Bosonic analogs of the fractional quantum Hall state in the vicinity of Mott states

In this paper, the Bose-Hubbard model (BHM) with the nearest-neighbor (NN) repulsions is studied from the viewpoint of possible bosonic analogs of the fractional quantum Hall (FQH) state in the vicinity of the Mott insulator (MI). First, by means of the Gutzwiller approximation, we obtain the phase diagram of the BHM in a magnetic field. Then, we introduce an effective Hamiltonian describing excess particles on a MI and calculate the vortex density, momentum distribution, and the energy gap. These calculations indicate that the vortex solid forms for small NN repulsions, but a homogeneous featureless “Bose metal” takes the place of it as the NN repulsion increases. We consider particular filling factors at which the bosonic FQH state is expected to form. Chern-Simons (CS) gauge theory to the excess particle is introduced, and a modified Gutzwiller wave function, which describes bosons with attached flux quanta, is introduced. The energy of the excess particles in the bosonic FQH state is calculated using that wave function, and it is compared with the energy of the vortex solid and Bose metal. We found that the energy of the bosonic FQH state is lower than that of the Bose metal and comparable with the vortex solid. Finally, we clarify the condition that the composite fermion appears by using CS theory on the lattice that we previously proposed for studying the electron FQH effect.

Figure 1

Obtained phase diagram of the BHM subject to a strong magnetic field. The yellow line is the phase boundary separating the MI and SF states for vanishing magnetic field $f=0$ obtained by the Gutzwiller numerical method. The green line is an elongated phase boundary as a result of the applied magnetic flux per plaquette, $2\pi f=\pi$. The red line represents the states with the average particle density $\rho =1.25$.

Figure 2

Numerical results for the $V/U=0$ and $J/U=0.05$ cases. The upper panels (a) correspond to the $f=\frac{1}{2}$ case. The phase diagram in the leftmost panel exhibits the point of ${\nu }_{\mathrm{ep}}=\frac{1}{2}$ by the cross $\times$ at which the measurements of vortex density $\mathrm{\Omega }\left(r\right)$ and the momentum distribution $n\left(\mathbf{k}\right)$ were performed. The middle and right panels show the calculations of $\mathrm{\Omega }\left(r\right)$ and $n\left(\mathbf{k}\right)$, respectively. The lower panels (b) correspond to the $f=\frac{1}{3}$ case (${\nu }_{\mathrm{ep}}=\frac{1}{2}$). In both cases, the results obviously show that the stable vortex solid forms.

Figure 3

Numerical results for the $V/U=0.2$ and $J/U=0.05$ cases. As in Fig. 2, the upper panels correspond to the $f=\frac{1}{2}$ case and the lower panels to the $f=\frac{1}{3}$ case. Calculations of $\mathrm{\Omega }\left(r\right)$ and $n\left(\mathbf{k}\right)$ are shown. In both cases, the signals of the vortex solid are weakened, in particular, in the $f=\frac{1}{2}$ case.

Figure 4

Energy gaps calculated by the single-mode approximation for the states described by the Gutzwiller wave functions in the case $J/U=0.05$. Applied magnetic field is $f=\frac{1}{2}$ and $\frac{1}{3}$ and $\rho \simeq 1.25$. As the NN repulsion $V$ increases, the energy gap increases from the vanishing value. At $V=0$, the stable vortex solid forms as the density profile in the inset indicates. As a result, the gapless Nambu-Goldstone boson exists. On the other hand for $V/U=0.2$, the vortex solid melts and the SF is destroyed, and then the excitations acquire a gap. We took 50 samples in each measurement because the value of the excitation gap depends on initial values of $\left\{{\mathrm{\Psi }}_i\right\}$.

Figure 5

Chern-Simons theory of lattice bosons in a strong magnetic field. Each boson is attached an even number of the flux quanta (orange arrow) of the CS gauge field. The external magnetic field (red arrow) is canceled out on the average by the CS gauge field.

Figure 6

Bogoliubov excitation spectrum of $\tilde{J}/U=0.1$ (a) for $V/U=0$ and (b) $V/U=0.2$. (c) Dispersion relations for $V/U=0$ and $V/U=0.2,E\left(k\right)$ and $E_V\left(k\right)$, in the plane $k_y=0$. $E\left(0\right)=E_V\left(0\right)$, however, $E_V\left(k\right)>E\left(0\right)$ for $\mathbf{k}\ne 0$. This indicates the stability of the CB picture for $V>0$.

Figure 7

Energies of the states described by the Gutzwiller wave function and the CS wave function, $E_{\mathrm{ex},\mathrm{GW}}$ and $E_{\mathrm{ex},\mathrm{CS}}$, respectively. For $V=0$, these two states have comparable energy. On the other hand for $V/U=0.2$, the CB state has a lower energy than the GW function state.

Figure 8

Parameters of BHM evaluated microscopically as a function of $V_0/E_{\mathrm{R}}$ and the $s$-wave scattering length $a_s$ for the experimental setup. The value of $a_s$ is obtained as $a_s\left(V_0\right)=[V/\alpha -U_d]/I$, where $\alpha =V/U=0.2$ or 0.3.