Instanton picture of the spin tunneling in the Lipkin model

A consistent theory of the ground state energy and its splitting due to the process of tunneling for the Lipkin model is presented. For the functional integral in terms of the spin coherent states for the partition function of the model we accurately calculate the trivial and the instanton saddle point contributions. We show that such calculation has to be perfomed very accurately taking into account the discrete nature of the functional integral. Such accurate consideration leads to finite corrections to a naive continous consideration. We present comparison with numerical calculation of the ground state energy and the tunneling splitting and with the results obtained by the quasiclassical method and get excellent agreement.