Nondivergent and negative susceptibilities around critical points of a long-range Hamiltonian system with two order parameters.

The linear response is investigated in a long-range Hamiltonian system from the viewpoint of dynamics, which is described by the Vlasov equation in the large-population limit. Because of the existence of the Casimir invariants of the Vlasov dynamics, an external field does not drive the system to the forced thermal equilibrium in general, and the linear response is suppressed. With the aid of a linear response theory based on the Vlasov dynamics, we compute the suppressed linear response in a system having two order parameters, which introduce the conjugate two external fields and the susceptibility matrix of size 2 accordingly. Moreover, the two order parameters bring three phases and there are three types of second-order phase transitions between them. For each type of phase transition, all the critical exponents for elements of the susceptibility matrix are computed. The critical exponents reveal that some elements of the matrices do not diverge even at critical points, while the mean-field theory predicts divergences. The linear response theory also suggests the appearance of negative off-diagonal elements; in other words, an applied external field decreases the value of an order parameter. These theoretical predictions are confirmed by direct numerical simulations of the Vlasov equation.