Quantum holonomy in the Lieb-Liniger model

We examine a parametric cycle in the $N$-body Lieb-Liniger model that starts from the free system and goes through Tonks-Girardeau and super-Tonks-Girardeau regimes and comes back to the free system. We show the existence of exotic quantum holonomy, whose detailed workings are analyzed with the specific sample of two- and three-body systems. The classification of eigenstates based on clustering structure naturally emerges from the analysis.

Figure 1

Parametric evolution of eigenenergies of the two-body Lieb-Liniger model, where the $x$ and $y$ axes indicate $(4/\pi )\arctan g$ and $\sqrt{E}$, respectively. The unit is chosen such that $\hslash$ and the mass of a particle are set to unity. The period of the position space is chosen to be $2\pi$. The thick (black) and thin (blue) lines correspond to the families specified by the minimal states $(0,0)$ and $(0,1)$, respectively. See Eqs. (21) and (22). Note that the eigenenergies are continuous at $g=\pm \infty$.

Figure 2

Eigenenergies of $N=3$ families, where the $x$ and $y$ axes are the same as in Fig. 1. The total momentum of all families shown here is zero. The thick (black) line corresponds to the family [Eq. (24)] specified by the minimal state $(0,0,0)$. The thin (blue) line corresponds to the $(-1,0,1)$ family. These two families are trimers. The dotted (red), dashed (brown), and dash-dotted (gray) lines are dimer families specified by minimal states $(-1,-1,2)$, $(-2,-2,4)$, and $(-3,-3,6)$, respectively. Although there are level crossings, the adiabatic theorem ensures that the adiabatic time evolution is confined within a family [25]. The choice of the unit is the same as in Fig. 1.

Figure 3

Parametric evolution of eigenenergies of the $N=4$ case, where the $x$ and $y$ axes are the same as in Fig. 1. The thick (black) line corresponds to the family $(0,0,0,0)\mapsto (-3,-1,1,3)\mapsto (-6,-2,2,6)\cdots .$ The thin (blue) and dotted (red) lines correspond to $(-1,0,0,1)$ and $(-1,-1,1,1)$ families, respectively. The choice of the unit is the same as in Fig. 1.

Figure 4

Contour plots of $\mathrm{Re}{E}_n\left(c\right)$: (a) $n=0$; (b) $n=1$; (c) $n=2$. Lighter (darker) color indicates larger (smaller) value of $\mathrm{Re}E$. Thin lines are the contours of $\mathrm{Re}{E}_n\left(c\right)$. The exceptional points are indicated by solid circles. Bold lines indicate the branch cuts. While all complex exceptional points appear in $E_0\left(c\right)$, each $E_n\left(c\right)$ ($n=1,2$) has a single exceptional point. (d) Schematic explanation of complex cycles that enclose exceptional points. We depict $\mathcal{C}$ by a thick (red) line. Dashed (blue) and dotted (black) curves indicate $C_1$ and $C_2$, respectively. See the main text. The choice of the unit is the same as in Fig. 1.