Intraband and interband conductivity in systems of strongly interacting bosons

Motivated by recent experimental progress on measuring various correlation functions in systems of ultracold atoms in optical lattices, we study properties of the Bose-Hubbard model in external synthetic magnetic field to describe transport phenomena in a multiband strongly interacting bosonic systems. We calculate the conductivity both in the Mott insulator and superfluid phases and investigate its two main contributions: intra- and interband. It appears that the interband processes dominate the transport properties by at least an order of magnitude. Also, at finite temperatures, additional transport channels appear due to coupling of the thermally excited particles or holes to the external field.

Figure 1

The conductivity calculated for a square lattice in staggered potential. The offset energy $\mathrm{\Delta }$ was added in order to differentiate between neighboring sites. For $\mathrm{\Delta }=0$ there is only intraband contribution to the conductivity. The interband channel appears only for nonzero $\mathrm{\Delta }$ and its value increases with increasing $\mathrm{\Delta }$.

Figure 2

Densities of states and generalized densities of states for three values of the magnetic field given by the parameter $\alpha$.

Figure 3

Density maps of $M_{12}^{xx}(k_x,k_y)$ and $M_{13}^{xx}(k_x,k_y)$ in the case of $\alpha =1/3$ (lighter color denotes higher values). The $\mathbf{k}$ domain spans over three Brillouin zones.

Figure 4

Phase diagrams of superfluid to Mott insulator phase transition for three values of the flux parameter $\alpha =1/2,1/3,1/4$ and for a range of values of temperature $\beta$. The Mott insulator is the ground state of the system below the critical line, while the superfluid is above. In finite temperature, the Mott insulator phase is replaced by the normal state, as the compressibility is no longer zero.

Figure 5

Real part of the conductivity as a function of frequency of the external field for $\alpha =1/2$. The conductivity is given in $1/{\mathrm{\Phi }}_0^2$ units. The graphs were calculated for $\mu /U=0.25$ and for $t/U=0.18$ (superfluid) and $t/U=0.08$ (Mott insulator).

Figure 6

Real part of the conductivity as a function of frequency of the external field for $\alpha =1/3,\mu /U=0.25,t/U=0.22$ (SF), and $t/U=0.09$ (MI) (in $1/{\mathrm{\Phi }}_0^2$ units).

Figure 7

Real part of the conductivity as a function of frequency of the external field for $\alpha =1/4,\mu /U=0.25,t/U=0.3$ (SF), and $t/U=0.09$ (MI) (in $1/{\mathrm{\Phi }}_0^2$ units).

Figure 8

Spectral function $\mathcal{A}(\mathbf{k},\omega )$ dependence on the frequency for $\alpha =1/3$ along the lines $\mathrm{\Gamma }\text{−}X,X\text{−}M$, and $M\text{−}\mathrm{\Gamma }$ in the reduced Brillouin zone (one third of the BZ) for $t/U=0.22$ in the SF phase. The quasiparticle bands are drawn with solid line, while hole bands with dotted lines. Arrows denote possible excitations: I corresponds to the intraband particle-hole pair generation, II corresponds to the interband particle-hole pair, and III corresponds to the excitation of thermally excited hole.

Figure 9

Influence of the temperature on the intraband conductivity for SF $\left(t/U=0.22\right)$ and MI $\left(t/U=0.09\right)$ phases with the chemical potential $\mu /U=0.25$.

Figure 10

Influence of the temperature on the interband conductivity for SF $\left(t/U=0.22\right)$ and MI $\left(t/U=0.09\right)$ phases with the chemical potential $\mu /U=0.25$.