fermion picture (with Weyl nodes) might be broken down in some strongly coupled WSM systems (thus no boundary state?), but it might be still possible to be in a topological state. – user9510 Dec 11 '15 at 21:51
• Well, our understanding of Weyl semimetals is based on non-interacting electrons, so is the material realizations. Electromagnetic field is not part of the system, we only turn on a background EM field to probe the EM response of the system. You can definitely imagine some exotic states obtained from coupling Weyl fermions to dynamical U(1) gauge fields, and free fermion picture could break down, but it does not mean that there is no boundary state. – Meng Cheng Dec 11 '15 at 22:02
• I am totally from outside:) Is there any material example in which all the transport physics are very similar to WsM (can be explained by chiral anomaly, e.g. negative magneto resistance, anomalous Hall effect etc.) but there is no similar ARPES observations (Weyl nodes + Fermi Arc)? If one realise such material, would it be topological matter? – user9510 Dec 11 '15 at 22:19