We study different scenarios of topological phase transitions in the vicinity of the \pi^0 condensation point in neutron matter. The transitions occur between the Fermi-liquid state and a topologically different one with two sheets of the Fermi surface. Two possibilities of a rearrangement of quasiparticle degrees of freedom are shown: the first-order topological phase transition and the second-order one. The order of the phase transition is found to be strongly dependent on the value of the critical wave vector of the soft \pi^0 mode. The thermodynamics of the system is also studied. It is shown that the topology of the quasiparticle momentum distribution is mainly determined by the neutron matter density, while the temperature T is essential in a narrow density region. A simple explanation of the first-order topological phase transition at T=0 is given.Comment: 8 pages, 11 figures. The English of the text was revised; 2 references were added; minor changes in figures were made; all results remained unchange
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