## Ising Hysteresis

When the magnetic field applied to a ferromagnet changes direction, the spins within the ferromagnet do not immediately flip to reach their typical equilibrium state. The magnetization is not just a function of the applied magnetic field, but also depends on the current state and the rate at which the field changes. This non-equilibrium behavior is called hysteresis.

In this Monte Carlo Metropolis program the spins are initially aligned with the magnetic field. Then slowly the field is reduced to zero and then changes sign and is increased in the opposite direction. The spins eventually align with the magnetic field in the opposite direction, and then the field is reduced to zero again, and again changes sign. You can explore hysteresis as a function of temperature as well as the rate at which the field is changed.

Problem: The Ising model in an external magnetic field
1. Use Program IsingHysteresis with $T= 1.8$, the initial magnetic field $H = 1$, $\Delta H=0.01$, and 10\,mcs for each value of $H$. The program plots the mean magnetization for each value of $H$, and changes $H$ by $\Delta H$ until $H$ reaches $H=-1$, when it changes $\Delta H$ to $-\Delta H$. Describe what you obtain and why it occurred. The resulting curve is called a hysteresis loop, and is characteristic of discontinuous phase transitions.
2. Change the value of mcs for each value of $H$ to 1 and view the resulting plot for $m$ versus $H$. Repeat for mcs equal to 100. Explain the differences you see.

## Resource

Problem 5.19 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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