Four-Coloring Model and Frustrated Superfluidity in the Diamond Lattice

We propose a novel four-coloring model which describes “frustrated superfluidity” of $p$-band bosons in the diamond optical lattice. The superfluid phases of the condensate wave functions on the diamond-lattice bonds are mapped to four distinct colors at low temperatures. The fact that a macroscopic number of states satisfy the constraints that four differently colored bonds meet at the same site leads to an extensive degeneracy in the superfluid ground state at the classical level. We demonstrate that the phase of the superfluid wave function as well as the orbital angular momentum correlations exhibit a power-law decay in the degenerate manifold that is described by an emergent magnetostatic theory with three independent flux fields. Our results thus provide a novel example of critical superfluid phase with algebraic order in three dimensions. We further show that quantum fluctuations favor a Néel ordering of orbital angular moments with broken sublattice symmetry through the order-by-disorder mechanism.

Figure 1

Mapping between the orbital angular momentum and color configuration on the diamond lattice. The red, green, blue, and yellow bonds have phases $\varphi =0°$, 90°, 180°, and 270°, respectively, when projected to the base plane perpendicular the direction of angular momentum $\mathbf{L}$. The mapping is not one to one, as each $\mathbf{L}$ corresponds to 4 different cyclic permutations of the colors which preserve the same chirality.

Figure 2

An example of valid coloring configuration on the diamond lattice. Here the constraint requires that four bonds attached to the same vertex must have different colors.

Figure 3

(a) The angular averaged superfluidity phase correlation function $C(r)=⟨e^{-i\phi (r)}{e}^{i\phi (0)}⟩$ on a system with linear size $L=10$ in the ground state. The inset in (a) shows the four unit vectors ${\mathit{\tau }}_s$ ($s=0\sim 3$) used in the definition of the emergent magnetic fields. (b) Phase correlation vs linear system size $L$. Here $C(L)$ denotes the color correlation between two points separated by $(L/2,L/2,0)$ in a finite lattice containing $L^3$ cubic unit cells. The inset shows the probability distribution $P(s)$ of flippable loop lengths in the critical phase obtained from simulations on a $L=50$ lattice.

Figure 4

(a) Néel ordering of orbital moments $\mathbf{L}$ and the corresponding 4-color configuration. The orbital moments ${\mathbf{L}}_i$ on the two sublattices (labeled by dark and cyan balls) are antiparallel to each other. (b) shows the band structure of the Bogoliubov quasiparticles in the orbital Néel state ($U/t_{\parallel }=0.1$).