Spin-Liquid Condensate of Spinful Bosons

We introduce the concept of a bosonic spin liquid condensate (SLC), where spinful bosons in a lattice form a zero-temperature spin disordered charge condensate that preserves the spin rotation symmetry, but breaks the U(1) symmetry due to a spinless order parameter with charge one. It has an energy gap to all the spin excitations. We show that such SLC states can be realized in a system of spin $S\ge 2$ bosons. In particular, we analyze the SLC phase diagram in the spin 2 case using a mean-field variational wave function method. We show there is a direct analogy between the SLC and the resonating-valence-bond state.

Figure 1

The on-site energy levels computed for $U_4=3U_2=30U_0$ and $\mu ={\mu }_0-\mathrm{\Delta }$, where ${\mu }_0=3U_2-U_0$. The $x$ and $y$ axes are the particle number $n$ and the energy $E$ in units of $U_0$, respectively. $E_{nl}$ represents the energy of the on-site state with particle number $n$ and total spin $l$. The arrows show the transition process from $|2,0,0{⟩}_i$ to $|3,0,0{⟩}_i$ via three time hoppings, which must overcome an activation energy $E_a$.

Figure 2

The phase diagram for $U_4=3U_2=30U_0$ and $z=4$ with respect to the chemical potential $\mu$ and the hopping $zt$. $E_a\approx 10U_0$ as is shown in Fig. 1. Dimer MI and trimer MI stands for the singlet Mott insulators with $n=2$ and $n=3$, respectively. In the large $t$ limit the system becomes a spinor BEC (nematic phase according to Refs. [5, 6, 7]).

Figure 3

(a). How a particle moves in SLC. Suppose initially two nearby sites are in states $|3,0,0{⟩}_i$ (left) and $|2,0,0{⟩}_i$ (right), respectively. Hopping once induces a singlet valence bond between the two sites and costs energy of order $E_a$. By two more hoppings, the two sites become spin singlets again and the valence bond is erased, while a particle moves to the right site. (b) Illustration of various resonating valence bonds created and annihilated due to the motion of particles in SLC.