One-dimensional Bose gas in optical lattices of arbitrary strength

One-dimensional Bose gas with contact interaction in optical lattices at zero temperature is investigated by means of the exact diffusion Monte Carlo algorithm. The results obtained from the fundamental continuous model are compared with those obtained from the lattice (discrete) Bose-Hubbard model, using exact diagonalization, and from the quantum sine-Gordon model. We map out the complete phase diagram of the continuous model and determine the regions of applicability of the Bose-Hubbard model. Various physical quantities characterizing the systems are calculated, and it is demonstrated that the sine-Gordon model used for shallow lattices is inaccurate.

Figure 1

The zero-temperature phase diagram of the continuous model compared with various approximate theories, as a function of the $s$-wave scattering length $a_{1\mathrm{D}}/a_0$ and the optical lattice intensity $V_0/E_{\mathrm{rec}}$. The position of the Mott-insulator–superfluid phase transition is defined by $K=2$. Black circles, DMC results; green solid line, exact diagonalization of the BHM; dashed red line, sine-Gordon predictions; squares with error bars (pink, amplitude modulation; blue, transport measurements), experimental results of Ref. [24]. DMC and exact diagonalization results are obtained for $N=12$.

Figure 2

Parameter $K=2S\left(k_{\min }\right)k_{\mathrm{F}}/k_{\min }$ calculated at the first point, $k_{\min }=2\pi /\left(L{a}_0\right)$, of the static structure factor for $N=12$ particles as a function of interaction strength $|a_{1\mathrm{D}}|/a$ for lattices of different heights $V_0=0,1,2,3,4,5E_{\mathrm{rec}}$ (top to bottom curves). Solid symbols connected with solid lines, DMC results; open symbols, BHM; dashed line, Lieb-Liniger thermodynamic result for $V_0=0$. The $K=2$ short-dashed line separates the superfluid regime (above) from the insulating one (below). Notice that $K$ can be identified with the Luttinger parameter in the region with $K>2$.

Figure 3

Static structure factor $S\left(k\right)$. Solid symbols, continuous model; solid lines, $S_{\mathrm{BH}}\left(k\right)$ defined by (4); dashed line, $S_0\left(k\right)$ defined by (5); dashed-dotted line, $S\left(k\right)=\left|k\right|/k_{\mathrm{F}}$ corresponds to a linear slope with the critical value of the Luttinger parameter $K=2$ and separates superfluid (SF) and Mott-insulator (MI) phases. The following parameters are used: short-dashed line, Tonks-Girardeau gas (i.e., $V_0=0$ and $a_{1\mathrm{D}}=0$); continuous and Bose-Hubbard models (from top to bottom) $V_0/E_{\mathrm{rec}}=0$, 4, 8, and $|a_{1\mathrm{D}}|=4a_0$.

Figure 4

OBDM ${\rho }_1(L/2)$ obtained from the pure DMC estimator for $V_0=1E_{\mathrm{rec}}$ and two characteristic values of the interaction strength, $|a_{1\mathrm{D}}|/a_0=0.1$ (insulating phase, left panel) and $|a_{1\mathrm{D}}|/a_0=1$ (superfluid phase, right panel), for different system sizes $L$. Solid symbols, continuous model; open symbols, BHM. For $|a_{1\mathrm{D}}|/a_0=0.1$ one observes an exponential decay (insulating phase), while for $|a_{1\mathrm{D}}|/a_0=1$ it is much better fit by a power-law form (superfluid phase).

Figure 5

Energy gap. Squares with error bars, experimental data from Ref. [24]; solid circles, DMC prediction obtained from the fit to $S(k\to 0)$; open circle, DMC prediction obtained according to Eq. (7); solid line, sine-Gordon model [24]; dashed line, upper bound given by $V_0/2$. $a_{1\mathrm{D}}/a_0=0.18182$.