Gapless bosonic excitation without symmetry breaking: An algebraic spin liquid with soft gravitons

A quantum ground state of matter is realized in a bosonic model on a three-dimensional fcc lattice with emergent low energy excitations. The phase obtained is a stable gapless boson liquid phase, with algebraic boson density correlations. The stability of this phase is protected against the instanton effect and superfluidity by self-duality and large gauge symmetries on both sides of the duality. The gapless collective excitations of this phase closely resemble the graviton, although they have a soft $\omega \sim k^2$ dispersion relation. There are three branches of gapless excitations in this phase, one of which is gapless scalar trace mode, the other two have the same polarization and gauge symmetries as the gravitons. The dynamics of this phase is described by a set of Maxwell’s equations. The defects carrying gauge charges can drive the system into the superfluid order when the defects are condensed; also the topological defects are coupled to the dual gauge field in the same manner as the charge defects couple to the original gauge field, after the condensation of the topological defects, the system is driven into the Mott insulator phase. In the two-dimensional case, the gapless soft graviton as well as the algebraic liquid phase are destroyed by the vertex operators in the dual theory, and the stripe order is most likely to take place close to the two-dimensional quantum critical point at which the vertex operators are tuned to zero.

Figure 1

The flipping term of the dimer model.

Figure 2

The sign convention of mapping from rotor number onto the vector electric field $\vec{E}$ in the $XY$ plane.

Figure 3

The structure of the 2D lattice. On each site there are two orbital levels, the occupation number is $(n_1,n_2)$. On each face center there is one orbital level, with occupation number $n$.

Figure 4

The sign convention in the definition of symmetric tensor $E_{ij}$. After introducing the signs on the lattice, the constraint imposed by $H_0$ can be written compactly as Eq. (13).

Figure 5

The ring exchange of the two-dimensional lattice. Elementary hoppings are between each site and its nearest neighbor face center. The gauge invariant ring exchange term can be represented as $\cos \left(R_{xyxy}\right)$, it involves eight single hoppings of bosons.

Figure 6

The distribution of the dual variable $h$. $h$ cannot all be integers over the whole two-dimensional plane, instead, one-quarter of $h$ have to be half-integers.

Figure 7

The distribution of boson numbers on fcc lattice. The link (in bold font) is shared by four plaquettes and two sites. Every site is occupied by three orbital levels, and on every plaquette there is one orbital level.

Figure 8

One of the defects carrying gauge charges. If $n_1$ on one site is increased by one, it is equivalent to creating a pair of opposite gauge charges on two $\widehat{x}$ links sharing this site.

Figure 9

One of the defects carrying gauge charges. If on one plaquette center, $n_{\overline{r}}$ is increased by one, gauge charges are created for four links sharing this plaquette.