Generalized Scaling and the Critical Eigenvector in Ideal Bose Condensation

Abstract

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 ${\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, $\ensuremath{\psi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$ to a fictitious external field $C(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$, 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 $\ensuremath{\psi}$) 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 ${〈{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(1) {\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(2)\ensuremath{\cdots}{\ensuremath{\psi}}^{\ifmmode\dagger\else\textdagger\fi{}}(M) \ensuremath{\psi}({1}^{\ensuremath{'}})\ensuremath{\cdots}\ensuremath{\psi}({N}^{\ensuremath{'}})〉}_{\ensuremath{\beta}}$ 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.