Effects of quantum instantons on the thermodynamics of the $\mathbb{C}{P}^{N-1}$ model

Using the $1/N$ expansion, we study the influence of quantum instantons on the thermodynamics of the $\mathbb{C}{P}^{N-1}$ model in $1+1$ dimensions. We do this by calculating the pressure to next-to-leading order in $1/N$, without quantum instanton contributions. The fact that the $\mathbb{C}{P}^1$ model is equivalent to the $\mathrm{O}(3)$ nonlinear sigma model allows for a comparison to the full pressure up to $1/N^2$ corrections for $N=3$. Assuming validity of the $1/N$ expansion for the $\mathbb{C}{P}^1$ model makes it possible to argue that the pressure for intermediate temperatures is dominated by the effects of quantum instantons. A similar conclusion can be drawn for general $N$ values by using the fact that the entropy should always be positive.

Figure 1

${\tilde{F}}_1(\beta m)/T^2$ and ${\tilde{F}}_2(\beta m)/T^2$ plotted as functions of $\beta m$.

Figure 2

$F_3(\beta m)/T^2$ and $F_4(\beta m)/T^2$ as functions of $\beta m$.

Figure 3

Contribution from the zero-winding-number configurations to the pressure $\mathcal{P}$ of the $\mathbb{C}{P}^{N-1}$ model normalized to $N{T}^2$, as a function of temperature and different values of $N$.

Figure 4

The normalized pressure $\mathcal{P}/T^2$ of the $\mathrm{O}(N)$ nonlinear sigma model to NLO in $1/N$ for $N=3$ compared to the contribution to the pressure of the $\mathbb{C}{P}^1$ model from the configurations with zero winding number.

Figure 5

Estimate of the contribution from the configurations with $Q\ne 0$ to the normalized pressure $\mathcal{P}/T^2$ of the $\mathbb{C}{P}^1$ model.