Exploring universal and nonuniversal regimes of trimers from three-body interactions in one-dimensional lattices

We investigate the formation of trimers in an infinite one-dimensional lattice model of hard-core particles with single-particle hopping $t$ and and nearest-neighbor two-body $U$ and three-body $V$ interactions of relevance to Rydberg atoms and polar molecules. For sufficiently attractive $U\le -2t$ and positive $V>0$ a large trimer is stabilized, which persists as $V\to \infty$, while both sufficiently attractive $U$ and $V$ bind a small trimer. The excited state above this small trimer is also bound and has a large extent; its behavior as $V\to -\infty$ resembles that of the large ground-state trimer. These large bound states appear to admit a continuum description. Furthermore, we find that in the limit $V\gg t,U<-2t$ the bound-state behavior qualitatively evolves with larger $\left|U\right|$ from a state described by the scattering of three far separated particles to a state of a compact dimer scattering with a single particle.

Figure 1

Trimer stability diagram at $K=0$ as a function of $V$ and $U$ in units of $t$. The gray area indicates the continuum. The solid blue line identifies the boundary of the stability region of attractively bound states, while dashed red lines identify regions where an excited-state (ES) trimer coexists with the ground-state (GS) trimer. The crossover from small ($\langle M\rangle \le 3$) to large ($\langle M\rangle >3$) GS trimers is indicated by the dotted blue line. In the inset, we show the spectral function $A\left(\omega \right)=-\frac{1}{\pi }\Im G(1,1;1,1;0,\omega )$ for the parameter values indicated by the cross: $V=-2.5t,U=-2.5t$, demonstrating the appearance of the GS (blue) and ES (red) trimer peaks below the edge of the continuum (dashed line).

Figure 2

Analysis of the size of trimers: The probability $P\left(M\right)={\sum }_{n_1+n_2=M}{\left|\langle 0,n_1,n_2|0,{\alpha }_{\mathcal{T}}\rangle \right|}^2$ for the two outer particles in a trimer to be $M=n_1+n_2$ sites apart at $U=-2.5t$ and various values of $V$ for the (a) ground-state (GS) trimers (lower-right quadrant of Fig. 1) and (b) excited-state (ES) trimers (lower-left quadrant of Fig. 1). The two trimers exhibit qualitatively similar behavior with increasing V [compare lines of the same colors and symbols in (a) and (b)]. Further analysis of the $M=8$ component of the GS trimer is presented in Fig. 7.

Figure 3

The expectation value of the GS trimer size $\langle M\rangle$ as a function of $U$ for several values of $V$. The size of the trimer diverges as $U$ approaches $-2t$ for all $V\ge 0$, implying emergent universal behavior.

Figure 4

Binding of ground-state (GS) (blue) and excited-state (ES) (salmon) trimers at $U=-2.5t$ as a function of $V$. We plot the binding energy $E_B$ (solid and dashed lines) and the average trimer size $\langle M\rangle$ (dotted lines) with $\langle M\rangle \pm \sigma$ (boundary of the shaded regions), where $\sigma$ is the standard deviation of $P\left(M\right)$. $E_B$ approaches the horizontal black line in the asymptotic limit $V\to -\infty \left(\infty \right)$ for the ES (GS) trimer.

Figure 5

Binding mechanism of the large GS trimer at $V=1000t$ as a function of $U$. We plot the binding energy $E_B$ (solid green line), the average trimer size $\langle M\rangle$ (dashed green line), and $\langle M\rangle \pm \sigma$ (boundary of shaded regions), where $\sigma$ is the standard deviation of $P\left(M\right)$. The shaded region shows the spread of $P\left(M\right)$ about the average $\langle M\rangle$.

Figure 6

Spatial structure of GS trimer shown as a color plot of the probability $P\left(M\right)={\sum }_{n_1+n_2=M}{\left|\langle 0,n_1,n_2|0,{\alpha }_{\mathcal{T}}\rangle \right|}^2$ for the two outer particles in a trimer to be $M=n_1+n_2$ sites apart for $V=1000t$ at (a) $U=-2.1t$ and (b) $U=-10t$. Analysis of the trimer's $M=8$ component (white line) is presented in Fig. 7.

Figure 7

Analysis of the internal structure of the large GS trimer through the probability $P\left(\mathrm{\Delta }\right)=\frac{|\langle 0,\mathrm{\Delta },M-\mathrm{\Delta }|0,{\alpha }_{\mathcal{T}}\rangle {|}^2}{P\left(M\right)}$ to find the central particle $\mathrm{\Delta }$ sites apart from the outermost left particle for the $M=8$ component of the trimer wave function at $V=1000t$ and for different values of $U$.