One-body density matrix and momentum distribution of strongly interacting one-dimensional spinor quantum gases

The one-body density matrix (OBDM) and the momentum distribution of quantum many-body systems are usually very difficult to calculate. Here we develop a technique to calculate the OBDM and the momentum distribution of a general one-dimensional (1D) spinor quantum gas in the strong interaction regime. This technique relies on a remarkable connection between the OBDM of the spinor gas and that of a spinless 1D hard-core anyon gas, which allows us to efficiently calculate the OBDM of the spinor system with particle numbers much larger than what was previously possible. Given the OBDM, we can easily calculate the momentum distribution of the spinor system, which is also related to the momentum distribution of the hard-core anyon gas. Our study not only provides a practical method for the calculation of the OBDM, but also provides significant insights into the properties of 1D strongly interacting spinor quantum gases.

Figure 1

The tensor contraction geometry for calculating $S_r({\sigma }^{\prime },\sigma )$ for an even $r$ case. $A$ and $B$ tensors, which are building blocks in MPSs (two sites in a unit cell), are calculated using the iTEBD method. Note that for a finite periodic boundary condition system, we also need to contract the remaining tensors outside the correlation range $m$ to $m+r$. Starting from the $m\mathrm{th}$ site with either $A$ tensor or $B$ tensor gives the same result.

Figure 2

Spin-correlation function and momentum distribution of translational invariant system. (a) $S_r$ calculated by iTEBD for an infinite chain. (b) $S^{\kappa }$ obtained by Fourier transform of $S_r$ with $r$ up to 10 000. (c) Momentum distribution of hard-core anyon gas ${\rho }^{\kappa }\left(p\right)$ for $N=201$. (d) Momentum distribution (summed over all spin components) of the spinor gases for $N=201$ particles. Note that in Eq. (14), ${\rho }^{\kappa }\left(p\right)$ and $S^{\kappa }$ are not generally real valued as in (b) and (c), but we can rearrange ${\rho }_r\left(y\right)$ and $S_r$ to make them real [43].

Figure 3

Left panel: evolution of the momentum distribution of the impurity atom. Here we take $N=60,{\gamma }_i=12$, and $\mathcal{F}=1$. $t_F=\hslash /E_F=1/N$, and $k_F=n_{1\text{D}}\pi =\sqrt{2N}$ is the Fermi momentum. All quantities are expressed in the dimensionless unit system defined by $\hslash =m=\omega =1$. Right panel: the solid lines replot the momentum distribution of the impurity atom from the left panel at four different times; the dash-dotted line is the momentum distribution of a homogeneous hard-core anyon gas, ${\rho }^{\kappa }\left(p\right)/N$, with statistical parameter $\kappa =Ft/\hslash k_F$. The anyon gas consists of $N$ particles confined in a region with length $L$ (periodic boundary condition is assumed) such that its density is given by $N/L=n_{1\text{D}}$.