Static and dynamic properties of interacting spin-1 bosons in an optical lattice

We study the physics of interacting spin-1 bosons in an optical lattice using a variational Gutzwiller technique. We compute the mean-field ground state wave function and discuss the evolution of the condensate, spin, nematic, and singlet order parameters across the superfluid-Mott transition. We then extend the Gutzwiller method to derive the equations governing the dynamics of low energy excitations in the lattice. Linearizing these equations, we compute the excitation spectra in the superfluid and Mott phases for both ferromagnetic and antiferromagnetic spin-spin interactions. In the superfluid phase, we recover the known excitation spectrum obtained from Bogoliubov theory. In the nematic Mott phase, we obtain gapped, quadratically dispersing particle and hole-like collective modes, whereas in the singlet Mott phase, we obtain a nondispersive gapped mode, corresponding to the breaking of a singlet pair. For the ferromagnetic Mott insulator, the Gutzwiller mean-field theory only yields particle-hole-like modes but no Goldstone mode associated with long-range spin order. To overcome this limitation, we supplement the Gutzwiller theory with a Schwinger boson mean-field theory which captures superexchange-driven fluctuations. In addition to the gapped particle-hole-like modes, we obtain a gapless quadratically dispersing ferromagnetic spin-wave Goldstone mode. We discuss the evolution of the singlet gap, particle-hole gap, and the effective mass of the ferromagnetic Goldstone mode as the superfluid-Mott phase boundary is approached from the insulating side. We discuss the relevance and validity of Gutzwiller mean-field theories to spinful systems, and potential extensions of this framework to include more exotic physics which appears in the presence of spin-orbit coupling or artificial gauge fields.

Figure 1

Density plot showing the evolution of the superfluid order parameter in the ferromagnetic $\left(U_2=-0.1U_0\right)$ spin-1 Bose gas. The superfluid order parameter goes to zero at the superfluid-Mott phase boundary as shown in the inset. The total spin $\mathcal{S}/n=1$ throughout the phase diagram. The numbers label the density in the Mott lobes. All transitions here are second order.

Figure 2

Density plot showing the evolution of the nematic (left) and singlet (right) order parameters in the antiferromagnetic $\left(U_2=0.1U_0\right)$ spin-1 Bose gas. The nematic order parameter is computed by taking the maximum eigenvalue of the nematic tensor. In the singlet phases, labeled $I$, nematic order vanishes, and the superfluid-Mott transition is first order, as indicated by the finite jump in the nematic order, as shown in the inset. In contrast, the transition into the lobe labeled $\text{II}$ is a continuous second order quantum phase transition. On the right, the singlet number takes on an integer value equal to $1\left(2\right)$ in the $n=2\left(4\right)$ Mott lobes, corresponding to the number of local singlets per site. The inset shows the singlet order parameter for two different cuts across the phase diagram: the cuts traverse the $n=2$ (dashed) and $n=1$ (solid) Mott-superfluid phase boundaries. At the $n=2$ Mott-superfluid phase boundary, the singlet order parameter displays a finite jump, which is evidence of the first order nature of the transition.

Figure 3

Evolution of the excitation spectrum from the nematic (polar) superfluid into the singlet phase. Throughout we fix $U_2=0.1U_0$ and vary $t$ and $\mu$ such that we access the deep superfluid (left), superfluid-Mott phase boundary (center), and the deep Mott phase (right). The polar superfluid has two linearly dispersing modes corresponding to density and spin modes, originally derived in Refs. [5, 6]. As the superfluid-Mott phase boundary is approached, the sound speeds associated with the density and spin modes become identical. Additionally, there is a quadratically dispersing gapped, particle-hole mode and a nondispersive mode. This nondispersive mode corresponds to the breaking of a local singlet pair and has no analog in the spinless case. This is therefore a new feature of the spin-1 Bose gas. Deep in the Mott insulator, the singlet pair breaking mode has the lowest energy and the particle-hole mode is pushed to higher energies.

Figure 4

Evolution of the singlet gap (solid) and the particle-hole gap (points) across the superfluid $n=2$ singlet Mott-insulator boundary for antiferromagnetic interactions. $\left(U_2/U_0=0.1\right)$. The singlet gap is completely independent of the hopping in the Mott insulator and and can be obtained by diagonalizing the on-site Hamiltonian. The particle-hole gap discontinuously goes to zero as the Mott-superfluid phase boundary is approached from the superfluid side, indicative of a first order transition. It approaches $0.5U_0$ deep in the Mott-insulator phase.

Figure 5

Evolution of the excitation spectrum from the ferromagnetic superfluid into the Mott insulating phase. Throughout we fix $U_2=-0.1U_0$ and vary $t$ and $\mu$ such that we access the deep superfluid (left), superfluid-Mott phase boundary (center), and the deep Mott phase (right). The ferromagnetic superfluid has one linearly dispersing mode corresponding to density fluctuations and a quadratically dispersing gapless and a gapped spin mode, corresponding to fluctuations of the spin in the direction parallel and perpendicular to the easy axis. As the superfluid-Mott phase boundary is approached, the spin gap associated with the spin mode goes to zero. Additionally, there is a gapped, quadratically dispersing particle-hole-like mode. The sound speed associated with the density mode vanishes at the transition, and deep in the Mott insulator there are two nondegenerate gapped modes corresponding to particle- and hole-like excitations. The gapped spin mode is effectively nondispersive in the Mott insulator as its effective mass is proportional to $t$, which is exponentially small.

Figure 6

Evolution of the particle-hole gap across the superfluid-$n=2$ ferromagnetic Mott insulator boundary for $U_2=-0.1U_0$.