kunal1729verma / ising-2d

Metropolis Markov Chain Monte Carlo simulation in C++ to compute expectation values of observables and use that to extract the critical exponents of the 2D Ising model.

Ising Model implementation in C++

The above code uses the Metropolis–Hastings MCMC algorithm to simulate the 2D Ising Model. We first perform a series of measurements of various observables, namely - $E, E^2, |M|, M^2, M^4, M$, where $E$ is the energy density and $M$ is the magnetization density in the system. On collecting these measurements, we calculate the autocorrelation times of all the observables (except $M$), and define the maximum of them as the autocorrelation time.

Once we obtain the autocorrelation time $\tau$, we run the Metropolis algorithm again for $2\tau \times N_\text{out}$ times where the $N_\text{out}$ is our desired number of measurements. We now take measurements of the observables every $2\tau$ MC steps to ensure uncorrelated data.

On obtaining the uncorrelated measurements for the observales, we calculate the errors in the observables and the derived physical quantities ($\chi, C_V, U_L$) using the Jackknife method.

As of now, we have used a very naive version of the data-collapse method to evaluate the critical exponents without taking into account the Jackknife errors evaluated for our physical quantities. Anders Sandvik's document on Computational Studies of Quantum Spin Systems is an excellent resource on all things related to spin system simulations, including finite size scaling analysis and error analysis as well.