## Ising 2d

Program Ising2d simulates $N$ Ising model spins on a $L \times L$ square lattice with periodic boundary conditions. The Metropolis Monte Carlo algorithm is used.

The goal of this simulation is to explore the properties of the 2d Ising model, including high and low temperature behavior and the nature of the phase transition between a ferromagnetic state at low temperatures and a paramagnetic state at high temperatures.

Problem: Simulation of the two-dimensional Ising model

Use Program Ising2d to explore some of the properties of the two-dimensional Ising model on a square lattice at a given temperature $T$ and external magnetic field $H$. First choose $N=L^2 = 32^2$ and set $H=0$. The initial orientation of the spins is all spins up.

1. Choose $T = 10$ and run until equilibrium has been established. Is the orientation of the spins random and the mean magnetization approximately equal to zero? What is a typical size of a domain, a region of parallel spins?
2. Choose a low temperature such as $T = 0.5$. Are the spins still random or is there a preferred direction? You will notice that $\overline{M} \approx 0$ for sufficient high $T$ and $\overline{M} \neq 0$ for sufficiently low $T$. Hence, there is an intermediate value of $T$ at which $\overline{M}$ first becomes nonzero.
3. Choose $L=4$ and $T=2.0$. Does the sign of $M$ change during the simulation? Choose a larger value of $L$ and observe if the sign of $M$ changes. Will the sign of $M$ change for $L \gg 1$? Should a theoretical calculation of $\langle M\rangle$ yield $\langle M \rangle \neq 0$ or $\langle M \rangle = 0$ for $T < T_c$?
4. Start at $T=4$ and determine the temperature dependence of $m$, the zero-field susceptibility $\chi$, the mean energy per spin $E/N$, and the specific heat $C$. Decrease the temperature in intervals of 0.2 until $T \approx 1.6$, equilibrating for at least $1000\,$mcs before collecting data at each value of $T$. Describe the qualitative temperature dependence of these quantities. (When the simulation is stopped, the mean magnetization and the mean of the absolute value of the magnetization is returned.) Because $M$ sometimes changes sign for small systems, the value of $\langle |m| \rangle$ is a more accurate representation of the magnetization. For this reason the susceptibility is given by \begin{equation} \chi = \frac{1}{kT}\big[\langle M^2\rangle -\langle|M|\rangle^2\big], \label{eq:5/chiabs} \end{equation} rather than by using $\langle M \rangle$.
5. Set $T = T_c \approx 2.269$ and choose $L \geq 128$. Obtain $\langle m \rangle$ for $H = 0.01$, 0.02, 0.04, 0.08, and 0.16. Make sure you equilibrate the system at each value of $H$ before collecting data. Make a log-log plot of $m$ versus $H$ and estimate the critical exponent $\delta$ using \begin{equation} |m| \sim |H|^{1/\delta} \qquad (T = T_c). \label{eq:5/delta} \end{equation}
Problem: The Ising model in an external magnetic field

Use Program Ising2d to simulate the Ising model on a square lattice at a given temperature. Choose $N=L^2=32^2$. Run for at least 200 Monte Carlo steps per spin at each value of the field.

1. Set $H=0.2$ and $T = 3$ and estimate the approximate value of the magnetization. Then change the field to $H = 0.1$ and continue updating the spins (do not press New) so that the simulation is continued from the last microstate. Note the value of $m$. Continue this procedure with $H=0$, then $H = -0.1$ and then $H=-0.2$. Do your values of $m$ change abruptly as you change the field? Is there any indication of a phase transition as you change $H$?
2. Repeat the same procedure as in part (a) at $T = 1.8$ which is below the critical temperature. What happens now? Is there evidence of a sudden change in $m$ as you change the direction of the field?
Problem: Simulations of metastability
1. Use Program Ising2d to simulate the Ising model in a magnetic field. Choose $L=64$, $T=1$, and $H=0.7$. Run until the system reaches equilibrium. You will notice that most of the spins are aligned with the magnetic field.
2. Pause the simulation and let $H=-0.7$; we say that we have “flipped” the field. Continue the simulation after the flip of the field and watch the spins. What is the equilibrium state of the system after the flip of the field? Do the spins align themselves with the magnetic field immediately after the flip? Monitor $m$ as a function of time. Is there a time interval during which the mean value of $m$ does not change appreciably? At what time does $m$ change sign? Is the time when $m$ becomes negative the same each time you do the simulation? (The program uses a different random number seed each time it is run.)
3. Keep the temperature fixed at $T=1$, set $H = 0.6$, and flip the field as in part (b). How does the time that $m$ changes sign compare to the mean time for $|H|=0.7$?
Problem: Finite-size scaling and critical exponents

We expect that if the correlation length $\xi(T)$ is less than the linear dimension $L$ of the system, our simulations will yield results comparable to an infinite system. However, for $T$ close to $T_c$, the results of simulations will be limited by finite-size effects. Because we can only simulate finite systems, it is difficult to obtain estimates for the critical exponents $\alpha$, $\beta$, and $\gamma$ by varying the temperature.

The effects of finite system size can be made quantitative by the following argument, which assumes that the only important length near the critical point is the correlation length. Consider, for example, the critical behavior of $\chi$. If the correlation length $\xi \gg 1$, (all lengths are measured in terms of the lattice spacing) but is much less than $L$, a power law behavior is expected to hold. However, if $\xi$ is comparable to $L$, $\xi$ cannot change appreciably and \begin{equation}\chi \sim |\epsilon|^{-\gamma}\label{eq:5/gamma} \end{equation} is no longer applicable. This qualitative change in the behavior of $\chi$ and other physical quantities occurs for \begin{equation} \xi \sim L \sim |T - T_c|^{-\nu}.\label{eq:5/lenscale} \end{equation} We invert (\ref{eq:5/lenscale}) and write \begin{equation} |T - T_c| \sim L^{-1/\nu}. \label{eq:5/distance} \end{equation} If $\xi$ and $L$ are approximately the same size, we can substitute \eqref{eq:5/distance} into \eqref{eq:5/gamma} to obtain \begin{equation} \chi(T=T_c) \sim [L^{-1/\nu}]^{-\gamma} \sim L^{\gamma/\nu}. \label{eq:5/finitechi} \end{equation} The relation (\ref{eq:5/finitechi}) between $\chi$ and $L$ at $T=T_c$ is consistent with the fact that a phase transition is defined only for infinite systems. We can use the relation (\ref{eq:5/finitechi}) to determine the ratio $\gamma/\nu$. This method of analysis is known as finite-size scaling.

1. Use Program Ising2d to estimate $\chi$ at $T=T_c$ for different values of $L$. Make a log-log plot of $\chi$ versus $L$ and use the scaling relation \eqref{eq:5/finitechi} to determine the ratio $\gamma/\nu$. Use the exact result $\nu=1$ to estimate $\gamma$. Then use the same reasoning to determine the exponent $\beta$ and compare your estimates for $\beta$ and $\gamma$ with the exact values $\beta = 1/8$ and $\gamma = 7/4$.
2. Make a log-log plot of $C$ versus $L$. If your data for $C$ are sufficiently accurate, you will find that the log-log plot of $C$ versus $L$ is not a straight line but shows curvature. The reason is that the exponent $\alpha$ equals zero for the two-dimensional Ising model, and $C \sim C_0 \ln L$. Is your data for $C$ consistent with this form? The constant $C_0$ is approximately 0.4995.

## Resources

Problems 5.14, 5.19, 5.22 and 5.38 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).

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