# Conjugate field and fluctuation-dissipation relation for the dynamic phase transition in the two-dimensional kinetic Ising model

## TL;DR

This study investigates the dynamic phase transition in the 2D kinetic Ising model under oscillating magnetic fields, revealing conjugate field relations and a fluctuation-dissipation relation that extends understanding of nonequilibrium critical phenomena.

## Contribution

It demonstrates that the bias magnetic field acts as a conjugate field to the dynamic order parameter and establishes a fluctuation-dissipation relation in the nonequilibrium setting.

## Key findings

- The scaling exponent of the bias field matches the equilibrium critical isotherm exponent.
- A fluctuation-dissipation relation holds for the dynamic order parameter and bias field.
- Finite-size scaling supports the conjugate field relationship.

## Abstract

The two-dimensional kinetic Ising model, when exposed to an oscillating applied magnetic field, has been shown to exhibit a nonequilibrium, second-order dynamic phase transition (DPT), whose order parameter Q is the period-averaged magnetization. It has been established that this DPT falls in the same universality class as the equilibrium phase transition in the two-dimensional Ising model in zero applied field. Here we study for the first time the scaling of the dynamic order parameter with respect to a nonzero, period-averaged, magnetic `bias' field, H_b, for a DPT produced by a square-wave applied field. We find evidence that the scaling exponent, \delta_d, of H_b at the critical period of the DPT is equal to the exponent for the critical isotherm, \delta_e, in the equilibrium Ising model. This implies that H_b is a significant component of the field conjugate to Q. A finite-size scaling analysis of the dynamic order parameter above the critical period provides further support for this result. We also demonstrate numerically that, for a range of periods and values of H_b in the critical region, a fluctuation-dissipation relation (FDR), with an effective temperature T_{eff}(T, P, H_0) depending on the period, and possibly the temperature and field amplitude, holds for the variables Q and H_b. This FDR justifies the use of the scaled variance of Q as a proxy for the nonequilibrium susceptibility, \partial<Q> / \partial H_b, in the critical region.