Generalized Scaling and the Critical Eigenvector in Ideal Bose Condensation

The condensation of an ideal Bose gas, in which the phase symmetry of the boson states has been removed by a linear coupling of the creation and annihilation operators ${\psi }^{†}\left(\overrightarrow{\mathrm{r}}\right)$, $\psi \left(\overrightarrow{\mathrm{r}}\right)$ to a fictitious external field $C\left(\overrightarrow{\mathrm{r}}\right)$, is considered as a model of a second-order phase transition. In the vicinity of the transition, the spontaneous-order parameter (thermal expectation value of $\psi$) is related to the conjugate field by a series of terms of which the first represents a scaled equation of state that exhibits a power-law behavior with nonclassical exponents. The expectation value for the ordered product of boson operators ${〈{\psi }^{†}\mathrm{}\left(1\right)\mathrm{} {\psi }^{†}\mathrm{}\left(2\right)\mathrm{}\cdots {\psi }^{†}\mathrm{}\left(M\right)\mathrm{} \psi \left(1^{\prime }\right)\cdots \psi \left(N^{\prime }\right)〉}_{\beta }$ is determined and its behavior across the coexistence curve discussed. The existence of a conjectured critical eigenvector is demonstrated and a related asymptotic property of the ordered product determined.