Temperature in one-dimensional bosonic Mott insulators

The Mott insulating phase of a one-dimensional bosonic gas trapped in an optical lattice is described by a Bose-Hubbard model. A continuous unitary transformation is used to map this model onto an effective model conserving the number of elementary excitations. We obtain quantitative results for the kinetics and for the spectral weights of the low-energy excitations for a broad range of parameters in the insulating phase. By these results, recent Bragg spectroscopy experiments are explained. Evidence for a significant temperature of the order of the microscopic energy scales is found.

Figure 1

(Color online) Residual off-diagonality for values of $t∕U∊\{0.1,0.2\}$ for maximal $r=2$ and $r=3$. Inset: Magnification for small values of the flow parameter $l$. The ROD for $r=3$ and $t∕U=0.2$ displays nonmonotonic behavior. In this case, we stop the flow at the first minimum of the ROD. Its position is indicated by a vertical line.

Figure 2

(Color online) Phase diagram of the Mott insulating (MI) and the superfluid (SF) phase in the $(t∕U,\mu ∕U)$ plane. Dotted (dashed) lines show CUT results for maximal extension $r=2(r=3)$. Solid lines (symbols) show series (DMRG) results from Refs. 10, 9.

Figure 3

(Color online) The dispersions of a particle ${\omega }^{\mathrm{p}}\left(k\right)$ (black) and a hole ${\omega }^{\mathrm{h}}\left(k\right)$ (green or grey, respectively) for $t=0.1U$ (solid), $t=0.15U$ (dotted), and $t=0.2U$ (dashed).

Figure 4

Sketch of the distribution of spectral weight $S(k=0,\omega )$. The weight centered around $\omega =U$ is given by $S_1$, the weight around $\omega =2U$ by $S_2$.

Figure 5

Terms of the local observable $R(l,\mathbf{r})$. The arrow indicates the bond on which a term acts. Filled circles: sites on which a nontrivial operator acts; “nontrivial” means that the operator is different from the identity. Open circles: sites on which the term under study does not act, i.e., the term acts as identity on these sites. (a) At $l=0$ the observable is $H_t$. It is composed of local terms that act only on adjacent sites of the lattice. (b) and (c) More complicated terms appear during the flow. The sum of the distances of all operators in a term is our measure $r_{\mathcal{O}}$ for the extension of a term. It is 2 for (b) and 4 for (c). Terms beyond a certain extension are omitted.

Figure 6

(Color online) Spectral weights of various processes for truncations $r_{\mathcal{O}}∊\{2,3,4\}$. (a) $I_{(\mathrm{hp};0)}^{\mathrm{eff}}$ (black curves) and $I_{(2\mathrm{h}2\mathrm{p};0)}^{\mathrm{eff}}$ (green or grey, respecitvely). (b) $I_{(\mathrm{h}2\mathrm{p};\mathrm{p})}^{\mathrm{eff}}+I_{(2\mathrm{hp};\mathrm{h})}^{\mathrm{eff}}$ (black) and $I_{(\mathrm{hd};\mathrm{p})}^{\mathrm{eff}}$ (grey).

Figure 7

(Color online) Temperature dependence of the relative spectral weight $S_2∕S_1$ for various values of $t∕U$. Inset: $S_2∕S_1$ as function of $t∕U$ for various temperatures.