Conductivity of strongly correlated bosons in optical lattices in an Abelian synthetic magnetic field

Topological phase engineering of neutral bosons loaded in an optical lattice opens a new window for manipulating of transport phenomena in such systems. Exploiting the Bose-Hubbard model and using the magnetic Kubo formula proposed in this paper we show that the optical conductivity abruptly changes for different flux densities in the Mott phase. Especially, when the frequency of the applied field corresponds to the on-site boson interaction energy, we observe insulator or metallic behavior for a given Hofstadter spectrum. We also prove that for different synthetic magnetic-field configurations the critical conductivity at the tip of the lobe is nonuniversal and depends on the energy minima of the spectrum. In the case of $1/2$ and $1/3$ flux per plaquette, our results are in good agreement with those of the previous Monte Carlo study. Moreover, we show that for half magnetic flux through the cell the critical conductivity suddenly changes in the presence of a superlattice potential with uniaxial periodicity.

Figure 1

Real part of optical conductivity in the two-dimensional square lattice in the Mott phase (first lobe) sketched in ${\sigma }_Q$ units [${\sigma }_Q={\left(e^{*}\right)}^2/h$ is a quantum conductance]. (a) $f=0$. (b) $f=1/2$. (c) $f=1/4$. Conductivity is plotted at zero temperature for different values of $J/U$ within the first lobe.

Figure 2

The wave-vector $\mathbf{k}$ dependence of the quasiparticle energy dispersions $E_q^{\alpha \pm }(\mathbf{k};p)$ for different amplitudes of uniform magnetic field $f$ and staggered potential $\Delta$ in the first magnetic Brillouin zone. Plots a–c correspond to Figs. 1–1, respectively. Additionally, plots d and e correspond to Figs. 4 and 5, respectively. This correspondence is revealed when choosing the same physical parameters for the same plotting style in the relevant figures.

Figure 3

Optical conductivity for different values of the magnetic field. The parameters $\mu /U$ and $J/U$ are chosen on the phase boundary where Mott insulator-superfluid phase transition takes place. The plot shows that the critical value of conductivity for $f=1/2$ ($1/4$) when $\omega \to 0$ is two (four) times higher than in the case when the magnetic field is absent. $\mathrm{Re}{\sigma }_{xx}$ was plotted in ${\sigma }_Q$ units.

Figure 4

(a and b) $yy$ and (c and d) $xx$ components (respectively) of the real part of the optical conductivity for different values of $\Delta$ in the absence of the magnetic field. On the density sketch the lighter color indicates a higher amplitude of the conductivity (the black area corresponds to the insulator behavior). All zero-temperature plots show the situation in which the uniaxially staggered field is subjected to the $x$ axis. The ratio of the hopping amplitude and on-site interaction energy is $J/U=0.03$ and the conductivity was plotted in ${\sigma }_Q$ units.

Figure 5

Analogous physical situation as in Fig. 4 but with nonzero amplitude of the uniform magnetic field (half magnetic flux per unit cell).

Figure 6

Imaginary part of the current-current correlation function in the two-dimensional square lattice in the Mott phase (first lobe) sketched in ${\tilde{\sigma }}_Q{f}_x^2$ units. (a) $f=0$. (b) $f=\frac{1}{2}$. (c) $f=\frac{1}{4}$ (c). Figures are plotted at zero temperature for different values of $J/U$.

Figure 7

Density plot of the lowest band energy in the Mott phase [the white rectangle describes the first magnetic Brillouin zone (MBZ)]. The range from darker to brighter color is assigned to the lowest and highest value of the energy spectrum, respectively. $\Delta =0$ (left) and $0.7$ (right). We see explicitly the disappearance of one of the two minima (black color) on the center of the MBZ.