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

## TL;DR

This paper investigates the linear response of a long-range Hamiltonian system with two order parameters near critical points, revealing suppressed responses, non-diverging susceptibilities, and negative off-diagonal elements, confirmed by numerical simulations.

## Contribution

It develops a Vlasov-based linear response theory for systems with two order parameters, uncovering non-divergent susceptibilities and negative responses at critical points.

## Key findings

- Some susceptibility matrix elements do not diverge at critical points.
- Negative off-diagonal susceptibility elements can occur, indicating inverse responses.
- Theoretical predictions are validated by numerical Vlasov simulations.

## Abstract

The linear response is investigated in a long-range Hamiltonian system from the view point of dynamics, which is described by the Vlasov equation in the limit of large population. Due to 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 two accordingly. Moreover, the two order parameters bring three phases and the three types of second-order phase transitions between two of them. For each type of the phase transitions, 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 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.