Topological signature of first-order phase transitions in a mean-field model

We study a mean-field Hamiltonian system whose potential energy V ({q(i)} i= 1... N) is expressed as a sum of k-body interactions and we show that in the thermodynamic limit the presence and the energy position of first-order phase transitions can be inferred by the study of the topology of configuration space induced by V, without resorting to any statistical measure. The thermodynamics of our model is analytically solvable and - depending on the value of k - displays no transition ( k = 1), second-order ( k = 2) or first-order (k > 2) phase transition. This rich behaviour is quantitatively retrieved by the investigation of one of the topological invariants (the Euler characteristic chi(v)) of the subsets M-v defined by M-v = {(q(1),..., q(N)) | V ({q(i)})/ N less than or equal to v}.