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A consistent theory of the ground state energy and its splitting due to theprocess of tunneling for the Lipkin model is presented. For the functionalintegral in terms of the spin coherent states for the partition function of themodel we accurately calculate the trivial and the instanton saddle pointcontributions. We show that such calculation has to be perfomed very accuratelytaking into account the discrete nature of the functional integral. Suchaccurate consideration leads to finite corrections to a naive continousconsideration. We present comparison with numerical calculation of the groundstate energy and the tunneling splitting and with the results obtained by thequasiclassical method and get excellent agreement.Comment: REVTEX, 32 pages, 3 figure

Instanton picture of the spin tunnelling in the Lipkin - Meshkov - Glick model