So far we lack a complete framework to understand "exotic" physics, but we know states with quantum order can be remarkable in numerous ways:

• They can violate the "Bloch theory rule" for insulators: such states can be insulators with no broken symmetry and an odd (or even fractional) number of electrons per primitive unit cell.
• They can exhibit excitations with unexpected (even fractional) quantum numbers and statistics.
• They can have gapless excitations that are not "Goldstone modes" but are still robust to perturbations (sometimes all perturbations!).
• They may be described by non-trivial quantum field theories without any weakly-interacting description.
• They can describe QCPs which violate Landau's "rules" for allowed continuous phase transitions.
• They may be useful for quantum computation.

The skeptic (and most of us in condensed matter physics are!) will note the copious use of "can" and "may" in the previous topic ("Why this is cool theory!"). Indeed, there no proven experimental examples of exotic phases outside of the fractional quantum Hall effect (a zoo of exotic animals but relatively well understood).

Such deconfined states of matter are so bizarre and counter-intuitive that many respectable researchers believed until recently that they were impossible in principle outside of the quantum Hall effect. Thus it came as a surprise to many when were able to construct two explicit examples of simple spin-1/2 models that could be proven to have such ground states! This paper postulated that an easy-axis Kagome spin-1/2 antiferromagnet provides an explicit example of a so-called Z₂ spin liquid. In this model, we were able to find an exact ground state wavefunction, and more generally to define non-local "string" (dis-)order parameters that characterize the liquid state. By exact diagonalization calculations (carried out by Dong-Ning Sheng at CSUN), in this paper we were able to confirm this exotic state is stable over a wide range of phase space including a two-spin Heisenberg-type model. In this paper , we showed that a three-dimensional easy-axis spin-1/2 antiferromagnet on the pyrochlore lattice provides a simple and explicit example of a "U(1)" spin liquid in 3d. Like the Z₂ spin liquid, this state supports deconfined spinons, but has in addition an emergent gapless transverse propagating "photon"-like excitation.

With several examples, we showed that quantum critical points (QCPs) - phase transitions at zero temperature as a function of other parameters like pressure, magnetic field, etc. - can be qualitatively different from their classical counterparts. The understanding of classical critical phenomena was a triumph of theoretical physics in the 70s. It was understood then that essentially all phase transitions can be described fundamentally in terms of "order parameters" that describe spontaneous symmetry breaking across the transition. For instance, the magnetization of a ferromagnet is the order parameter for the Curie point. Lev Landau laid down the rules ("Landau theory") for constructing a description of these transitions, and for determining from the order parameters which transitions can be continuous (with smoothly varying order parameter) or discontinuous. In this paper , we showed that several QCPs are not described just in terms of their order parameters, and that they violate the Landau rules. Remarkably, instead of the order parameters, the natural variables to discuss these QCPs are instead fractional degrees of freedom (i.e. fields which carry fractional quantum numbers). Hence we dubbed these "deconfined" QCPs. Although these fractional excitations govern the criticality, they become confined in the two neighboring phases. Thus it appears that these exotic objects may be more common than naively expected in quantum critical systems.

Specifically, several such QCPs were described. The most interesting is the transition from a Neél antiferromagnet to a valence bond solid state on a square lattice. This provides an answer to the old question of how antiferromagnetism is destroyed by frustration in a spin-1/2 magnet (though we described the situation for general spin, and with magnetic anisotropy). The mechanism for this transition is novel: the topological defects of either phase (skyrmions/vortices in the antiferromagnet and domain walls and their intersections in the valence bond solid) can be shown to carry unusual quantum numbers, such that when they proliferate as their associated order is destroyed, they induce the order of the neighboring phase. Another such deconfined QCP is the transition from a valence bond solid state to a Z₂ spin liquid. We showed that at the QCP the system can be regarded as a deconfined U(1) (rather than Z₂) state, with specific associated properties. Here is a long detailed paper on this problem. A less detailed but more pedagogical presentation is here .

These ideas have rather broad scope, and the group is actively pursuing other applications with an eye to experimental implications. Some immediate follow-ups are cond-mat/0408329 and cond-mat/0409470 . These papers, and related work, are described elsewhere on this site. A puzzling commentary by Robert Laughlin can be found in Science (Science, Vol 303, Issue 5663, 1475-1477) .

• disclaimer: this list, from 2005, is collective "work" of several people, without careful effort!

1. Are quantum effects important for physics of hexagonal manganites (Eg: YMnO3)? What is the mechanism of coupling between electric polarization and spin order?
2. Theory of field-induced transition from heavy fermi liquid to fermi liquid with polarized local moments? Application to CeRu2Si2 or URu2Si2? Is there significant Fermi surface reconstruction at the metamagnetic transition in these materials?
3. In amorphous films undergoing a field-tuned ``superconductor-insulator” transition, to what extent can the vortices be regarded as quantum particles?
4. Experiments on a number of heavy fermion critical points (Fermi liquid to AF metal/spin glass) see non-trivial exponents for dynamical spin correlations. Thus far these exponents cluster around 2 values (2/3 and 1/3). Is there any systematics to these exponent values? Is there any theory?
5. Can non-trivial two dimensional quantum paramagnets be accessed by weakly coupling together spin-1/2 chains?
6. Can the Oshikawa/Hastings arguments on structure of paramagnetic states of easy plane/axis magnets be generalized to SU(2) invariant systems? (Eg S = 1 on 2d square lattice believed not to have trivial paramagnetic ground state with SU(2) symmetry: can this be understood at the same level as generality as Oshikawa/Hastings?)
7. Are there any clear demonstrable instances of criticality in quantum systems where spatial correlations are mean-field like but time correlations are anamolous? (analogous to proposal of Si et al for heavy fermion critical points)
8. Is there a microscopic model which can be demonstrated to have a deconfined Landau-forbidden deconfined quantum critical point? Spin systems, bosons on various lattices?
9. Can one understand the theoretical problem of Fermi surface coupled to a gauge field in some controlled approximation? (Do better than work from 1990's) Fate of monopoles? 2kf correlations? Luttinger theorem?
10. Does the sigma model formulation for 2d deconfined quantum critical points have any power for analytic/numerical calculations? Is there a `sigma model’ description of stable two dimensional algebraic spin liquids in terms of a bosonic field theory perhaps with topological terms?
11. Is there a `solution’ to the sign problem in simulating Hamiltonians of quantum many particle systems? How well-defined is the sign problem? Are there classes of problems that have an `intrinsic' problem that will not disappear in any useful reformulation? Will such systems have properties different from those without an `intrinsic' sign problem?
12. How do we describe the single impurity Kondo effect in a bosonic decription of the impurity spin?
13. Is there really `reduced' dimensionality for spin fluctuations at heavy fermion critical points that show non-fermi liquid behavior? Does the reduced dimensionality play any direct role in giving the non-Fermi liquid physics?
14. Is gauge theory useful (necessary?) for understanding high-Tc cuprates? Is there a viable alternate to implement Mott/valence bond physics in doped Mott insulators?
15. Is there a quasi-realistic spin/Hubbard model that can be shown to be in a spin liquid phase?
16. How correct is it to integrate out fermions in a Hertz-Millis theory of a metallic quantum phase transition? (Action in ordered and disordered phases is different; is this a problem?)
17. Is there interesting physics at a quantum first order transition in a metallic system?
18. Cs₂CuCl₄:
    • Where in q-space are there "gapless" excitations associated with power law tails in inelastic neutron scattering?
    • To what extent is the scattering in these tails polarized in easy plane?
    • What is the theoretically expected answer for 1d-2d crossover?
19. What is the phase diagram of the 2d Hubbard model at half-filling on a frustrated lattice? Eg: Triangular or Kagome lattices
20. Do there exist non-Fermi liquid states of matter with Fermi arcs? Can we construct an example?