Description of composite bosons in discrete models

The understanding of the behavior of systems of identical composite bosons has progressed significantly in connection with the analysis of the entanglement between constituents and the development of coboson theory. The basis of these treatments is a coboson ansatz for the ground state of a system of $N$ pairs, stating that in appropriate limits this state is well approximated by the account of Pauli exclusion in what would otherwise be the product state of $N$ independent pairs, each described by the single-pair ground state. In this work we study the validity of this ansatz for particularly simple problems, and show that short-range attractive interactions in very dilute limits and a single-pair ground state with very large entanglement are not enough to render the ansatz valid. On the contrary, we find that the dimensionality of the problem plays a crucial role in the behavior of the many-body ground state.

Figure 1

Fidelity between the analytical ground state and the coboson ansatz for two pairs in the very strongly bound limit, in a one-dimensional lattice with $L$ sites and periodic boundary conditions. Inset: Probability to find a pair in each site of the lattice conditional on having found one pair in the first site, for $L=20$, evaluated over the analytical ground state (20) and over the coboson ansatz, displayed with violet $+$ and turquoise $\times$ symbols, respectively. All depicted quantities are dimensionless.

Figure 2

Fidelity between the ground state found numerically and the coboson ansatz for two pairs in an $n\times L$ lattice with torus boundary conditions, and where the effective tunneling strength is the same in both directions. For fixed $n$, in all cases the fidelity decreases with $L$. All depicted quantities are dimensionless.

Figure 3

Correlations between two pairs in the ground state of the $4\times 18$ lattice with equal tunneling in both directions. Given the position of one pair at site (1,1), the plot shows the probability to find the other pair as a function of the position in the same sublattice, $j_y=1$ (violet dashes) and in the other three sublattices ($j_y=2,3,4$, where cases 2 and 4 are equal for symmetry reasons). Apart from the region which is closest to (1,1), the probabilities for all four sublattices are very similar and resemble the sinusoidal distribution found for the $1\times L$ lattice. All depicted quantities are dimensionless.

Figure 4

Fidelity between the ground state found numerically and the coboson ansatz for an $L\times L$ torus, as a function of $L$. The anisotropies are, from top to bottom, given by $J_x^{\mathrm{eff}}/J_y^{\mathrm{eff}}=\xi$, for $\xi =1$, 2, 5, 10, 100. All depicted quantities are dimensionless.

Figure 5

Probability of finding a pair as a function of position given that there is a pair on site (1,1), for an isotropic lattice with $51\times 51$ sites. As one moves away from the occupied site the probability becomes relatively flat, allowing for a good description in terms of the coboson ansatz. All depicted quantities are dimensionless.

Figure 6

Exact energy and energy as given by the coboson ansatz for an $n\times L$ lattice with equal tunnelings in both directions, as a function of $L$. All depicted quantities are dimensionless.

Figure 7

Exact energy and energy as given by the coboson ansatz for an $L\times L$ lattice with equal tunnelings in both directions, as a function of $L$. Inset: Exact energy and energy as given by the ansatz for a one-dimensional lattice. All depicted quantities are dimensionless.