Incompressible Quantum Liquids and New Conservation Laws

In this Letter, we investigate a class of Hamiltonians which, in addition to the usual center-of-mass momentum conservation, also have center-of-mass position conservation. We find that, regardless of the particle statistics, the energy spectrum is at least $q$-fold degenerate when the filling factor is $p/q$, where $p$ and $q$ are coprime integers. Interestingly, the simplest Hamiltonian respecting this type of symmetry encapsulates two prominent examples of novel states of matter, namely, the fractional quantum Hall liquid and the quantum dimer liquid. We discuss the relevance of this class of Hamiltonian to the search for featureless Mott insulators.

Figure 1

The density wave order parameter and the energy gap of Eq. (6) as a function of $\kappa$. Data are obtained for $L\le 30$. (a) The crystalline order parameter. Inset: The quantity $\Gamma =(-\log \mathcal{O}{)}^{-1/2}$, extrapolated to $L=\infty$ using the alternating epsilon algorithm [28]. The smooth curve is a polynomial fit of the data for $0.25<\kappa <0.45$ (crosses). (b) The energy gap for data restricted to $\kappa L>12$ and even particle number (see text). Inset: The energy gap is plotted versus $\mathcal{O}f(1{)}^2$. Dark gray (blue) and light gray (red) symbols are for $\kappa >1.1$ and $\kappa <1.1$, respectively. Two distinct branches are clearly visible.