Truncated Calogero-Sutherland models

A one-dimensional quantum many-body system consisting of particles confined in a harmonic potential and subject to finite-range two-body and three-body inverse-square interactions is introduced. The range of the interactions is set by truncation beyond a number of neighbors and can be tuned to interpolate between the Calogero-Sutherland model and a system with nearest and next-nearest neighbors interactions discussed by Jain and Khare. The model also includes the Tonks-Girardeau gas describing impenetrable bosons as well as an extension with truncated interactions. While the ground state wave function takes a truncated Bijl-Jastrow form, collective modes of the system are found in terms of multivariable symmetric polynomials. We numerically compute the density profile, one-body reduced density matrix, and momentum distribution of the ground state as a function of the range $r$ and the interaction strength.

Figure 1

Local quantum correlations of the TCSM. The density profile is plotted as a function of distance with respect to the center of the trap for $\mathrm{N}=5$. (a) Increasing the interaction strength while keeping the range fixed ($r=2$) leads to spatial antibunching reflected in the fringes of the density profile. (b) For a given strength of the interactions ($\lambda =1$), the TCSM interpolates between the Jain-Khare model ($r=1$) and the rational Calogero-Sutherland model ($r=\mathrm{N}-1$) as the range is increased, enhancing spatial antibunching. The case $\lambda =1$ shown here can be thought of as a truncated Tonks-Girardeau (TTG) gas.

Figure 2

One-body reduced density matrix of the symmetrized TCSM. Plot of $\rho (x,x^{\prime })$ as a function of space for $\mathrm{N}=5$ particles and interaction strength $\lambda =1$, i.e., the TTG gas. The case $r=1$ in (a) is contrasted with $r=2$ in (b) and $r=3$ in (b), showing that off-diagonal long-range order is suppressed as the interaction range $r$ increases. The color coding runs from dark to light with increasing value of $\rho (x,x^{\prime })$.

Figure 3

Nonlocal quantum correlations of the TCSM ground state. (a) Decay of the one-body reduced density matrix $\rho (x,0)$ as a function of the distance away from the center of the trap. $\mathrm{N}=5,\lambda =1$, and $r=\{1,2,3,4\}$, from top to bottom. Away from the origin, a power-law fit $\rho (x,0)=\frac{\gamma }{{\left|x\right|}^p}$ yields an exponent $p$ that increases with the interaction range $r$. This is consistent with the loss of off-diagonal long-range order with increasing $r$. (b) Momentum distribution $n\left(k\right)$ for the symmetrized TTG. While the momentum distribution is sharply peaked at $k=0$, its tails are broadened as the range of the interaction increases until recovering the full-range model, the conventional TG gas with $r=\mathrm{N}-1$.

Figure 4

Density profile of excited states. These three plots show $n_{nk}\left(x\right)$ for $\mathrm{N}=4$ and $\lambda =1$ of the first three excited states: (a) $(n,k)=(0,1)$, (b) $(n,k)=(1,0)$, and (c) $(n,k)=(0,2)$. In these three cases, increasing $r$ both increases the spatial extent of the distribution and the degree of spatial antibunching, leading to a higher visibility of the fringes in the density profile. Such behavior is also observed in the ground-state density $n\left(x\right)$ with increasing $r$, as shown in Fig. 1.