Orthogonality catastrophe and fractional exclusion statistics

We show that the $N$-particle Sutherland model with inverse-square and harmonic interactions exhibits orthogonality catastrophe. For a fixed value of the harmonic coupling, the overlap of the $N$-body ground state wave functions with two different values of the inverse-square interaction term goes to zero in the thermodynamic limit. When the two values of the inverse-square coupling differ by an infinitesimal amount, the wave function overlap shows an exponential suppression. This is qualitatively different from the usual power law suppression observed in the Anderson's orthogonality catastrophe. We also obtain an analytic expression for the wave function overlaps for an arbitrary set of couplings, whose properties are analyzed numerically. The quasiparticles constituting the ground state wave functions of the Sutherland model are known to obey fractional exclusion statistics. Our analysis indicates that the orthogonality catastrophe may be valid in systems with more general kinds of statistics than just the fermionic type.

Figure 1

Overlap between the ground state of a SM (1) with arbitrary $\omega$ and $\mu =3/4$ and the ground state of another model (1) obtained by a perturbation of the inverse-square coupling from $\lambda$ [recall $\lambda =(\sqrt{\mu +1}+1)/2\left)\right]$ to $\lambda +\delta \lambda$ for different $\delta \lambda$. The harmonic term does not change in any case. The points were obtained with the exact expression (11), and the solid lines correspond to the perturbative approximation (14).

Figure 2

Overlap between the ground state of a SM (1) with arbitrary $\omega$ and $\mu =3/4$ and the ground state of other SMs (1) with the same harmonic coupling $\omega$ and different inverse-square strength ${\mu }^{\prime }$. Observe that the results do not depend on the value of $\omega$. The points were computed employing the exact expression (11), and the solid lines correspond to the asymptotic expansion (16) obtained by applying the Stirling formula for the Gamma function when $N\to \infty$.