Off-diagonal correlations of the Calogero-Sutherland model

We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.

Figure 1

Ground state energy of the CSM as a function of the interaction parameter $\lambda$: solid line, total energy per particle (44); dashed line, kinetic energy per particle (46); dotted-dashed line, potential energy per particle (45). Energies are measured in units of ${\hslash }^2{n}^2∕m$.

Figure 2

(Color online) One-body density matrix $g_1\left(x\right)$ in the thermodynamic limit for different values of the interaction parameter $\lambda$. Solid lines, bosons (upper line); fermions (lower line). Long dashed lines, long range expansion for bosons, Eq. (30) (upper line); for fermions, Eq. (31) (lower line). Short dashed line (c,d,e,f): short-range expansion Eqs. (41, 47). Note that the fermionic long-range expansion, Eq. (31), is exact for $\lambda =1$ (c).

Figure 3

(Color online) Short-range behavior of the bosonic (a) and fermionic (b) one-body density matrix $g_1\left(x\right)$ in the thermodynamic limit for different values of the interaction parameter $\lambda$. Symbols, results of the Monte Carlo simulations; lines, analytic part of the short-range expansion $(\lambda >1)$, Eqs. (41, 47).

Figure 4

(Color online) Momentum distributions for bosonic (a) and fermionic (b) CSM for different values of the interaction strength. Lines, in descending value at $k=1.5k_F$, $\lambda =10;3;2;3∕2;1;3∕4;2∕3$.

Figure 5

(Color online) Pair distribution function for different values of the interaction strength $\lambda$. Lines, in descending order of the value at the first peak, $\lambda =10;3;2;1;1∕2,1∕5$.

Figure 6

(Color online) Static structure factor for different values of the interaction strength $\lambda$. Dashed lines, low momenta phononic behavior, Eq. (54). Interaction parameter (in order of decreasing slope at small $k$), $\lambda =1∕5;1∕2;1;2;3;10$.

Figure 7

(Color online) Phase diagram of the Calogero-Sutherland system. See text for the explanation.