Does the Mott insulator exist? The Mott insulator is a system that, due to strong electron-electron interactions, is an insulator but is expected to be a metal by formal charge counting of electrons in the unit cell.
Often, the Mott insulator is defined to be the ground state of the Hubbard model at half filling
$$H = -t \sum_{i} (c^{\dagger}_{i+1} c_{i} + c^{\dagger}_{i} c_{i+1}) + U\sum_{i} n_{i\uparrow} n_{i \downarrow}$$
Presumably, the Mott insulator is the ground state of the Hubbard model. Yet, in a paper by Robert Laughlin (Nobel prize in Condensed Matter Physics 1998), Phys. Rev. B 89, 035134 (2014), he states the following (emphasis mine):

Unfortunately, the phenomenological definition of a Mott insulator has always been somewhat difficult to state [...].


The enormous amount of theoretical work stimulated by the cuprate discovery has now built up a strong case that the Mott insulator does not exist as a distinct zero-temperature state of matter.

Laughlin says similar things in Phys. Rev. Lett. 112, 017004 (2014)

Unfortunately, the cuprates are so anomalous phenomenologically that they have thus far defied categorization as conventional metals or insulators. This has led to speculation that they might involve a new, and as-yet unidentified, parent vacuum. Proposals for such a vacuum include the Mott insulator, the resonating valence bond, the non-Fermi liquid, and the loop-current insulator[4–7].
However, there is a much simpler potential explanation: the zero-temperature phases of the cuprates are conventional,and the strange behaviors are just critical phenomena and glassiness associated with transitions among two or more of these phases [8–10]. This view is supported by the absence of experimental evidence for new states of matter at lowest temperature scales. It is also supported by theory, in that none of the proposed theoretical alternatives to conventional metals and insulators can be (1) written down in a straight forward way at zero temperature or (2) shown to be stabilized by any simple Hamiltonian. There is no mathematical case that any of them actually exist.

I know Laughlin has a reputation of being strong headed (to say the least), but I want to understand the basis for these statements. Clearly he has reasons to make these statements in such absolute terms, but mainstream physicists treat the Mott insulator as if its existence is unquestionable. The "textbook" examples of Mott insulators are NiO and La2CuO4, which definitely seem to be insulators at zero temperature, but presumably these examples "don't count" to Laughlin.
So, my question is: does the Mott insulator exist as a distinct ground state?
If the Mott insulator does exist, why does Laughlin argue it does not? If the Mott insulator does not exist, why is its nonexistence seemingly ignored by most physicists?
An aside: I disagree with the comments by user:wcc which claim ultracold atom simulations have "solved" this issue long ago and that Laughlin is only referring to superconducting copper oxides. First of all, ultracold atom simulators cannot reach the ground state of the Hubbard model (they are much too hot to be anywhere close). But more importantly, Laughlin claims the Mott insulator does not exist, and that is a much stronger statement which is about pure physics. Finally, Laughlin's papers were published is 2014, which is many years after the start of cold atoms on optical lattices which began in the late 90's, so I am confident the Novel laureate author is aware of cold atom simulations but finds them irrelevant to his points.
 A: According to Mott's original idea, a metal-Mott insulator phase transition the Coulombic repulsion leads to localization of single-electron wavefunction. In the purest situation this transition is not accompanied by any other symmetry breaking; the only thing that occurs is the localization of electron wavefunctions (an ideal Mott insulator must be then, among other things, a quantum spin liquid in its ground state).
This is a highly non-trivial idea and this is not surprising that Laughlin is not the first one to question whether Mott insulators are realized in real world (or, in a more mild form of it, whether the "canonical" Mott insulator such as Nichel Oxide or Vanadium Oxide are indeed insulator in Mott sense). In fact the most famous name here is John Slater. His argument was that there is no need to invoke such intricate arguments to explain the insulating behavior of Nickel- or Vanadium Oxide. A change in the unit cell like in a charge- or spin density wave would do the job without any exotica with localization involved (see e.g. Peierls transition).
So this is what Laughlin's point is about. There are undoubtedly many interaction driven metal-insulator transitions out there however none of them has been convincingly demonstrated to be a Mott insulator in the original sense of the term. For example in this survey doi.org/10.1080/00018737700101443 the author have analyzed the existing experimental data on the "classic" Mott insulator and shows that in all of them the Mott transition is accompanied by a break of translation symmetry of the ground state of some sort (Think of the antiferromagnetism of undoped La$_2$CuO$_4$).
This is a known problem in Condensed Matter community. For instance there is whole sub-field of ultrafast vanadium oxide (V$_2$O$_3$) studies in which the (potential) Mott insulating state is accompanied by a structural phase transition and people are pondering whether it is a Mott insulator in which the insulating state happens to facilitate the structural changes of the lattice or whether it is a Slater insulator after all. There has not been a conclusive answer as of yet, to the best of my knowledge (in this regard it is worthwhile to mention that Mott transition should in fact indeed lead to the structural changes. Indeed, following Mott, the electrons cease to be delocalized in the insulating phase, thus reducing the cohesion among the atoms in the material, assuming that sharing of valence electrons in the metallic phase has a contribution to the binding energy. On the other hand this would imply a reduction in the elastic modulus as a rule for any Mott insulator. To the best of my knowledge, such behavior is not reported for the common materials usually called Mott insulators).
