Spin-incoherent Luttinger liquid of one-dimensional spin-1 Tonks-Girardeau Bose gases: Spin-dependent properties

Spin-incoherent Luttinger liquid (SILL) is a different universal class from the Luttinger liquid. This difference results from the spin incoherence of the system when the thermal energy of the system is higher than the spin excitation energy. We consider one-dimensional spin-1 Bose gas in the SILL regime and investigate its spin-dependent many-body properties. In Tonks-Girardeau limit, we are able to write down the general wave functions in a harmonic trap. We numerically calculate the spin-dependent (spin-plus, minus, and zero) momentum distributions in the sector of zero magnetization which allows us to demonstrate the most significant spin-incoherent feature compared to the spinless or spin-polarized case. In contrast to the spinless Bose gas, the momentum distributions are broadened and in the large momentum limit follow the same asymptotic $1/p^4$ dependence but with reduced coefficients. While the density matrices and momentum distributions differ between different spin components for small $N$, at large $N$ they approach each other. We show these by analytic arguments and numerical calculations up to $N=16$.

Figure 1

Single-particle density matrix of spin-1 and spinless bosons for $N=10$ in the sector of $S_z=0$. The density matrix of spin-1 bosons is $\rho (x,x^{\prime })=2{\rho }_{+}(x,x^{\prime })+{\rho }_0(x,x^{\prime })$. The spatial correlations are plotted at chosen $x/x_{ho}=-1$. Spinless (spl) bosons ($+$) show broader spatial distributions than the spin-1 bosons ($\square$), indicating a narrower distribution in momentum space.

Figure 2

Momentum distributions of (a) spin-plus [${\rho }_{+}\left(p\right)$] and (b) zero components [${\rho }_0\left(p\right)$] in the SILL regime. In the sector of $S_z=0$, we numerically calculate the momentum distributions of 1D spin-1 TG gas up to $N=16$. They are uniformly broadened as $N$ increases, and the peaks of ${\rho }_0\left(p\right)$ are larger than ${\rho }_{+}\left(p\right)$ due to the spin function overlaps.

Figure 3

Asymptotics of high momentum distributions in Fig. 2. High momentum distributions are plotted in logarithmic scales and compared with analytic calculations (dash) in the spin-plus component of 1D spin-1 TG Bose gas. The analytical asymptotics are $(0.085$, 0.245, 0.694, 1.267, 2.131, 3.225, $8.305)/p^4$, respectively for $N=2–7$ and 10. The line symbols and colors are the same as Fig. 2, and a vertical line (dash-dot) at around $p{x}_{ho}\sim 7$ guides the eye for the limitation in accuracy of the numerical calculations.

Figure 4

Difference between spin-plus and spin-zero momentum distributions. The symbols and colors are plotted correspondingly as in Fig. 2, where we choose $N=2–6$ and 16. The case of $N=7$ is neglected here for it is too close to the one of $N=6$. The difference diminishes gradually as $N$ increases.

Figure 5

Relative deviations of Eq. (24) for the asymptotic forms ${\overline{f}}_j^{(N-j)}$ and ${\overline{f}}_0^{(N-j)}$ of the exact spin function overlaps ${(E,P_{12...j})}_{+}$ and ${(E,P_{12...j})}_0$, respectively. The relative deviations are plotted for $j=2$($\square$), 3($+$), 4($\times$), 5($\circ$), and 10($\diamond$). The deviations decrease as $N$ increases, which indicates that the asymptotic forms are approaching each other. It suggests that spin-plus and zero components of the momentum distributions coincide in the large $N$ limit.

Figure 6

Contributions from the integral regions for ${\rho }_{+}(x<x^{\prime })$ with $N=5$. The values for each integral region without and with the multiplications of spin function overlaps are shown in (a) and (b), respectively. Four spatial correlations are chosen at $(x,x^{\prime })$ as denoted in the legend of (a) along with the symbols $(\square ,+,\circ ,\times )$ for clarity. The inset in (b) displays 15 spin function overlaps for the case of five bosons.

Figure 7

Comparisons of the coefficients in the momentum distributions of spinless and spin-1 Bose gas. We plot the coefficients of (a) $c\left(N\right)$ ($\diamond$) and (b) $c_{+}\left(N\right)$ ($\square$) and $c_0\left(N\right)$ ($\times$), respectively for spinless bosons, spin-plus, and spin-zero components in high $p$ limit. These coefficients approach the asymptotic values $c\left(\infty \right)=2$ (dash) and $c_{+\left(0\right)}\left(\infty \right)=4/9$ (dash) when $N$ increases.

Figure 8

Integration paths for calculating the asymptotic form of $f_k^{\left(N\right)}$ traversing through three saddle points (denoted as $\times$). Starting from $w=-\pi$ in the complex plane of $w$, we choose a vertical path $D_{-1}$ to reach the first saddle $w_{-1}$ with increasing imaginary parts only. Paths $C_{\pm 1}$ traverse through the saddle points $w_{\pm 1}$ from the left tangent to the real axis, which connect to the path $C_0$ crossing the third one $w_0$, again tangent to the real axis. The last path $D_1$ reaches the end point of $w=\pi$ starting from $w_1$. Paths $C_{\pm 1}$ and $C_0$ follow the valleys of steepest descent while $D_{\pm 1}$ do not.

Figure 9

Exact spin function overlaps and their relative deviations from analytical derivations. Exact spin function overlaps, ${(E,P_{12...j})}_{+,0}$, for spin-plus and spin-zero components are plotted in (a) and (b) respectively, top to bottom, from $j/N=1/8, 2/8$, and $3/8$ (solid) to $j=N/2-1$ (dash). The values decrease rapidly as $j$ increases. The relative deviations of the exact spin function overlaps ${(E,P_{12...j})}_{+,0}$ from ${\overline{f}}_{j,0}^{(N-j)}/{\overline{f}}_0^{\left(N\right)}$ are plotted for $j=1/8, 2/8$, and $3/8$ (solid-$\square$ and $\circ$), and $j=N/2-1$ (dash), bottom to top, in (c) and (d), respectively. The deviations increase as $j$ increases. The relative deviations of ${(E,E)}_{+,0}$ from ${\overline{f}}_{1,0}^{(N-1)}/{\overline{f}}_0^{\left(N\right)}$ are denoted as $(+)$ in (c), which almost overlap with each other. ${(E,E)}_{+\left(0\right)}$ reaches $0.333\left(0.334\right)$ at $N=150$ and should approach $1/3$ for even larger $N$ as indicated by Eq. (A22).