Toda Darboux transformations and vacuum expectation values

Abstract

The determinant formulas for the vacuum expectation values $〈s+k+n-m,-s|e^{H\left(t\right)}{\beta }_m^{⁎}\cdots {\beta }_1^{⁎}{\beta }_n\cdots {\beta }_{1}g|k〉$ are obtained through the application of Toda Darboux transformations. Initially, it is noted that the 2–Toda hierarchy can be regarded as the 2–component bosonization of the fermionic KP hierarchy. Subsequently, two fundamental Toda Darboux transformation operators, namely $T_{+}\left(q\right)=\Lambda \left(q\right)\cdot \Delta \cdot q^{-1}$ and $T_{-}\left(r\right)={\Lambda }^{-1}{\left(r\right)}^{-1}\cdot {\Delta }^{-1}\cdot r$, are constructed from the changes in the Toda (adjoint) wave functions, by employing the 2–component boson–fermion correspondence. On this basis, the aforementioned vacuum expectation values can be realized as the multi–step Toda Darboux transformations. Therefore, the corresponding determinant formulas are derived from the determinant representations of these Toda Darboux transformations. Ultimately, by adopting similar methodologies, we also present the determinant formulas for $〈n-m|e^{H\left(x\right)}{\beta }_m^{⁎}\cdots {\beta }_1^{⁎}{\beta }_n\cdots {\beta }_{1}g|k〉$, which are associated with the KP tau functions.