3 ms·

In nature it does not matter if the spectral gap is really tiny. It's de facto without spectral gap if it's just small enough. Whereas for this particular deci

• selimthegrim 11y ago
  Who says? Graphene gaps, (it must gap) even though its so small it's impossible to observe in practice. This had very important consequences for topological materials theory...
  • sharpneli 11y ago
    This is exactly what I meant. Even if it has theoretical reasons to have a tiny gap in practice it works as zero gap.
    • tobycubitt 11y ago
      Except that in our result, the spectrum is either continuous, or the spectral gap is guaranteed to be >= 1 (in natural units). It does not become arbitrarily small in the gapped case. The reason you can't use this to compute the uncomputable is that real systems are finite, and the spectral gap is always computable in principle (maybe with a lot of effort) for any finite system. A real (possibly very large but still finite) system will either have a gap or not, and you'll be able to measure it. This definitely doesn't solve an undecidable problem. However, the undecidability in the idealised infinite lattice limit "shows through" to the experimentally accessible finite-size case, in the form of some rather unusual finite-size physics. This is discussed in more detail in the paper itself (the relevant section is quoted verbatim here http://mathoverflow.net/a/225905 http://mathoverflow.net/a/225905) and in the comments on Scott Aaronson's blog.