Understand
We investigate, by numerical simulations on a lattice, the $\theta$-dependence of 2$d$ $CP^{N-1}$ models for a range of $N$ going from 9 to 31, combining imaginary $\theta$ and simulated tempering techniques to improve the signal-to-noise ratio and alleviate the critical slowing down of the topological modes.
- We provide continuum extrapolations for the second and fourth order coefficients in the Taylor expansion in $\theta$ of the vacuum energy of the theory, parameterized in terms of the topological susceptibility $\chi$ and of the so-called $b_2$ coefficient.
- Those are then compared with available analytic predictions obtained within the $1/N$ expansion, pointing out that higher order corrections might be relevant in the explored range of $N$, and that this fact might be related to the non-analytic behavior expected for $N = 2$.
- We also consider sixth-order corrections in the $\theta$ expansion, parameterized in terms of the so-called $b_4$ coefficient: in this case our present statistical accuracy permits to have reliable non-zero continuum estimations only for $N \leq 11$, while for larger values we can only set upper bounds.